/* A collection of about 730 integrals with power functions. The integrals are taken from functions.wolfram.com OK - The integral is solved by Maxima NEW - Solution after extension of the integrator (rev. 1.34) NOUN - No solution a noun form is returned. */ (kill(all), declare(z,complex,n,integer), assume(n>0), assume(notequal(v,0)), domain:complex, radexpand:true, /* This function checks if we get back the integrand. * * integral is the result of an integration, * var the integration variable, and * simplifier a simplifying function. */ diffcheck(integral,var,simplifier):= if op(integral)=nounify(integrate) then integral else simplifier(diff(integral,var)), factor_radcan(expr):=factor(radcan(expr)), done); done; /* ----- Involving only one direct function --------------------------------- */ /* Problem 2: OK */ integrate(a^(c*z),z); a^(c*z)/(log(a)*c); /* Problem 3: OK */ integrate(a^z,z); a^z/log(a); /* Problem 4: OK */ integrate(%e^(a*z+b),z); %e^(a*z+b)/a; /* Problem 5: OK */ integrate(%e^(a*z),z); %e^(a*z)/a; /* Problem 6: OK */ integrate(%e^z,z); %e^z; /* ----- Power arguments and base a ----------------------------------------- */ /* Problem 7: NEW Type 1 */ integrate(a^(b*z^r),z); z*gamma_incomplete(1/r,-b*z^r*log(a))*(-b*z^r*log(a))^(-1/r)/-r; /* Problem 9: OK */ assume(a>1,b>0); [a>1,b>0]; integrate(a^(b*z^2),z),erf_representation:erfi,trigsign:true; -sqrt(%pi)*%i*erfi(%i*sqrt(-log(a)*b)*z)/(2*sqrt(-log(a)*b))$ forget(a>1,b>0); [a>1,b>0]; /* Problem 11: OK */ integrate(a^(b*sqrt(z)),z),radcan,factor; 2*(log(a)*b*sqrt(z)-1)*a^(b*sqrt(z))/(log(a)^2*b^2); /* Problem 12: NEW Type 1 */ integrate(a^(b*(z^r)^p),z); z*gamma_incomplete(1/(p*r),-log(a)*b*(z^r)^p) /(-p*r*(-b*log(a)*(z^r)^p)^(1/(p*r))); /* Problem 13: Expected failure * (z^r)^(1/r) simplifies to z which is not correct in general. * rischint does not respect the complex character of z. * The result of rischint is only correct for z>0. */ integrate(a^(b*(z^r)^(1/r)),z), domain:real, radcan; a^(b*z)/(log(a)*b)$ /* Problem 14: Expected failure * sqrt(z^2) simplifies to z, which is not correct in general. * rischint does not respect the complex character of z. * The result of rischint is only correct for z>0. */ integrate(a^(b*sqrt(z^2)),z), radcan; a^(b*z)/(log(a)*b)$ /* ----- Power arguments and base %e ---------------------------------------- */ /* Problem 15: NEW Type 1 */ integrate(%e^(a*z^r),z); -z*gamma_incomplete(1/r,-a*z^r)/(r*(-a*z^r)^(1/r)); /* Problem 17: OK */ assume(a>0); [a>0]; integrate(%e^(a*z^2),z),erf_representation:erfi; -sqrt(%pi)*%i*erfi(%i*sqrt(-a)*z)/(2*sqrt(-a))$ forget(a>0); [a>0]; /* Problem 19: OK */ integrate(%e^(a*sqrt(z)),z); 2*(a*sqrt(z)-1)*%e^(a*sqrt(z))/a^2; /* Problem 20: NEW Type 1 */ integrate(%e^(a*(z^r)^p),z); -z*gamma_incomplete(1/(p*r),-a*(z^r)^p)/(p*r*(-a*(z^r)^p)^(1/(p*r))); /* Problem 21: Expected failure * See problem 13 */ integrate(%e^(a*(z^r)^(1/r)),z), domain:real; %e^(a*z)/a$ /* Problem 22: Expected failure * See problem 14 */ integrate(%e^(a*sqrt(z^2)),z); %e^(a*z)/a$ /* ----- Involving z^(v-1), arguments b*z and base a ------------------------ */ /* Problem 23: NEW Type 2 */ integrate(z^(v-1)*a^(b*z),z); -z^v*gamma_incomplete(v,-log(a)*b*z)/(-b*z*log(a))^v; /* Problem 24: NEW Type 2 */ integrate(a^(b*z)/z,z),gamma_expand:true,logcontract; expintegral_ei(b*z*log(a)); /* Problem 25: NEW Type 2 */ integrate(z^(v-1)*a^z,z); -gamma_incomplete(v,-log(a)*z)*z^v/(-log(a)*z)^v; /* Problem 26: NEW Type 2 */ integrate(a^z/z,z),gamma_expand:true,logcontract; expintegral_ei(z*log(a)); /* ----- Involving z^(v-1), arguments a*z and base %e ----------------------- */ /* Problem 27: NEW Type 2 */ integrate(z^(v-1)*%e^(a*z),z); -gamma_incomplete(v,-a*z)*z^v/(-a*z)^v; /* Problem 28: NEW Type 2 */ integrate(z^(v-1)*%e^z,z); -gamma_incomplete(v,-z)*z^v/(-z)^v; /* Problem 29: NEW Type 2 */ integrate(z^n*%e^(a*z),z),factor; a^(-n-1)*(-1)^n*gamma_incomplete(n+1,-a*z); /* Problem 30: New Type 2 */ integrate(%e^(a*z)/z^n,z),factor; a^(n-1)*(-1)^n*gamma_incomplete(1-n,-a*z); /* Problem 31: OK */ integrate(z*%e^(a*z),z); (a*z-1)*%e^(a*z)/a^2; /* Problem 32: Expected failure * Simplifies sqrt(z^2) -> z, which is correct only for z>0. * The problem is in rischint. */ integrate(sqrt(z^2)*%e^(a*z),z); (a*z-1)*%e^(a*z)/a^2$ /* Problem 33: OK */ integrate(z^2*%e^(a*z),z); (a^2*z^2-2*a*z+2)*%e^(a*z)/a^3; /* Problem 34: OK */ integrate(z^3*%e^(a*z),z); (a^3*z^3-3*a^2*z^2+6*a*z-6)*%e^(a*z)/a^4; /* Problem 35: OK */ integrate(z^4*%e^(a*z),z); (a^4*z^4-4*a^3*z^3+12*a^2*z^2-24*a*z+24)*%e^(a*z)/a^5; /* Problem 36: OK */ integrate(z^5*%e^(a*z),z); (a^5*z^5-5*a^4*z^4+20*a^3*z^3-60*a^2*z^2+120*a*z-120)*%e^(a*z)/a^6; /* Problem 37: OK */ integrate(z^6*%e^(a*z),z); (a^6*z^6-6*a^5*z^5+30*a^4*z^4-120*a^3*z^3+360*a^2*z^2-720*a*z+720) *%e^(a*z) /a^7; /* Problem 38: OK */ integrate(z^7*%e^(a*z),z); (a^7*z^7-7*a^6*z^6+42*a^5*z^5-210*a^4*z^4+840*a^3*z^3-2520*a^2*z^2 +5040*a*z-5040) *%e^(a*z) /a^8; /* Problem 39: OK */ integrate(z^8*%e^(a*z),z); (a^8*z^8-8*a^7*z^7+56*a^6*z^6-336*a^5*z^5+1680*a^4*z^4-6720*a^3*z^3 +20160*a^2*z^2-40320*a*z+40320) *%e^(a*z) /a^9; /* Problem 40: NEW Type 2 */ integrate(%e^(a*z)/z,z),gamma_expand:true,logcontract; expintegral_ei(a*z); /* Problem 41: NEW Type 2 */ integrate(%e^(a*z)/z^2,z),gamma_expand:true,logcontract,ratsimp; (a*z*expintegral_ei(a*z)-%e^(a*z))/z; /* Problem 42: NEW Type 2 */ integrate(%e^(a*z)/z^3,z),gamma_expand:true,logcontract,ratsimp; (a^2*z^2*expintegral_ei(a*z)+(-a*z-1)*%e^(a*z))/(2*z^2); /* Problem 43: NEW Type 2 */ integrate(%e^(a*z)/z^4,z),gamma_expand:true,logcontract,ratsimp; (a^3*z^3*expintegral_ei(a*z)+(-a^2*z^2-a*z-2)*%e^(a*z))/(6*z^3); /* Problem 44: NEW Type 2 */ integrate(%e^(a*z)/z^5,z),gamma_expand:true,logcontract,ratsimp; (a^4*z^4*expintegral_ei(a*z)+(-a^3*z^3-a^2*z^2-2*a*z-6)*%e^(a*z))/(24*z^4); /* Problem 45: NEW Type 2 (OK) * The following gives sqrt(z)/sqrt(-z) and not sqrt(-z)/sqrt(z), * but the phase is correct. */ integrate(z^(1/2+n)*%e^(a*z),z),ratsimp; a^(-n-1)*(-1)^n*gamma_incomplete((2*n+3)/2,-a*z)*sqrt(z)/sqrt(-a*z); /* Problem 47: OK */ assume(a>0); [a>0]; integrate(sqrt(z)*%e^(a*z),z),erf_representation:erfi,expand; sqrt(z)*%e^(a*z)/a+sqrt(%pi)*%i*erfi(%i*sqrt(-a)*sqrt(z))/(2*sqrt(-a)*a)$ forget(a>0); [a>0]; /* Problem 50: OK */ assume(a>0); [a>0]; multthru(integrate(z^(3/2)*%e^(a*z),z)),erf_representation:erfi; (2*a*z^(3/2)-3*sqrt(z))*%e^(a*z)/(2*a^2) -3*sqrt(%pi)*%i*erfi(%i*sqrt(-a)*sqrt(z))/(4*sqrt(-a)*a^2)$ forget(a>0); [a>0]; /* Problem 53: OK */ assume(a>0); [a>0]; multthru(integrate(z^(5/2)*%e^(a*z),z)),erf_representation:erfi; (4*a^2*z^(5/2)-10*a*z^(3/2)+15*sqrt(z))*%e^(a*z)/(4*a^3) +15*sqrt(%pi)*%i*erfi(%i*sqrt(-a)*sqrt(z))/(8*sqrt(-a)*a^3); forget(a>0); [a>0]; /* Problem 56: OK */ assume(a>0); [a>0]; multthru(integrate(z^(7/2)*%e^(a*z),z)),erf_representation:erfi; (8*a^3*z^(7/2)-28*a^2*z^(5/2)+70*a*z^(3/2)-105*sqrt(z))*%e^(a*z)/(8*a^4) -105*sqrt(%pi)*%i*erfi(%i*sqrt(-a)*sqrt(z))/(16*sqrt(-a)*a^4); forget(a>0); [a>0]; /* Problem 59: OK */ assume(a>0); [a>0]; multthru(integrate(z^(9/2)*%e^(a*z),z)),erf_representation:erfi; (16*a^4*z^(9/2)-72*a^3*z^(7/2)+252*a^2*z^(5/2)-630*a*z^(3/2)+945*sqrt(z)) *%e^(a*z) /(16*a^5) +945*sqrt(%pi)*%i*erfi(%i*sqrt(-a)*sqrt(z))/(32*sqrt(-a)*a^5)$ forget(a>0); [a>0]; /* Problem 62: OK */ assume(a>0); [a>0]; integrate(%e^(a*z)/sqrt(z),z),erf_representation:erfi; -sqrt(%pi)*%i*erfi(%i*sqrt(-a)*sqrt(z))/sqrt(-a)$ forget(a>0); [a>0]; /* Problem 64: NEW Type 2 (OK) * We have a problem to simplify this integral to an erfi representation, * because of expression like sqrt(-z), which do not simplify to %i*sqrt(z). * We give the results in term of the erfc function. * The same for No. 48, 49, 50, 51 */ multthru(integrate(%e^(a*z)/z^(3/2),z)),gamma_expand:true; 2*sqrt(%pi)*sqrt(-a*z)*erfc(sqrt(-a*z))/sqrt(z)-2*%e^(a*z)/sqrt(z); /* Problem 65: NEW Type 2 (OK) * Maxima simplifies z/sqrt(z^3) -> 1/sqrt(z). Therefore, we get the same * result for Problem 64 and 65. * Because of revision 1.125 of simp.lisp Maxima no longer simplifies * z/sqrt(z^3) -> 1/sqrt(z). We get a noun form. */ multthru(integrate(%e^(a*z)/sqrt(z^3),z)),gamma_expand:true, domain:real; 2*sqrt(%pi)*sqrt(-a*z)*erfc(sqrt(-a*z))/sqrt(z)-2*%e^(a*z)/sqrt(z)$ /*'integrate(%e^(a*z)/sqrt(z^3),z);*/ /* Problem 66: NEW Type 2 (OK) */ multthru(integrate(%e^(a*z)/z^(5/2),z)),gamma_expand:true; (-4*a*z/3-2/3)*%e^(a*z)/z^(3/2)-4*sqrt(%pi)*(-a*z)^(3/2)*erfc(sqrt(-a*z)) /(3*z^(3/2)); /* Problem 67: NEW Type 2 (OK) */ multthru(integrate(%e^(a*z)/z^(7/2),z)),gamma_expand:true; 8*sqrt(%pi)*(-a*z)^(5/2)*erfc(sqrt(-a*z))/(15*z^(5/2)) +(-8*a^2*z^2/15-4*a*z/15-2/5)*%e^(a*z)/z^(5/2); /* Problem 68: NEW Type 2 (OK) */ multthru(integrate(%e^(a*z)/z^(9/2),z)),gamma_expand:true; (-16*a^3*z^3/105-8*a^2*z^2/105-4*a*z/35-2/7)*%e^(a*z)/z^(7/2) -16*sqrt(%pi)*(-a*z)^(7/2)*erfc(sqrt(-a*z))/(105*z^(7/2)); /* ----- Involving z^(v-1), arguments b*z+d and base a ---------------------- */ /* Problem 69: NEW Type 2 */ integrate(z^(v-1)*a^(b*z+d),z); -a^d*gamma_incomplete(v,-log(a)*b*z)*z^v/(-log(a)*b*z)^v$ /* Problem 70: NEW 2 */ integrate(a^(b*z+d)/z,z),gamma_expand:true,logcontract; a^d*expintegral_ei(b*z*log(a)); /* ----- Involving z^(v-1), arguments b*z+d and base %e --------------------- */ /* Problem 71: NEW 2 */ diffcheck(integrate(z^(v-1)*%e^(a*z+b),z),z,factor); z^(v-1)*%e^(a*z+b); /* Problem 72: NEW 2 */ integrate(z^n*%e^(a*z+b),z),factor; a^(-n-1)*%e^b*(-1)^n*gamma_incomplete(n+1,-a*z); /* Problem 73: OK */ integrate(z*%e^(a*z+b),z),factor; (a*z-1)*%e^(a*z+b)/a^2; /* Problem 74: NEW 2 */ integrate(%e^(a*z+b)/z,z),gamma_expand:true,logcontract; %e^b*expintegral_ei(a*z); /* Problem 76: OK */ assume(a>0); [a>0]; integrate(%e^(a*z+b)/sqrt(z),z),erf_representation:erfi; -sqrt(%pi)*%i*%e^b*erfi(%i*sqrt(-a)*sqrt(z))/sqrt(-a)$ forget(a>0); [a>0]; /* ----- Involving z^(v-1), arguments b*z^r and base a ---------------------- */ /* Problem 77: NEW 2 */ integrate(z^(v-1)*a^(b*z^r),z); -gamma_incomplete(v/r,-log(a)*b*z^r)*z^v/(r*(-log(a)*b*z^r)^(v/r))$ /* Problem 79: NEW 2 */ integrate(a^(b*z^r)/z,z),gamma_expand:true,logcontract; expintegral_ei(log(a)*b*z^r)/r; /* ----- Involving z^(v-1), arguments a*z^r and base %e --------------------- */ /* Problem 80: NEW 2 */ integrate(z^(v-1)*%e^(a*z^r),z); -gamma_incomplete(v/r,-a*z^r)*z^v/(r*(-a*z^r)^(v/r))$ /* Problem 81: NEW 2 */ integrate(%e^(a*z^r)/z,z),gamma_expand:true,logcontract; expintegral_ei(a*z^r)/r; /* Problem 82: NEW 2 */ integrate(z^(2*n)*%e^(a*z^2),z),factor; -(-1)^n*gamma_incomplete((2*n+1)/2,-a*z^2)*z/(2*a^n*sqrt(-a*z^2))$ /* Problem 83: NEW 2 */ integrate(z^(2*n-1)*%e^(a*z^2),z),factor; -(-1)^n*gamma_incomplete(n,-a*z^2)/(2*a^n); /* Problem 84: OK */ integrate(z*%e^(a*z^2),z); %e^(a*z^2)/(2*a); /* Problem 86: OK */ assume(a>0); [a>0]; integrate(z^2*%e^(a*z^2),z),erf_representation:erfi; sqrt(%pi)*%i*erfi(%i*sqrt(-a)*z)/(4*sqrt(-a)*a)+z*%e^(a*z^2)/(2*a)$ forget(a>0); [a>0]; /* Problem 88: OK */ integrate(z^3*%e^(a*z^2),z); (a*z^2-1)*%e^(a*z^2)/(2*a^2); /* Problem 90: OK */ assume(a>0); [a>0]; integrate(z^4*%e^(a*z^2),z),erf_representation:erfi; (2*a*z^3-3*z)*%e^(a*z^2)/(4*a^2)-3*sqrt(%pi)*%i*erfi(%i*sqrt(-a)*z) /(8*sqrt(-a)*a^2)$ forget(a>0); [a>0]; /* Problem 92: OK */ integrate(z^5*%e^(a*z^2),z); (a^2*z^4-2*a*z^2+2)*%e^(a*z^2)/(2*a^3); /* Problem 93: NEW 2 */ integrate(%e^(a*z^2)/z,z),gamma_expand:true,logcontract; expintegral_ei(a*z^2)/2; /* Problem 94: NEW 2 */ multthru(integrate(%e^(a*z^2)/z^2,z)), gamma_expand:true; sqrt(%pi)*sqrt(-a*z^2)*erfc(sqrt(-a*z^2))/z-%e^(a*z^2)/z; /* Problem 95: NEW 2 */ integrate(%e^(a*z^2)/z^3,z),gamma_expand:true,logcontract,ratsimp; (a*z^2*expintegral_ei(a*z^2)-%e^(a*z^2))/(2*z^2); /* Problem 96: NEW 2 */ integrate(%e^(a*z^2)/z^4,z), gamma_expand:true,ratsimp; (2*sqrt(%pi)*a^2*z^4*erfc(sqrt(-a*z^2))+sqrt(-a*z^2)*(2*a*z^2+1)*%e^(a*z^2)) /(3*z^3*sqrt(-a*z^2))/-1; /* Problem 97: NEW 2 */ integrate(%e^(a*z^2)/z^5,z),gamma_expand:true,logcontract; (2*a^2*z^4*expintegral_ei(a*z^2)+(-2*a*z^2-2)*%e^(a*z^2))/(8*z^4); /* Problem 98: NEW 2 */ integrate(z^n*%e^(a*sqrt(z)),z); -2*gamma_incomplete(2*(n+1),-a*sqrt(z))/a^(2*(n+1)); /* Problem 99: OK */ integrate(z*%e^(a*sqrt(z)),z); 2*(a^3*z^(3/2)-3*a^2*z+6*a*sqrt(z)-6)*%e^(a*sqrt(z))/a^4; /* Problem 100: OK */ integrate(z^2*%e^(a*sqrt(z)),z); 2*(a^5*z^(5/2)-5*a^4*z^2+20*a^3*z^(3/2)-60*a^2*z+120*a*sqrt(z)-120) *%e^(a*sqrt(z)) /a^6; /* Problem 101: OK */ integrate(z^3*%e^(a*sqrt(z)),z); 2*(a^7*z^(7/2)-7*a^6*z^3+42*a^5*z^(5/2)-210*a^4*z^2+840*a^3*z^(3/2) -2520*a^2*z+5040*a*sqrt(z)-5040)*%e^(a*sqrt(z)) /a^8; /* Problem 102: OK */ integrate(z^4*%e^(a*sqrt(z)),z); 2*(a^9*z^(9/2)-9*a^8*z^4+72*a^7*z^(7/2)-504*a^6*z^3+3024*a^5*z^(5/2) -15120*a^4*z^2+60480*a^3*z^(3/2)-181440*a^2*z +362880*a*sqrt(z)-362880)*%e^(a*sqrt(z)) /a^10; /* Problem 103: OK */ integrate(z^5*%e^(a*sqrt(z)),z); 2*(a^11*z^(11/2)-11*a^10*z^5+110*a^9*z^(9/2)-990*a^8*z^4 +7920*a^7*z^(7/2)-55440*a^6*z^3+332640*a^5*z^(5/2) -1663200*a^4*z^2+6652800*a^3*z^(3/2)-19958400*a^2*z +39916800*a*sqrt(z)-39916800)*%e^(a*sqrt(z)) /a^12; /* Problem 104: NEW 2 */ integrate(%e^(a*sqrt(z))/z,z),gamma_expand:true,logcontract; 2*expintegral_ei(a*sqrt(z)); /* Problem 105: NEW 2 */ integrate(%e^(a*sqrt(z))/z^2,z),gamma_expand:true,logcontract; ((2*a*z+2*sqrt(z))*%e^(a*sqrt(z))-2*a^2*expintegral_ei(a*sqrt(z))*z^(3/2)) /-(2*z^(3/2)); /* Problem 106: NEW 2 */ integrate(%e^(a*sqrt(z))/z^3,z),gamma_expand:true,logcontract; ((2*a^3*z^3+2*a^2*z^(5/2)+4*a*z^2+12*z^(3/2))*%e^(a*sqrt(z)) -2*a^4*expintegral_ei(a*sqrt(z))*z^(7/2)) /-(24*z^(7/2)); /* Problem 107: NEW 2 */ integrate(%e^(a*sqrt(z))/z^4,z),gamma_expand:true,logcontract; ((2*a^5*z^5+2*a^4*z^(9/2)+4*a^3*z^4+12*a^2*z^(7/2)+48*a*z^3+240*z^(5/2)) *%e^(a*sqrt(z)) -2*a^6*expintegral_ei(a*sqrt(z))*z^(11/2)) /-(720*z^(11/2)); /* Problem 108: NEW 2 */ integrate(%e^(a*sqrt(z))/z^5,z),gamma_expand:true,logcontract; ((2*a^7*z^7+2*a^6*z^(13/2)+4*a^5*z^6+12*a^4*z^(11/2)+48*a^3*z^5 +240*a^2*z^(9/2)+1440*a*z^4+10080*z^(7/2)) *%e^(a*sqrt(z)) -2*a^8*expintegral_ei(a*sqrt(z))*z^(15/2)) /-(40320*z^(15/2)); /* ----- Involving z^(v-1), arguments b*z^r and base a ---------------------- */ /* Problem 109: NEW 2 */ diffcheck(integrate(z^(v-1)*a^(d+b*z^r),z),z,radcan); z^(v-1)*a^(d+b*z^r); /* Problem 110: NEW 2 */ integrate(a^(d+b*z^r)/z,z),gamma_expand:true,logcontract; a^d*expintegral_ei(log(a)*b*z^r)/r; /* ----- Involving z^(v-1), arguments b*z^r and base %e --------------------- */ /* Problem 111: NEW 2 */ diffcheck(integrate(z^(v-1)*%e^(b+a*z^r),z),z,factor); z^(v-1)*%e^(b+a*z^r); /* Problem 112: NEW 2 */ integrate(%e^(b+a*z^r)/z,z),gamma_expand:true,logcontract; %e^b*expintegral_ei(a*z^r)/r; /* ----- Involving (a*z+b)^-n ----------------------------------------------- */ /* Problem 113: NEW 3 */ integrate(%e^(c*z)/(b+a*z),z),expintexpand:true,logcontract; %e^-(b*c/a)*expintegral_ei((a*c*z+b*c)/a)/a; /* Problem 114: NEW 3 */ integrate(%e^(c*z)/(b+a*z)^2,z); -expintegral_e(2,-c*(a*z+b)/a)*%e^-(b*c/a)/(a*(a*z+b)); /* Problem 115: NEW 3 */ integrate(%e^(c*z)/(b+a*z)^3,z); -expintegral_e(3,-c*(a*z+b)/a)*%e^-(b*c/a)/(a*(a*z+b)^2); /* Problem 116: NEW 3 */ integrate(%e^(c*z)/(b+a*z)^4,z); -expintegral_e(4,-c*(a*z+b)/a)*%e^-(b*c/a)/(a*(a*z+b)^3); /* Problem 117: NEW 3 */ integrate(%e^(c*z)/(b+a*z)^5,z); -expintegral_e(5,-c*(a*z+b)/a)*%e^-(b*c/a)/(a*(a*z+b)^4); /* Problem 118: NEW 3 */ integrate(%e^(c*z)/(b+a*z)^6,z); -expintegral_e(6,-c*(a*z+b)/a)*%e^-(b*c/a)/(a*(a*z+b)^5); /* Problem 119: NOUN * This integral and the integrals 120, 121, 122, 123, 124 return an * unevaluated integral. This is caused by rischint. Some of the integrals * can be integrated in a second step. The algorithm can be improved. */ integrate(z*%e^(c*z)/(b+a*z),z); z*%e^(c*z)/(a*c*z+b*c)-b*'integrate(%e^(c*z)/(a^2*c*z^2+2*a*b*c*z+b^2*c),z); /* Problem 120: NOUN */ integrate(z^2*%e^(c*z)/(b+a*z),z); (c*z^2-z)*%e^(c*z)/(a*c^2*z+b*c^2)-'integrate( (b*c*z-b)*%e^(c*z) /(a^2*c^2*z^2+2*a*b*c^2*z+b^2*c^2),z); /* Problem 121: NOUN */ integrate(z*%e^(c*z)/(b+a*z)^2,z); 'integrate((a*z-b)*%e^(c*z)/(a^3*c*z^3+3*a^2*b*c*z^2+3*a*b^2*c*z+b^3*c),z) +z*%e^(c*z)/(a^2*c*z^2+2*a*b*c*z+b^2*c); /* Problem 122: NOUN */ integrate(z^2*%e^(c*z)/(b+a*z)^2,z); z^2*%e^(c*z)/(a^2*c*z^2+2*a*b*c*z+b^2*c) -2*b*'integrate(z*%e^(c*z)/(a^3*c*z^3+3*a^2*b*c*z^2+3*a*b^2*c*z+b^3*c),z); /* Problem 123: NOUN */ integrate(z^3*%e^(c*z)/(b+a*z)^2,z); 'integrate((2*b^2*c*z+2*b^2)*%e^(c*z)/(a^4*c^2*z^3+3*a^3*b*c^2*z^2 +3*a^2*b^2*c^2*z+a*b^3*c^2),z) +(a*c*z^3-a*z^2-2*b*z)*%e^(c*z)/(a^3*c^2*z^2+2*a^2*b*c^2*z+a*b^2*c^2); /* Problem 124: NOUN */ integrate(z^4*%e^(c*z)/(b+a*z)^2,z); 'integrate(((2*a*b^2*c-2*b^3*c^2)*z-2*b^3*c-4*a*b^2)*%e^(c*z) /(a^5*c^3*z^3+3*a^4*b*c^3*z^2+3*a^3*b^2*c^3*z+a^2*b^3*c^3),z) +(a^2*c^2*z^4-2*a^2*c*z^3+(2*a^2-2*a*b*c)*z^2+(2*b^2*c+4*a*b)*z)*%e^(c*z) /(a^4*c^3*z^2+2*a^3*b*c^3*z+a^2*b^2*c^3); /* ----- Involving (a*z^2+b)^-n and base %e --------------------------------- */ /* Problem 125: NOUN */ integrate(%e^(c*z)/(b+a*z^2),z); 'integrate(%e^(c*z)/(a*z^2+b),z); /* Problem 126: NOUN */ integrate(z*%e^(c*z)/(b+a*z^2),z); 'integrate((a*z^2-b)*%e^(c*z)/(a^2*c*z^4+2*a*b*c*z^2+b^2*c),z) +z*%e^(c*z)/(a*c*z^2+b*c); /* Problem 127: NOUN */ integrate(%e^(c*z)/(b+a*z^2)^2,z); 'integrate(%e^(c*z)/(a*z^2+b)^2,z); /* Problem 128: NOUN */ integrate(z*%e^(c*z)/(b+a*z^2)^2,z); 'integrate((3*a*z^2-b)*%e^(c*z)/(a^3*c*z^6+3*a^2*b*c*z^4+3*a*b^2*c*z^2+b^3*c),z) +z*%e^(c*z)/(a^2*c*z^4+2*a*b*c*z^2+b^2*c); /* ----- Involving (a*z^2+b*z+c)^-n ----------------------------------------- */ /* Problem 129: NOUN */ integrate(%e^(d*z)/(c+b*z+a*z^2),z); 'integrate(%e^(d*z)/(a*z^2+b*z+c),z); /* Problem 130: NOUN */ integrate(z*%e^(d*z)/(c+b*z+a*z^2),z); 'integrate((a*z^2-c)*%e^(d*z)/(a^2*d*z^4+2*a*b*d*z^3+(2*a*c*d+b^2*d)*z^2 +2*b*c*d*z+c^2*d),z) +z*%e^(d*z)/(a*d*z^2+b*d*z+c*d); /* Problem 131: NOUN */ integrate(%e^(d*z)/(c+b*z+a*z^2)^2,z); 'integrate(%e^(d*z)/(a*z^2+b*z+c)^2,z); /* Problem 132: NOUN */ integrate(z*%e^(d*z)/(c+b*z+a*z^2)^2,z); 'integrate((3*a*z^2+b*z-c)*%e^(d*z)/(a^3*d*z^6+3*a^2*b*d*z^5 +(3*a^2*c*d+3*a*b^2*d)*z^4 +(6*a*b*c*d+b^3*d)*z^3 +(3*a*c^2*d+3*b^2*c*d)*z^2 +3*b*c^2*d*z+c^3*d),z) +z*%e^(d*z)/(a^2*d*z^4+2*a*b*d*z^3+(2*a*c*d+b^2*d)*z^2+2*b*c*d*z+c^2*d); /* ----- Involving (a*z*b)^w and base %e ------------------------------------ */ /* Problem 133: NEW 3 */ integrate((a*z+b)^w*%e^(c*z+d),z); -%e^(d-b*c/a)*expintegral_e(-w,-c*(a*z+b)/a)*(a*z+b)^(w+1)/a; /* Problem 134: NEW 3 */ integrate((a*z+b)^w*%e^(c*z),z); -%e^-(b*c/a)*expintegral_e(-w,-c*(a*z+b)/a)*(a*z+b)^(w+1)/a; /* Problem 135: OK */ multthru(integrate((a*z+b)^(3/2)*%e^(c*z),z)); 3*sqrt(%pi)*a*%e^-(b*c/a)*erf(sqrt(-c/a)*sqrt(a*z+b))/(4*c^2*sqrt(-c/a)) +(2*a*c*(a*z+b)^(3/2)-3*a^2*sqrt(a*z+b))*%e^(c*(a*z+b)/a-b*c/a)/(2*a*c^2); /* Problem 136: OK */ multthru(integrate(sqrt(a*z+b)*%e^(c*z),z)); sqrt(a*z+b)*%e^(c*(a*z+b)/a-b*c/a)/c-sqrt(%pi)*%e^-(b*c/a) *erf(sqrt(-c/a)*sqrt(a*z+b)) /(2*c*sqrt(-c/a)); /* Problem 137: OK */ integrate(%e^(c*z)/sqrt(a*z+b),z); sqrt(%pi)*%e^-(b*c/a)*erf(sqrt(-c/a)*sqrt(a*z+b))/(a*sqrt(-c/a)); /* Problem 138: NEW 2 * This example is first transformed to an example of type 2. */ diffcheck(integrate(%e^(c*z)/(a*z+b)^(3/2),z),z,factor); %e^(c*z)/(a*z+b)^(3/2); /* ----- Involving a*z^2+b*z+c ---------------------------------------------- */ /* Problem 139: OK */ integrate(d^(c+b*z+a*z^2),z),erf_representation:erfi,radexpand:all,factor; sqrt(%pi)*d^c*%e^-(b^2*log(d)/(4*a))*erfi(sqrt(log(d))*(2*a*z+b)/(2*sqrt(a))) /(2*sqrt(a)*sqrt(log(d))); /* Problem 140: OK */ integrate(%e^(c+b*z+a*z^2),z),erf_representation:erfi,radexpand:all,factor; sqrt(%pi)*%e^(c-b^2/(4*a))*erfi((2*a*z+b)/(2*sqrt(a)))/(2*sqrt(a)); /* ----- Involving a*z^2+b -------------------------------------------------- */ /* Problem 141: OK */ integrate(d^(b*z+a*z^2),z),erf_representation:erfi,radexpand:all,factor; sqrt(%pi)*%e^-(b^2*log(d)/(4*a))*erfi(sqrt(log(d))*(2*a*z+b)/(2*sqrt(a))) /(2*sqrt(a)*sqrt(log(d))); /* Problem 142: OK */ integrate(%e^(b*z+a*z^2),z),erf_representation:erfi,radexpand:all,factor; sqrt(%pi)*%e^-(b^2/(4*a))*erfi((2*a*z+b)/(2*sqrt(a)))/(2*sqrt(a))$ /* Involving a*z^2+c -------------------------------------------------------- */ /* Problem 143: OK */ integrate(d^(c+a*z^2),z),erf_representation:erfi,radexpand:all; sqrt(%pi)*d^c*erfi(sqrt(a)*sqrt(log(d))*z)/(2*sqrt(a)*sqrt(log(d)))$ /* Problem 144: OK */ integrate(%e^(c+a*z^2),z),erf_representation:erfi,radexpand:all; sqrt(%pi)*%e^c*erfi(sqrt(a)*z)/(2*sqrt(a)); /* Involving a*z^2+b/z^2 ---------------------------------------------------- */ /* Problem 145: NEW 4 */ diffcheck(integrate(d^(b/z^2+a*z^2),z),z,factor_radcan); d^(b/z^2+a*z^2); /* Problem 146: NEW 4 */ diffcheck(integrate(%e^(b/z^2+a*z^2),z),z,factor); %e^(b/z^2+a*z^2); /* ----- Involving az^2+b/z^2+c --------------------------------------------- */ /* Problem 147: NEW 4 */ diffcheck(integrate(d^(c+b/z^2+a*z^2),z),z,factor_radcan); d^(c+b/z^2+a*z^2); /* Problem 148: NEW 4 */ diffcheck(integrate(%e^(c+b/z^2+a*z^2),z),z,factor); %e^(c+b/z^2+a*z^2); /* ----- Involving a*z+b*sqrt(z)+c ------------------------------------------ */ /* Problem 149: NEW 5 * This integral is first transformed to an integral of type 5. */ diffcheck(integrate(d^(c+a*z+sqrt(z)*b),z),z,radcan); d^(c+a*z+sqrt(z)*b); /* Problem 150: NEW 5 * This integral is first transformed to an integral of type 5. */ diffcheck(integrate(%e^(c+a*z+sqrt(z)*b),z),z,factor), domain:real; %e^(c+a*z+sqrt(z)*b); /* ----- Involving a*z+b*sqrt(z) -------------------------------------------- */ /* Problem 151: NEW 5 */ diffcheck(integrate(d^(a*z+sqrt(z)*b),z),z,radcan); d^(a*z+sqrt(z)*b); /* Problem 152: NEW 5 */ diffcheck(integrate(%e^(a*z+sqrt(z)*b),z),z,factor), domain:real; %e^(a*z+sqrt(z)*b); /* ----- Involving a*z^r+c -------------------------------------------------- */ /* Problem 153: NEW 1 */ diffcheck(integrate(d^(c+a*z^r),z),z,radcan); d^(c+a*z^r); /* Problem 154: NEW 1 */ diffcheck(integrate(%e^(c+a*z^r),z),z,factor); %e^(c+a*z^r); /* Problem 155: OK */ integrate(%e^(c+a*z^2),z),erf_representation:erfi,radexpand:all; sqrt(%pi)*%e^c*erfi(sqrt(a)*z)/(2*sqrt(a))$ /* Problem 156: OK */ integrate(%e^(c+a*sqrt(z)),z); 2*(a*%e^c*sqrt(z)-%e^c)*%e^(a*sqrt(z))/a^2; /* ----- Arguments involving exponential functions -------------------------- */ /* Problem 157: NEW 2 */ integrate(a^b^z,z),gamma_expand:true,logcontract; expintegral_ei(log(a)*b^z)/log(b); /* Problem 158: NEW 2 */ integrate(%e^a^z,z),gamma_expand:true,logcontract,domain:complex; (expintegral_ei(a^z)+log(-1))/log(a); /* Problem 159: NEW 2 */ integrate(%e^%e^z,z),gamma_expand:true,logexpand:all; expintegral_ei(%e^z)+log(-1); /* ----- Involving trigonometric functions ---------------------------------- */ /* Problem 160: NOUN */ integrate(%e^tan(z),z); 'integrate(%e^tan(z),z); /* Problem 161: NOUN */ integrate(%e^(a*tan(z)),z); 'integrate(%e^(a*tan(z)),z); /* Problem 162: NOUN */ integrate(%e^cot(z),z); 'integrate(%e^cot(z),z); /* Problem 163: NOUN */ integrate(%e^(a*cot(z)),z); 'integrate(%e^(a*cot(z)),z); /* ----- Involving hyperbolic functions ------------------------------------- */ /* Problem 164: NOUN */ integrate(%e^tanh(z),z); 'integrate(%e^tanh(z),z); /* Problem 165: NOUN */ integrate(%e^(a*tanh(z)),z); 'integrate(%e^(a*tanh(z)),z); /* Problem 166: NOUN */ integrate(%e^coth(z),z); 'integrate(%e^coth(z),z); /* Problem 167: NOUN */ integrate(%e^(a*coth(z)),z); 'integrate(%e^(a*coth(z)),z); /* ----- Involving inverse trigonometric functions -------------------------- */ /* Problem 168: NOUN */ integrate(%e^asin(z),z); 'integrate(%e^asin(z),z); /* Problem 169: NOUN */ integrate(%e^(a*asin(z)),z); 'integrate(%e^(a*asin(z)),z); /* Problem 170: NOUN */ integrate(%e^acos(z),z); 'integrate(%e^acos(z),z); /* Problem 171: NOUN */ integrate(%e^(a*acos(z)),z); 'integrate(%e^(a*acos(z)),z); /* Problem 172: NOUN */ integrate(%e^atan(z),z); 'integrate(%e^atan(z),z); /* Problem 173: NOUN */ integrate(%e^(a*atan(z)),z); 'integrate(%e^(a*atan(z)),z); /* Problem 174: NOUN */ integrate(%e^acot(z),z); 'integrate(%e^acot(z),z); /* Problem 175: NOUN */ integrate(%e^(a*acot(z)),z); 'integrate(%e^(a*acot(z)),z); /* Problem 176: NOUN */ integrate(%e^acsc(z),z); 'integrate(%e^acsc(z),z); /* Problem 177: NOUN */ integrate(%e^(a*acsc(z)),z); 'integrate(%e^(a*acsc(z)),z); /* Problem 178: NOUN */ integrate(%e^asec(z),z); 'integrate(%e^asec(z),z); /* Problem 179: NOUN */ integrate(%e^(a*asec(z)),z); 'integrate(%e^(a*asec(z)),z); /* Arguments invovling inverse hyperbolic functions ------------------------- */ /* Problem 180: NEW 12/2009 */ integrate(%e^asinh(z),z); log(sqrt(z^2+1)+z)/2+z*sqrt(z^2+1)/2+z^2/2; /* Problem 181: NOUN */ integrate(%e^(a*asinh(z)),z); 'integrate(%e^(a*asinh(z)),z); /* Problem 182: NEW 12/2009 */ integrate(%e^acosh(z),z); -log(2*sqrt(z^2-1)+2*z)/2+z*sqrt(z^2-1)/2+z^2/2; /* Problem 183: NOUN */ integrate(%e^(a*acosh(z)),z); 'integrate(%e^(a*acosh(z)),z); /* Problem 184: NOUN */ integrate(%e^atan(z),z); 'integrate(%e^atan(z),z); /* Problem 185: NOUN */ integrate(%e^(a*atan(z)),z); 'integrate(%e^(a*atan(z)),z); /* Problem 186: NOUN */ integrate(%e^acot(z),z); 'integrate(%e^acot(z),z); /* Problem 187: NOUN */ integrate(%e^(a*acot(z)),z); 'integrate(%e^(a*acot(z)),z); /* Problem 189: NEW 08/2010 * This example works nicely for x>0. For z complex we get abs(z) expressions. */ assume(x>0); [x>0]; integrate(%e^acsch(x),x),radcan; -log(sqrt(x^2+1)+1)+2*log(x)+sqrt(x^2+1); forget(x>0); [x>0]; /* Problem 191: NOUN */ integrate(%e^(a*acsch(z)),z); 'integrate(%e^(a*acsch(z)),z); /* Problem 193: NEW 12/2009 */ assume(x>0); [x>0]; integrate(%e^asech(x),x),ratsimp; -log((2*sqrt(1-x^2)+2)/x)+log(x)+sqrt(1-x^2)$ forget(x>0); [x>0]; /* Problem 195: NOUN */ integrate(%e^(a*asech(z)),z); 'integrate(%e^(a*asech(z)),z); /* ----- Involving z^n*d^(a*z^r+b*z) ---------------------------------------- */ /* Problem 196: NEW 5 */ niceindices(integrate(z^n*d^(b*z+a*z^2),z)); -a^(-n-1)*('sum(b^(n-i)*gamma_incomplete((i+1)/2,-log(d)*(2*a*z+b)^2/(4*a)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*a*z+b)^(i+1) /(-log(d)*(2*a*z+b)^2/a)^((i+1)/2),i,0,n)) /(2*d^(b^2/(4*a))); /* ----- Prove algorithm type 5 for some special case ----- */ diffcheck(integrate(z*d^(b*z+a*z^2),z),z,radcan); z*d^(b*z+a*z^2); diffcheck(integrate(z^2*d^(b*z+a*z^2),z),z,radcan); z^2*d^(b*z+a*z^2); diffcheck(integrate(z^3*d^(b*z+a*z^2),z),z,radcan); z^3*d^(b*z+a*z^2); /* Problem 200: NEW 5 */ niceindices(integrate(z^n*%e^(b*z+a*z^2),z)); -a^(-n-1)*%e^-(b^2/(4*a)) *('sum(b^(n-i)*gamma_incomplete((i+1)/2,-(2*a*z+b)^2/(4*a)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*a*z+b)^(i+1) /(-(2*a*z+b)^2/a)^((i+1)/2),i,0,n)) /2; /* ----- Prove algorithm type 5 for some special case ----- */ diffcheck(integrate(z*%e^(b*z+a*z^2),z),z,factor), domain:real; z*%e^(a*z^2+b*z)$ diffcheck(integrate(z^2*%e^(b*z+a*z^2),z),z,factor), domain:real; z^2*%e^(a*z^2+b*z)$ diffcheck(integrate(z^3*%e^(b*z+a*z^2),z),z,factor), domain:real; z^3*%e^(a*z^2+b*z)$ /* Problem 204: NEW 6 */ niceindices(niceindices(integrate(z^n*d^(b*z+a*sqrt(z)),z))); log(d)^(-2*(n+1)+2*n+1)*2^(-n-1) *('sum(4^j*binomial(n,j) *'sum(a^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*b*sqrt(z)+a)^(j+i) *(a*log(d) *gamma_incomplete( (j+i+1)/2, -log(d) *(2*b*sqrt(z)+a)^2 /(4*b))*(2*b*sqrt(z)+a) +2*b *gamma_incomplete( (j+i+2)/2, -log(d) *(2*b*sqrt(z)+a)^2 /(4*b)) *sqrt( -log(d) *(2*b*sqrt(z)+a)^2 /b)) /(-log(d)*(2*b*sqrt(z)+a)^2/b) ^((j+i+1)/2),i,0,j),j,0,n)) /(b^(2*(n+1))*d^(a^2/(4*b))); /* --- For special cases n the integral is transformed to type 5 ------------ */ diffcheck(integrate(z*d^(b*z+a*sqrt(z)),z),z,radcan); z*d^(b*z+a*sqrt(z)); diffcheck(integrate(z^2*d^(b*z+a*sqrt(z)),z),z,radcan); z^2*d^(b*z+a*sqrt(z)); diffcheck(integrate(z^3*d^(b*z+a*sqrt(z)),z),z,radcan); z^3*d^(b*z+a*sqrt(z)); /* Problem 208: NEW 6 */ niceindices(niceindices(integrate(z^n*%e^(b*z+a*sqrt(z)),z))); %e^-(a^2/(4*b))*2^(-n-1) *('sum(4^j*binomial(n,j) *'sum(a^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*b*sqrt(z)+a)^(j+i) *(a *gamma_incomplete( (j+i+1)/2, -(2*b*sqrt(z)+a)^2/(4*b)) *(2*b*sqrt(z)+a) +2*b *gamma_incomplete( (j+i+2)/2, -(2*b*sqrt(z)+a)^2/(4*b)) *sqrt(-(2*b*sqrt(z)+a)^2/b)) /(-(2*b*sqrt(z)+a)^2/b)^((j+i+1)/2),i,0,j),j, 0,n)) /b^(2*(n+1)); /* --- Again the problem is integrated with type 5 --- */ diffcheck(integrate(z*%e^(b*z+a*sqrt(z)),z),z,factor), domain:real; z*%e^(b*z+a*sqrt(z)); diffcheck(integrate(z^2*%e^(b*z+a*sqrt(z)),z),z,factor), domain:real; z^2*%e^(b*z+a*sqrt(z)); diffcheck(integrate(z^3*%e^(b*z+a*sqrt(z)),z),z,factor), domain:real; z^3*%e^(b*z+a*sqrt(z)); /* ----- Involving z^n*d^(a*z^r+b*z+c) -------------------------------------- */ /* Problem 212: NEW 5 */ niceindices(integrate(z^n*d^(c+b*z+a*z^2),z)); -a^(-n-1)*d^(c-b^2/(4*a)) *('sum(b^(n-i)*gamma_incomplete((i+1)/2,-log(d)*(2*a*z+b)^2/(4*a)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*a*z+b)^(i+1) /(-log(d)*(2*a*z+b)^2/a)^((i+1)/2),i,0,n)) /2; /* Problem 213: NEW 5 */ niceindices(integrate(z^n*%e^(c+b*z+a*z^2),z)); -a^(-n-1)*%e^(c-b^2/(4*a)) *('sum(b^(n-i)*gamma_incomplete((i+1)/2,-(2*a*z+b)^2/(4*a)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*a*z+b)^(i+1) /(-(2*a*z+b)^2/a)^((i+1)/2),i,0,n)) /2; /* Problem 214: NEW 6 */ niceindices(niceindices(integrate(z^n*d^(c+b*z+a*sqrt(z)),z))); d^(c-a^2/(4*b))*log(d)^(-2*(n+1)+2*n+1)*2^(-n-1) *('sum(4^j*binomial(n,j) *'sum(a^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*b*sqrt(z)+a)^(j+i) *(a*log(d) *gamma_incomplete( (j+i+1)/2, -log(d)*(2*b*sqrt(z)+a)^2/(4*b)) *(2*b*sqrt(z)+a) +2*b *gamma_incomplete( (j+i+2)/2, -log(d)*(2*b*sqrt(z)+a)^2/(4*b)) *sqrt( -log(d)*(2*b*sqrt(z)+a)^2/b)) /(-log(d)*(2*b*sqrt(z)+a)^2/b)^((j+i+1)/2),i, 0,j),j,0,n)) /b^(2*(n+1)); /* Problem 215: NEW 6 */ niceindices(niceindices(integrate(z^n*%e^(c+b*z+a*sqrt(z)),z))); %e^(c-a^2/(4*b))*2^(-n-1) *('sum(4^j*binomial(n,j) *'sum(a^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*b*sqrt(z)+a)^(j+i) *(a *gamma_incomplete( (j+i+1)/2, -(2*b*sqrt(z)+a)^2/(4*b)) *(2*b*sqrt(z)+a) +2*b *gamma_incomplete( (j+i+2)/2, -(2*b*sqrt(z)+a)^2/(4*b)) *sqrt(-(2*b*sqrt(z)+a)^2/b)) /(-(2*b*sqrt(z)+a)^2/b)^((j+i+1)/2),i,0,j),j, 0,n)) /b^(2*(n+1)); /* ----- Involving z^(v-1)*d^(c+a*z^r) -------------------------------------- */ /* Problem 216: NEW 2 */ integrate(z^(v-1)*d^(c+a*z^r),z); -d^c*gamma_incomplete(v/r,-a*log(d)*z^r)*z^v/(r*(-a*log(d)*z^r)^(v/r)); /* Problem 217: NEW 2 */ integrate(z^(v-1)*%e^(c+a*z^r),z); -%e^c*gamma_incomplete(v/r,-a*z^r)*z^v/(r*(-a*z^r)^(v/r)); /* Problem 218: NEW 2 */ integrate(z^(v-1)*d^(c+a*z^2),z); -d^c*gamma_incomplete(v/2,-a*log(d)*z^2)*z^v/(2*(-a*log(d)*z^2)^(v/2)); /* Problem 219: NEW 2 */ integrate(z^(v-1)*%e^(c+a*z^2),z); -%e^c*gamma_incomplete(v/2,-a*z^2)*z^v/(2*(-a*z^2)^(v/2)); /* Problem 220: NEW 2 */ integrate(z^(v-1)*d^(c+a*sqrt(z)),z); -2*d^c*gamma_incomplete(2*v,-a*log(d)*sqrt(z))*z^v/(-a*log(d)*sqrt(z))^(2*v); /* Problem 221: NEW 2 */ integrate(z^(v-1)*%e^(c+a*sqrt(z)),z); -2*%e^c*gamma_incomplete(2*v,-a*sqrt(z))*z^v/(-a*sqrt(z))^(2*v); /* ----- Involving z^(2*n)*d^(a*z^2+b/Z^2) ---------------------------------- */ /* Problem 222: NOUN */ integrate(z^(2*n)*d^(b/z^2+a*z^2),z); 'integrate(d^(a*z^2+b/z^2)*z^(2*n),z); /* Problem 223: NOUN */ integrate(z^(2*n)*%e^(b/z^2+a*z^2),z); 'integrate(z^(2*n)*%e^(a*z^2+b/z^2),z); /* ----- Involving z^(2*n)*d^(a*z^2+b/Z^2+c) -------------------------------- */ /* Problem 224: NOUN */ integrate(z^(2*n)*d^(c+b/z^2+a*z^2),z); 'integrate(d^(a*z^2+b/z^2+c)*z^(2*n),z); /* Problem 225: NOUN */ integrate(z^(2*n)*%e^(c+b/z^2+a*z^2),z); 'integrate(z^(2*n)*%e^(a*z^2+b/z^2+c),z); /* ----- Involving powers of exp with arguments c*z and base a -------------- */ /* Problem 226: OK */ integrate((a^(c*z))^v,z); (a^(c*z))^v/(c*v*log(a)); /* Problem 227: OK */ integrate((a^z)^v,z); (a^z)^v/(log(a)*v); /* ----- with arguments c*z and base %e ------------------------------------- */ /* The following examples 217 - 224 are not really correct, because Maxima * simplifies (%e^(c*z))^v -> %e^(v*c*z), which is not correct in general. */ /* Problem 228: OK */ integrate((%e^(c*z))^v,z); (%e^(c*z))^v/(c*v); /* Problem 229: OK */ integrate((%e^z)^v,z); (%e^z)^v/v; /* Problem 230: OK */ integrate(sqrt(%e^(c*z)),z); 2*sqrt(%e^(c*z))/c; /* Problem 231: OK */ integrate((%e^(c*z))^(3/2),z); 2*(%e^(c*z))^(3/2)/(3*c); /* Problem 232: OK */ integrate((%e^(c*z))^(5/2),z); 2*(%e^(c*z))^(5/2)/(5*c); /* Problem 233: OK */ integrate(1/sqrt(%e^(c*z)),z); -2/(c*sqrt(%e^(c*z))); /* Problem 234: OK */ integrate(1/(%e^(c*z))^(3/2),z); -2*(%e^(c*z))^(-3/2)/(3*c); /* Problem 235: OK */ integrate(1/(%e^(c*z))^(5/2),z); -2*(%e^(c*z))^(-5/2)/(5*c); /* ----- with arguments c*z+b and base a ------------------------------------ */ /* Problem 236: OK */ integrate((a^(c*z+b))^v,z); (a^(c*z+b))^v/(log(a)*c*v); /* ----- with arguments c*z+b and base %e ---------------------------------- */ /* Problem 237: (OK) wrong simplification */ integrate((%e^(c*z+b))^v,z); (%e^(c*z+b))^v/(c*v); /* ----- with arguments c*z^r and base a ------------------------------------ */ /* Problem 238: NEW 1a */ integrate((a^(c*z^r))^v,z); -gamma_incomplete(1/r,-log(a)*c*v*z^r)*(a^(c*z^r))^v*z /(a^(c*v*z^r)*r*(-log(a)*c*v*z^r)^(1/r)); /* Problem 239: (OK) wrong simplification * The examples 229 and 230 are not really correct. * e.g. (a^(c*z^2))^v is wrongly simplified to a^(v*c*z^2) and integrated. */ integrate((a^(c*z^2))^v,z), domain:real; sqrt(%pi)*erf(sqrt(-log(a)*c*v)*z)/(2*sqrt(-log(a)*c*v))$ /* Problem 240: (OK) wrong simplification */ integrate((a^(c*sqrt(z)))^v,z), domain:real; 2*(log(a)*c*v*sqrt(z)-1)*%e^(log(a)*c*v*sqrt(z))/(log(a)^2*c^2*v^2)$ /* ----- with arguments c*z^r ----------------------------------------------- */ /* Problem 241: NEW 1a */ integrate((%e^(c*z^r))^v,z); -gamma_incomplete(1/r,-c*v*z^r)*z*(%e^(c*z^r))^v*%e^-(c*v*z^r) /(r*(-c*v*z^r)^(1/r)); /* Problem 242: (OK) wrong simplification*/ integrate((%e^(c*z^2))^v,z), domain:real; sqrt(%pi)*erf(sqrt(-c*v)*z)/(2*sqrt(-c*v))$ /* Problem 243: (OK) wrong simplification*/ integrate((%e^(c*sqrt(z)))^v,z), domain:real; 2*(c*v*sqrt(z)-1)*%e^(c*v*sqrt(z))/(c^2*v^2)$ /* ----- with arguments c*z^r+d and base a ---------------------------------- */ /* Problem 244: NEW 1a */ integrate((a^(d+c*z^r))^v,z); -gamma_incomplete(1/r,-log(a)*c*v*z^r)*(a^(c*z^r+d))^v*z /(a^(c*v*z^r)*r*(-log(a)*c*v*z^r)^(1/r)); /* Problem 245: (OK) wrong simplification */ integrate((a^(d+c*z^2))^v,z), domain:real; sqrt(%pi)*a^(d*v)*erf(sqrt(-log(a)*c*v)*z)/(2*sqrt(-log(a)*c*v))$ /* Problem 246: (OK) wrong simplification */ integrate((a^(d+c*sqrt(z)))^v,z), domain:real; 2*(a^(d*v)*log(a)*c*v*sqrt(z)-a^(d*v))*%e^(log(a)*c*v*sqrt(z)) /(log(a)^2*c^2*v^2)$ /* ----- with arguments c*z^r+d --------------------------------------------- */ /* Problem 247: New 1a */ integrate((%e^(d+c*z^r))^v,z); -gamma_incomplete(1/r,-c*v*z^r)*z*%e^-(c*v*z^r)*(%e^(c*z^r+d))^v /(r*(-c*v*z^r)^(1/r)); /* Problem 248: (OK) wrong simplification */ integrate((%e^(d+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(d*v)*erf(sqrt(-c*v)*z)/(2*sqrt(-c*v))$ /* Problem 249: (OK) wrong simplification */ integrate((%e^(d+c*sqrt(z)))^v,z), domain:real; 2*(c*v*%e^(d*v)*sqrt(z)-%e^(d*v))*%e^(c*v*sqrt(z))/(c^2*v^2)$ /* ----- with arguments c*(z^r)^p and base a -------------------------------- */ /* Problem 250: NEW 1a * The minus factor is a workaround to get a comparable answer. */ -integrate((a^(c*(z^r)^p))^v,z); gamma_incomplete(1/(p*r),-log(a)*c*v*(z^r)^p)*(a^(c*(z^r)^p))^v*z /(a^(c*v*(z^r)^p)*p*r*(-log(a)*c*v*(z^r)^p)^(1/(p*r))); /* Problem 251: (OK) wrong simplification */ integrate((a^(c*(z^r)^(1/r)))^v,z), domain:real; %e^(log(a)*c*v*z)/(log(a)*c*v)$ /* Problem 252: (OK) wrong simplification */ integrate((a^(c*sqrt(z^2)))^v,z), domain:real; %e^(log(a)*c*v*z)/(log(a)*c*v)$ /* Problem 253: NEW 1a */ integrate((%e^(c*(z^r)^p))^v,z); -gamma_incomplete(1/(p*r),-c*v*(z^r)^p)*z*(%e^(c*(z^r)^p))^v*%e^-(c*v*(z^r)^p) /(p*r*(-c*v*(z^r)^p)^(1/(p*r))); /* Problem 254: (OK) wrong simplification */ integrate((%e^(c*(z^r)^(1/r)))^v,z), domain:real; %e^(c*v*z)/(c*v)$ /* Problem 255: (OK) wrong simplification */ integrate((%e^(c*sqrt(z^2)))^v,z), domain:real; %e^(c*v*z)/(c*v)$ /* ----- with arguments c*(z^r)^p+d and base a ------------------------------ */ /* Problem 256: NEW 1a */ -integrate((a^(d+c*(z^r)^p))^v,z); gamma_incomplete(1/(p*r),-log(a)*c*v*(z^r)^p)*(a^(c*(z^r)^p+d))^v*z /(a^(c*v*(z^r)^p)*p*r*(-log(a)*c*v*(z^r)^p)^(1/(p*r)))$ /* Problem 257: (OK) wrong simplification */ integrate((a^(d+c*(z^r)^(1/r)))^v,z), domain:real; a^(d*v)*%e^(log(a)*c*v*z)/(log(a)*c*v); /* Problem 258: (OK) wrong simplification */ integrate((a^(d+c*sqrt(z^2)))^v,z), domain:real; a^(d*v)*%e^(log(a)*c*v*z)/(log(a)*c*v); /* Problem 259: NEW 1a */ integrate((%e^(d+c*(z^r)^p))^v,z); -gamma_incomplete(1/(p*r),-c*v*(z^r)^p)*z*%e^-(c*v*(z^r)^p) *(%e^(c*(z^r)^p+d))^v /(p*r*(-c*v*(z^r)^p)^(1/(p*r))); /* Problem 260: (OK) wrong simplification */ integrate((%e^(d+c*(z^r)^(1/r)))^v,z), domain:real; %e^(c*v*z+d*v)/(c*v); /* Problem 261: (OK) wrong simplification */ integrate((%e^(d+c*sqrt(z^2)))^v,z), domain:real; %e^(c*v*z+d*v)/(c*v); /* ----- with arguments a*z^2+b/z^2 ----------------------------------------- */ /* Problem 262: Noun */ integrate((%e^(b/z^2+a*z^2))^v,z); 'integrate((%e^(b/z^2+a*z^2))^v,z); /* ----- with arguments a*z^2+b/z^2+c --------------------------------------- */ /* Problem 263: Noun */ integrate((%e^(c+b/z^2+a*z^2))^v,z); 'integrate((%e^(c+b/z^2+a*z^2))^v,z); /* ----- Involving products of power functions ------------------------------ */ /* ----- a^(d*z)*h^(c*z) ---------------------------------------------------- */ /* Problem 265: OK * In this example we have a substitution of a^(d*z). * We get a question about the sign of an expression. The expand(XXX,0,0) is due to a bug--the antiderivative is unsimplified. */ (assume(notequal((c * log(h))/(d * log(a)),-1)),0); 0$ expand(integrate(a^(d*z)*h^(c*z),z),0,0); (a^(d*z))^(c*log(h)/(log(a)*d)+1)/(log(a)*(c*log(h)/(log(a)*d)+1)*d)$ (forget(notequal((c * log(h))/(d * log(a)),-1)),0); 0$ /* ----- a^(d*z)*h^(c*z+g) -------------------------------------------------- */ /* Problem 267: OK * But not for this example. We get no question about a sign. * With the help of radcan we get the expected answer. */ integrate(a^(d*z)*h^(c*z+g),z),radcan; a^(d*z)*h^(c*z+g)/(c*log(h)+log(a)*d)$ /* ----- a^(d*z+e)*h^(c*z+g) ------------------------------------------------ */ /* Problem 268: OK */ integrate(a^(d*z+e)*h^(c*z+g),z),radcan; a^(d*z+e)*h^(c*z+g)/(c*log(h)+log(a)*d)$ /* ----- a^(d*z)*h^(c*z^r) -------------------------------------------------- */ /* Problem 269: OK */ -integrate(a^(d*z)*h^(c*z^2),z),factor; sqrt(%pi)*%e^-(log(a)^2*d^2/(4*c*log(h))) *erf((2*c*log(h)*z+log(a)*d)/(2*sqrt(-c*log(h)))) /(2*sqrt(-c*log(h))); /* Problem 270: NEW 8 */ diffcheck(integrate(a^(d*z)*h^(c*sqrt(z)),z),z,radcan); a^(d*z)*h^(c*sqrt(z)); /* ----- a^(d*z+e)*h^(c*z^r) ------------------------------------------------ */ /* Problem 271: OK */ -integrate(a^(d*z+e)*h^(c*z^2),z),factor; sqrt(%pi)*a^e*%e^-(log(a)^2*d^2/(4*c*log(h))) *erf((2*c*log(h)*z+log(a)*d)/(2*sqrt(-c*log(h)))) /(2*sqrt(-c*log(h))); /* Problem 272: NEW 8 */ diffcheck(integrate(a^(d*z+e)*h^(c*sqrt(z)),z),z,radcan); a^(d*z+e)*h^(c*sqrt(z)); /* ----- a^(b*z^r)*h^(c*z^r) ------------------------------------------------ */ /* Problem 273: NEW 7 */ diffcheck(integrate(a^(b*z^r)*h^(c*z^r),z),z,radcan); a^(b*z^r)*h^(c*z^r); /* Problem 274: OK */ integrate(a^(b*z^2)*h^(c*z^2),z); sqrt(%pi)*erf(sqrt(-c*log(h)-log(a)*b)*z)/(2*sqrt(-c*log(h)-log(a)*b)); /* Problem 275: OK */ integrate(a^(b*sqrt(z))*h^(c*sqrt(z)),z),radcan, factor; 2*a^(b*sqrt(z))*h^(c*sqrt(z))*(c*log(h)*sqrt(z)+log(a)*b*sqrt(z)-1) /(c*log(h)+log(a)*b)^2; /* ----- a^(d*z)*h^(c*z^r+g) ------------------------------------------------ */ /* Problem 276: OK */ integrate(a^(d*z)*h^(g+c*z^2),z),factor; -sqrt(%pi)*h^g*%e^-(log(a)^2*d^2/(4*c*log(h))) *erf((2*c*log(h)*z+log(a)*d)/(2*sqrt(-c*log(h)))) /(2*sqrt(-c*log(h))); /* Problem 277: NEW 8 */ diffcheck(integrate(a^(d*z)*h^(g+c*sqrt(z)),z),z,radcan); a^(d*z)*h^(g+c*sqrt(z)); /* ----- a^(d*z+e)*h^(c*z^r+g) ---------------------------------------------- */ /* Problem 278: OK */ integrate(a^(d*z+e)*h^(g+c*z^2),z); sqrt(%pi)*a^e*h^g*%e^-(log(a)^2*d^2/(4*c*log(h))) *erf(sqrt(-c*log(h))*z-log(a)*d/(2*sqrt(-c*log(h)))) /(2*sqrt(-c*log(h))); /* Problem 279: NEW 8 */ diffcheck(integrate(a^(d*z+e)*h^(g+c*sqrt(z)),z),z,radcan); a^(d*z+e)*h^(g+c*sqrt(z)); /* ----- a^(b*z^r)*h^(c*z^r+g) ---------------------------------------------- */ /* Problem 280: NEW 7 */ diffcheck(integrate(a^(b*z^r)*h^(g+c*z^r),z),z,radcan); a^(b*z^r)*h^(g+c*z^r); /* Problem 281: OK */ integrate(a^(b*z^2)*h^(g+c*z^2),z); sqrt(%pi)*h^g*erf(sqrt(-c*log(h)-log(a)*b)*z) /(2*sqrt(-c*log(h)-log(a)*b)); /* Problem 282: OK */ integrate(a^(b*sqrt(z))*h^(g+c*sqrt(z)),z),radcan,factor; 2*a^(b*sqrt(z))*h^(c*sqrt(z)+g)*(c*log(h)*sqrt(z)+log(a)*b*sqrt(z)-1) /(c*log(h)+log(a)*b)^2; /* ----- a^(b*z^r+e)*h^(c*z^r+g) -------------------------------------------- */ /* Problem 283: NEW 7 */ expand(integrate(a^(e+b*z^r)*h^(g+c*z^r),z) - -a^e*h^g*gamma_incomplete(1/r,-(c*log(h)+log(a)*b)*z^r)*z /(r*(-(c*log(h)+log(a)*b)*z^r)^(1/r))); 0; /* Problem 284: OK */ integrate(a^(e+b*z^2)*h^(g+c*z^2),z); sqrt(%pi)*a^e*h^g*erf(sqrt(-c*log(h)-log(a)*b)*z) /(2*sqrt(-c*log(h)-log(a)*b)); /* Problem 285: OK */ integrate(a^(e+b*sqrt(z))*h^(g+c*sqrt(z)),z),radcan,factor; 2*a^(b*sqrt(z)+e)*h^(c*sqrt(z)+g)*(c*log(h)*sqrt(z)+log(a)*b*sqrt(z)-1) /(c*log(h)+log(a)*b)^2; /* ----- a^(d*z)*h^(c*z^r+f*z) ---------------------------------------------- */ /* Problem 286: OK */ integrate(a^(d*z)*h^(f*z+c*z^2),z); sqrt(%pi)*%e^-((-f*log(h)-log(a)*d)^2/(4*c*log(h))) *erf(sqrt(-c*log(h))*z+(-f*log(h)-log(a)*d) /(2*sqrt(-c*log(h)))) /(2*sqrt(-c*log(h))); /* Problem 287: NEW 8 */ multthru(integrate(a^(d*z)*h^(f*z+c*sqrt(z)),z)); a^(d*z)*h^(f*z+c*sqrt(z))/(f*log(h)+log(a)*d) -sqrt(%pi)*c*%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))*log(h) *erfi((2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h)) /(2*sqrt(f*log(h)+log(a)*d))) /(2*(f*log(h)+log(a)*d)^(3/2)); /* ----- a^(d*z+e)*h^(c*z^r+f*z) -------------------------------------------- */ /* Problem 288: OK */ integrate(a^(d*z+e)*h^(f*z+c*z^2),z); sqrt(%pi)*a^e*%e^-((-f*log(h)-log(a)*d)^2/(4*c*log(h))) *erf(sqrt(-c*log(h))*z+(-f*log(h)-log(a)*d) /(2*sqrt(-c*log(h)))) /(2*sqrt(-c*log(h))); /* Problem 289: NEW 8 */ multthru(integrate(a^(d*z+e)*h^(f*z+c*sqrt(z)),z)); a^(d*z+e)*h^(f*z+c*sqrt(z))/(f*log(h)+log(a)*d) -sqrt(%pi)*a^e*c*%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))*log(h) *erfi((2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h)) /(2*sqrt(f*log(h)+log(a)*d))) /(2*(f*log(h)+log(a)*d)^(3/2)); /* ----- a^(b*z^r)*h^(c*z^r+f*z) -------------------------------------------- */ /* Problem 290: OK */ integrate(a^(b*z^2)*h^(f*z+c*z^2),z); sqrt(%pi)*%e^(f^2*log(h)^2/(4*(-c*log(h)-log(a)*b))) *erf(sqrt(-c*log(h)-log(a)*b)*z -f*log(h)/(2*sqrt(-c*log(h)-log(a)*b))) /(2*sqrt(-c*log(h)-log(a)*b)); /* Problem 291: NEW 9 * This problem does not simplify to a nice expression in terms of the erf * function and a term a^b*sqrt(z)*h^(sqrt(z)+f*z), but it is correct. */ diffcheck(integrate(a^(b*sqrt(z))*h^(f*z+c*sqrt(z)),z),z,radcan); a^(b*sqrt(z))*h^(f*z+c*sqrt(z)); /* ----- a^(b*z^r+e)*h^(c*z^r+f*z) ------------------------------------------ */ /* Problem 292: OK */ integrate(a^(e+b*z^2)*h^(f*z+c*z^2),z); sqrt(%pi)*a^e*%e^(f^2*log(h)^2/(4*(-c*log(h)-log(a)*b))) *erf(sqrt(-c*log(h)-log(a)*b)*z -f*log(h)/(2*sqrt(-c*log(h)-log(a)*b))) /(2*sqrt(-c*log(h)-log(a)*b)); /* Problem 293: NEW 9 */ diffcheck(integrate(a^(e+b*sqrt(z))*h^(f*z+c*sqrt(z)),z),z,radcan); a^(e+b*sqrt(z))*h^(f*z+c*sqrt(z)); /* ----- a^(b*z^r+d*z)*h^(c*z^r+f*z) ---------------------------------------- */ /* Problem 294: OK */ integrate(a^(d*z+b*z^2)*h^(f*z+c*z^2),z); sqrt(%pi)*%e^((-f*log(h)-log(a)*d)^2/(4*(-c*log(h)-log(a)*b))) *erf(sqrt(-c*log(h)-log(a)*b)*z +(-f*log(h)-log(a)*d)/(2*sqrt(-c*log(h)-log(a)*b))) /(2*sqrt(-c*log(h)-log(a)*b)); /* Problem 295: NEW 9 * Is this example correct? Verify it again. * The problem is that Maxima first does some substitution, the integral * is solvable too, but the result becomes more complicated. */ multthru(integrate(a^(d*z+b*sqrt(z))*h^(f*z+c*sqrt(z)),z)); gamma_incomplete(1, -(log(h)*(2*f*sqrt(z)+c)+log(a)*(2*d*sqrt(z)+b))^2 /(4*(f*log(h)+log(a)*d))) *%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d))) /(f*log(h)+log(a)*d) -gamma_incomplete(1/2, -(log(h)*(2*f*sqrt(z)+c)+log(a)*(2*d*sqrt(z)+b))^2 /(4*(f*log(h)+log(a)*d))) *%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))*(-c*log(h)-log(a)*b) *(log(h)*(2*f*sqrt(z)+c)+log(a)*(2*d*sqrt(z)+b)) /(2*(f*log(h)+log(a)*d)^2 *sqrt(-(log(h)*(2*f*sqrt(z)+c)+log(a)*(2*d*sqrt(z)+b))^2 /(f*log(h)+log(a)*d)))$ /* ----- a^(d*z)*h^(c*z^r+f*z+g) -------------------------------------------- */ /* Problem 296: OK */ integrate(a^(d*z)*h^(g+f*z+c*z^2),z); sqrt(%pi)*h^g*%e^-((-f*log(h)-log(a)*d)^2/(4*c*log(h))) *erf(sqrt(-c*log(h))*z+(-f*log(h)-log(a)*d) /(2*sqrt(-c*log(h)))) /(2*sqrt(-c*log(h))); /* Problem 297: NEW 8 */ multthru(integrate(a^(d*z)*h^(g+f*z+c*sqrt(z)),z)); a^(d*z)*h^(f*z+c*sqrt(z)+g)/(f*log(h)+log(a)*d) -sqrt(%pi)*c*h^g*%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))*log(h) *erfi((2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h)) /(2*sqrt(f*log(h)+log(a)*d))) /(2*(f*log(h)+log(a)*d)^(3/2)); /* ----- a^(d*z+e)*h^(c*z^r+f*z+g) ------------------------------------------ */ /* Problem 298: OK */ integrate(a^(d*z+e)*h^(g+f*z+c*z^2),z); sqrt(%pi)*a^e*h^g*%e^-((-f*log(h)-log(a)*d)^2/(4*c*log(h))) *erf(sqrt(-c*log(h))*z+(-f*log(h)-log(a)*d) /(2*sqrt(-c*log(h)))) /(2*sqrt(-c*log(h))); /* Problem 299: NEW 8 */ multthru(integrate(a^(d*z+e)*h^(g+f*z+c*sqrt(z)),z)); a^(d*z+e)*h^(f*z+c*sqrt(z)+g)/(f*log(h)+log(a)*d) -sqrt(%pi)*a^e*c*h^g*%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))*log(h) *erfi((2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h)) /(2*sqrt(f*log(h)+log(a)*d))) /(2*(f*log(h)+log(a)*d)^(3/2)); /* ----- a^(b*z^r)*h^(c*z^r+f*z+g) ------------------------------------------ */ /* Problem 300: OK */ integrate(a^(b*z^2)*h^(g+f*z+c*z^2),z); sqrt(%pi)*h^g*%e^(f^2*log(h)^2/(4*(-c*log(h)-log(a)*b))) *erf(sqrt(-c*log(h)-log(a)*b)*z -f*log(h)/(2*sqrt(-c*log(h)-log(a)*b))) /(2*sqrt(-c*log(h)-log(a)*b)); /* Problem 301: NEW 9 */ multthru(integrate(a^(b*sqrt(z))*h^(g+f*z+c*sqrt(z)),z)); gamma_incomplete(1,-(log(h)*(2*f*sqrt(z)+c)+log(a)*b)^2/(4*f*log(h))) *h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h))) /(f*log(h)) -gamma_incomplete(1/2,-(log(h)*(2*f*sqrt(z)+c)+log(a)*b)^2/(4*f*log(h))) *h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))*(-c*log(h)-log(a)*b) *(log(h)*(2*f*sqrt(z)+c)+log(a)*b) /(2*f^2*log(h)^2*sqrt(-(log(h)*(2*f*sqrt(z)+c)+log(a)*b)^2/(f*log(h)))); /* ----- a^(b*z^r+e)*h^(c*z^r+f*z+g) ---------------------------------------- */ /* Problem 302: OK */ integrate(a^(e+b*z^2)*h^(g+f*z+c*z^2),z); sqrt(%pi)*a^e*h^g*%e^(f^2*log(h)^2/(4*(-c*log(h)-log(a)*b))) *erf(sqrt(-c*log(h)-log(a)*b)*z -f*log(h)/(2*sqrt(-c*log(h)-log(a)*b))) /(2*sqrt(-c*log(h)-log(a)*b)); /* Problem 303: NEW 9 */ multthru(integrate(a^(e+b*sqrt(z))*h^(g+f*z+c*sqrt(z)),z)); gamma_incomplete(1,-(log(h)*(2*f*sqrt(z)+c)+log(a)*b)^2/(4*f*log(h))) *a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h))) /(f*log(h)) -gamma_incomplete(1/2,-(log(h)*(2*f*sqrt(z)+c)+log(a)*b)^2/(4*f*log(h))) *a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))*(-c*log(h)-log(a)*b) *(log(h)*(2*f*sqrt(z)+c)+log(a)*b) /(2*f^2*log(h)^2*sqrt(-(log(h)*(2*f*sqrt(z)+c)+log(a)*b)^2/(f*log(h)))); /* ----- a^(b*z^r+d*z)*h^(c*z^r+f*z+g) -------------------------------------- */ /* Problem 304: OK */ integrate(a^(d*z+b*z^2)*h^(g+f*z+c*z^2),z); sqrt(%pi)*h^g*%e^((-f*log(h)-log(a)*d)^2/(4*(-c*log(h)-log(a)*b))) *erf(sqrt(-c*log(h)-log(a)*b)*z +(-f*log(h)-log(a)*d)/(2*sqrt(-c*log(h)-log(a)*b))) /(2*sqrt(-c*log(h)-log(a)*b)); /* Problem 305: NEW 9 */ multthru(integrate(a^(d*z+b*sqrt(z))*h^(g+f*z+c*sqrt(z)),z)); gamma_incomplete(1, -(log(h)*(2*f*sqrt(z)+c)+log(a)*(2*d*sqrt(z)+b))^2 /(4*(f*log(h)+log(a)*d))) *h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d))) /(f*log(h)+log(a)*d) -gamma_incomplete(1/2, -(log(h)*(2*f*sqrt(z)+c)+log(a)*(2*d*sqrt(z)+b))^2 /(4*(f*log(h)+log(a)*d))) *h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d))) *(-c*log(h)-log(a)*b)*(log(h)*(2*f*sqrt(z)+c)+log(a)*(2*d*sqrt(z)+b)) /(2*(f*log(h)+log(a)*d)^2 *sqrt(-(log(h)*(2*f*sqrt(z)+c)+log(a)*(2*d*sqrt(z)+b))^2 /(f*log(h)+log(a)*d))); /* ----- a^(b*z^r+d*z+e)*h^(c*z^r+f*z+g) ------------------------------------ */ /* Problem 306: OK */ integrate(a^(e+d*z+b*z^2)*h^(g+f*z+c*z^2),z); sqrt(%pi)*a^e*h^g*%e^((-f*log(h)-log(a)*d)^2/(4*(-c*log(h)-log(a)*b))) *erf(sqrt(-c*log(h)-log(a)*b)*z +(-f*log(h)-log(a)*d)/(2*sqrt(-c*log(h)-log(a)*b))) /(2*sqrt(-c*log(h)-log(a)*b)); /* Problem 307: NEW 9 */ multthru(integrate(a^(e+d*z+b*sqrt(z))*h^(g+f*z+c*sqrt(z)),z)); gamma_incomplete(1, -(log(h)*(2*f*sqrt(z)+c)+log(a)*(2*d*sqrt(z)+b))^2 /(4*(f*log(h)+log(a)*d))) *a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d))) /(f*log(h)+log(a)*d) -gamma_incomplete(1/2, -(log(h)*(2*f*sqrt(z)+c)+log(a)*(2*d*sqrt(z)+b))^2 /(4*(f*log(h)+log(a)*d))) *a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d))) *(-c*log(h)-log(a)*b)*(log(h)*(2*f*sqrt(z)+c)+log(a)*(2*d*sqrt(z)+b)) /(2*(f*log(h)+log(a)*d)^2 *sqrt(-(log(h)*(2*f*sqrt(z)+c)+log(a)*(2*d*sqrt(z)+b))^2 /(f*log(h)+log(a)*d))); /* ----- a^(d*z)*b^(e*z)*c^(f*z) -------------------------------------------- */ /* Problem 309: OK * This example does not simplify in a product of three power functions. */ (assume(notequal(ratsimp((log(c)*f+log(b)*e)/(log(a)*d)),-1)),0); 0$ integrate(a^(d*z)*b^(e*z)*c^(f*z),z); (a^(d*z))^(log(c)*f/(log(a)*d)+log(b)*e/(log(a)*d)+1)/(log(a)*d*(log(c)*f/(log(a)*d)+log(b)*e/(log(a)*d)+1))$ (forget(notequal(log(c)*f/(log(a)*d)+log(b)*e/(log(a)*d),-1)),0); 0$ /* ----- %e^(c*z)*(%e^(a*z))^v ---------------------------------------------- */ /* Problem 311: (OK) wrong simplification for complex integration variable * Change example to a real variable x. */ integrate(%e^(c*x)*(%e^(a*x))^v,x); %e^(a*v*x+c*x)/(a*v+c)$ /* ----- %e^(c*z+d)*(%e^(a*z))^v -------------------------------------------- */ /* Problem 312: (OK) wrong simplification * Change example to a real variable x. */ integrate(%e^(c*x+d)*(%e^(a*x))^v,x); %e^(a*v*x+c*x+d)/(a*v+c)$ /* ----- %e^(c*z)*(%e^(a*z+b))^v -------------------------------------------- */ /* Problem 313: (OK) wrong simplification * Change example to a real variable x. */ integrate(%e^(c*x)*(%e^(a*x+b))^v,x); %e^(v*(a*x+b)+c*x)/(a*v+c)$ /* ----- %e^(c*z+d)*(%e^(a*z+b))^v ------------------------------------------ */ /* Problem 314: (OK) wrong simplifcation * Change example to a real variable x. */ integrate(%e^(c*x+d)*(%e^(a*x+b))^v,x); %e^(v*(a*x+b)+c*x+d)/(a*v+c)$ /* ----- %e^(b*z^r)*(%e^(c*z))^v -------------------------------------------- */ /* Problem 315: (OK) wrong simplification */ integrate(%e^(b*z^2)*(%e^(c*z))^v,z), domain:real; sqrt(%pi)*%e^-(c^2*v^2/(4*b))*erf(sqrt(-b)*z-c*v/(2*sqrt(-b)))/(2*sqrt(-b)); /* Problem 316: NEW 9-1 */ diffcheck(integrate(%e^(b*sqrt(z))*(%e^(c*z))^v,z),z,factor_radcan), domain:complex; %e^(b*sqrt(z))*(%e^(c*z))^v; /* ----- %e^(b*z^r+e)*(%e^(c*z))^v ------------------------------------------ */ /* Problem 317: (OK) wrong simplification */ integrate(%e^(e+b*z^2)*(%e^(c*z))^v,z), domain:real; sqrt(%pi)*%e^(e-c^2*v^2/(4*b))*erf(sqrt(-b)*z-c*v/(2*sqrt(-b)))/(2*sqrt(-b)); /* Problem 318: NEW 9-1 */ diffcheck(integrate(%e^(e+b*sqrt(z))*(%e^(c*z))^v,z),z,factor_radcan), domain:complex; %e^(e+b*sqrt(z))*(%e^(c*z))^v; /* ----- %e^(b*z^r+d*z)*(%e^(c*z))^v ---------------------------------------- */ /* Problem 319: (OK) wrong simplification */ integrate(%e^(d*z+b*z^2)*(%e^(c*z))^v,z), domain:real; sqrt(%pi)*%e^-((-c*v-d)^2/(4*b))*erf(sqrt(-b)*z+(-c*v-d)/(2*sqrt(-b))) /(2*sqrt(-b)); /* Problem 320: NEW 8-1 */ diffcheck(integrate(%e^(d*z+b*sqrt(z))*(%e^(c*z))^v,z),z,factor); %e^(d*z+b*sqrt(z))*(%e^(c*z))^v; /* ----- %e^(b*z^r+d*z+e)*(%e^(c*z))^v -------------------------------------- */ /* Problem 321: OK */ integrate(%e^(e+d*z+b*z^2)*(%e^(c*z))^v,z), domain:real; sqrt(%pi)*%e^(e-(-c*v-d)^2/(4*b))*erf(sqrt(-b)*z+(-c*v-d)/(2*sqrt(-b))) /(2*sqrt(-b)); /* Problem 322: NEW 8-1 */ diffcheck(integrate(%e^(e+d*z+b*sqrt(z))*(%e^(c*z))^v,z),z,factor); %e^(e+d*z+b*sqrt(z))*(%e^(c*z))^v; /* ----- %e^(b*z^r)*(%e^(f*z+g))^v ------------------------------------------ */ /* Problem 323: OK */ integrate(%e^(b*z^2)*(%e^(g+f*z))^v,z), domain:real; sqrt(%pi)*%e^(g*v-f^2*v^2/(4*b))*erf(sqrt(-b)*z-f*v/(2*sqrt(-b)))/(2*sqrt(-b)); /* Problem 324: NEW 9-1 */ diffcheck(integrate(%e^(b*sqrt(z))*(%e^(g+f*z))^v,z),z,factor), domain:real; %e^(f*v*z+b*sqrt(z)+g*v)$ /* ----- %e^(b*z^r+e)*(%e^(f*z+g))^v ---------------------------------------- */ /* Problem 325: OK */ integrate(%e^(e+b*z^2)*(%e^(g+f*z))^v,z), domain:real; sqrt(%pi)*%e^(-f^2*v^2/(4*b)+g*v+e)*erf(sqrt(-b)*z-f*v/(2*sqrt(-b))) /(2*sqrt(-b)); /* Problem 326: NEW 9-1 */ diffcheck(integrate(%e^(e+b*sqrt(z))*(%e^(g+f*z))^v,z),z,factor), domain:real; %e^(f*v*z+b*sqrt(z)+g*v+e)$ /* ----- %e^(b*z^r+d*z)*(%e^(f*z+g))^v -------------------------------------- */ /* Problem 327: OK */ integrate(%e^(d*z+b*z^2)*(%e^(g+f*z))^v,z), domain:real; sqrt(%pi)*%e^(g*v-(-f*v-d)^2/(4*b))*erf(sqrt(-b)*z+(-f*v-d)/(2*sqrt(-b))) /(2*sqrt(-b)); /* Problem 328: NEW 9-1 * This example is not correct simplified. */ diffcheck(integrate(%e^(d*z+b*sqrt(z))*(%e^(g+f*z))^v,z),z,radcan), domain:complex; (%e^(f*z))^v*%e^(d*z+b*sqrt(z)+g*v); /* ----- %e^(b*z^r+d*z+e)*(%e^(f*z+g))^v ------------------------------------ */ /* Problem 329: OK */ integrate(%e^(e+d*z+b*z^2)*(%e^(g+f*z))^v,z), domain:real; sqrt(%pi)*%e^(-(-f*v-d)^2/(4*b)+g*v+e)*erf(sqrt(-b)*z+(-f*v-d)/(2*sqrt(-b))) /(2*sqrt(-b)); /* Problem 330: NEW 9-1 * This example is not correct simplified. */ diffcheck(integrate(%e^(e+d*z+b*sqrt(z))*(%e^(g+f*z))^v,z),z,radcan), domain:complex; (%e^(f*z))^v*%e^(d*z+b*sqrt(z)+g*v+e); /* ----- %e^(b*z)*(%e^(c*z^r))^v -------------------------------------------- */ /* Problem 331: OK */ integrate(%e^(b*z)*(%e^(c*z^2))^v,z), domain:real; sqrt(%pi)*%e^-(b^2/(4*c*v))*erf(sqrt(-c*v)*z-b/(2*sqrt(-c*v)))/(2*sqrt(-c*v)); /* Problem 332: NEW 9-1 */ diffcheck(integrate(%e^(b*z)*(%e^(c*sqrt(z)))^v,z),z,radcan),domain:complex; %e^(b*z)*(%e^(c*sqrt(z)))^v; /* ----- %e^(d*z+e)*(%e^(c*z^r))^v ------------------------------------------ */ /* Problem 333: OK */ integrate(%e^(d*z+e)*(%e^(c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(e-d^2/(4*c*v))*erf(sqrt(-c*v)*z-d/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 334: NEW 9-1 */ diffcheck(integrate(%e^(d*z+e)*(%e^(c*sqrt(z)))^v,z),z,factor), domain:real; %e^(d*z+c*v*sqrt(z)+e)$ /* ----- %e^(b*z^r)*(%e^(c*z^r))^v ------------------------------------------ */ /* Problem 335: NEW 7-1 */ diffcheck(integrate(%e^(b*z^r)*(%e^(c*z^r))^v,z),z,factor); %e^(b*z^r)*(%e^(c*z^r))^v; /* Problem 336: OK */ integrate(%e^(b*z^2)*(%e^(c*z^2))^v,z), domain:real; sqrt(%pi)*erf(sqrt(-c*v-b)*z)/(2*sqrt(-c*v-b)); /* Problem 337: OK */ integrate(%e^(b*sqrt(z))*(%e^(c*sqrt(z)))^v,z), domain:real; 2*((c*v+b)*sqrt(z)-1)*%e^(c*v*sqrt(z)+b*sqrt(z))/(c^2*v^2+2*b*c*v+b^2); /* ----- %e^(b*z^r+e)*(%e^(c*z^r))^v ---------------------------------------- */ /* Problem 338: NEW 7-1 */ diffcheck(integrate(%e^(e+b*z^r)*(%e^(c*z^r))^v,z),z,factor); %e^(e+b*z^r)*(%e^(c*z^r))^v; /* Problem 339: OK */ integrate(%e^(e+b*z^2)*(%e^(c*z^2))^v,z), domain:real; sqrt(%pi)*%e^e*erf(sqrt(-c*v-b)*z)/(2*sqrt(-c*v-b)); /* Problem 340: OK */ integrate(%e^(e+b*sqrt(z))*(%e^(c*sqrt(z)))^v,z), domain:real; 2*((c*%e^e*v+b*%e^e)*sqrt(z)-%e^e)*%e^(c*v*sqrt(z)+b*sqrt(z)) /(c^2*v^2+2*b*c*v+b^2); /* ----- %e^(b*z^r+d*z)*(%e^(c*z^r))^v -------------------------------------- */ /* Problem 341: OK */ integrate(%e^(d*z+b*z^2)*(%e^(c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(d^2/(4*(-c*v-b)))*erf(sqrt(-c*v-b)*z-d/(2*sqrt(-c*v-b))) /(2*sqrt(-c*v-b)); /* Problem 342: NEW 9-1 */ diffcheck(integrate(%e^(d*z+b*sqrt(z))*(%e^(c*sqrt(z)))^v,z),z,radcan), domain:complex; %e^(d*z+b*sqrt(z))*(%e^(c*sqrt(z)))^v; /* ----- %e^(b*z^r+d*z+e)*(%e^(c*z^r))^v ------------------------------------ */ /* Problem 343: OK */ integrate(%e^(e+d*z+b*z^2)*(%e^(c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(d^2/(4*(-c*v-b))+e) *erf(sqrt(-c*v-b)*z-d/(2*sqrt(-c*v-b))) /(2*sqrt(-c*v-b)); /* Problem 344: NEW 9-1 */ diffcheck(integrate(%e^(e+d*z+b*sqrt(z))*(%e^(c*sqrt(z)))^v,z),z,radcan), domain:complex; %e^(e+d*z+b*sqrt(z))*(%e^(c*sqrt(z)))^v; /* ----- %e^(d*z)*(%e^(c*z^r+g))^v ------------------------------------------ */ /* Problem 345: OK */ integrate(%e^(d*z)*(%e^(g+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(g*v-d^2/(4*c*v))*erf(sqrt(-c*v)*z-d/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 346: NEW 9-1 */ expand(integrate(%e^(d*z)*(%e^(g+c*sqrt(z)))^v,z)- (-(-gamma_incomplete(1/2,-(2*d*sqrt(z)+c*v)^2/(4*d))*c*v*(2*d*sqrt(z)+c*v) /(2*d^(3/2)*sqrt(-(2*d*sqrt(z)+c*v)^2/d)) -gamma_incomplete(1,-(2*d*sqrt(z)+c*v)^2/(4*d))/sqrt(d)) *(%e^(c*sqrt(z)+g))^v*%e^(d*z-(d*sqrt(z)+c*v)*sqrt(z)-c^2*v^2/(4*d)) /sqrt(d))); 0; /* ----- %e^(d*z+e)*(%e^(c*z^r+g))^v ---------------------------------------- */ /* Problem 347: OK */ integrate(%e^(d*z+e)*(%e^(g+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(g*v-d^2/(4*c*v)+e)*erf(sqrt(-c*v)*z-d/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 348: NEW 9-1 */ expand(integrate(%e^(d*z+e)*(%e^(g+c*sqrt(z)))^v,z)- (-(-gamma_incomplete(1/2,-(2*d*sqrt(z)+c*v)^2/(4*d))*c*v*(2*d*sqrt(z)+c*v) /(2*d^(3/2)*sqrt(-(2*d*sqrt(z)+c*v)^2/d)) -gamma_incomplete(1,-(2*d*sqrt(z)+c*v)^2/(4*d))/sqrt(d)) *(%e^(c*sqrt(z)+g))^v*%e^(d*z-(d*sqrt(z)+c*v)*sqrt(z)-c^2*v^2/(4*d)+e) /sqrt(d))); 0; /* ----- %e^(b*z^r)*(%e^(c*z^r+g))^v ---------------------------------------- */ /* Problem 349: NEW 7-1 * The result is not correct simplified. */ diffcheck(integrate(%e^(b*z^r)*(%e^(g+c*z^r))^v,z),z,radcan), domain:complex; (%e^(c*z^r))^v*%e^(b*z^r+g*v); /* Problem 350: OK */ integrate(%e^(b*z^2)*(%e^(g+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(g*v)*erf(sqrt(-c*v-b)*z)/(2*sqrt(-c*v-b)); /* Problem 351: OK */ multthru(integrate(%e^(b*sqrt(z))*(%e^(g+c*sqrt(z)))^v,z)), domain:real; 2*(c*v*%e^(g*v)+b*%e^(g*v))*sqrt(z)*%e^(c*v*sqrt(z)+b*sqrt(z)) /(c^2*v^2+2*b*c*v+b^2) -2*%e^(c*v*sqrt(z)+b*sqrt(z)+g*v)/(c^2*v^2+2*b*c*v+b^2); /* ----- %e^(b*z^r+e)*(%e^(c*z^r+g))^v -------------------------------------- */ /* Problem 352: NEW 7-1 */ diffcheck(integrate(%e^(e+b*z^r)*(%e^(g+c*z^r))^v,z),z,factor); %e^(e+b*z^r)*(%e^(g+c*z^r))^v; /* Problem 353: OK */ integrate(%e^(e+b*z^2)*(%e^(g+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(g*v+e)*erf(sqrt(-c*v-b)*z)/(2*sqrt(-c*v-b)); /* Problem 354: OK */ multthru(integrate(%e^(e+b*sqrt(z))*(%e^(g+c*sqrt(z)))^v,z)), domain:real; 2*(c*v*%e^(g*v+e)+b*%e^(g*v+e))*sqrt(z)*%e^(c*v*sqrt(z)+b*sqrt(z)) /(c^2*v^2+2*b*c*v+b^2) -2*%e^(c*v*sqrt(z)+b*sqrt(z)+g*v+e)/(c^2*v^2+2*b*c*v+b^2); /* ----- %e^(b*z^r+d*z)*(%e^(c*z^r+g))^v ------------------------------------ */ /* Problem 355: OK */ integrate(%e^(d*z+b*z^2)*(%e^(g+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(d^2/(4*(-c*v-b))+g*v) *erf(sqrt(-c*v-b)*z-d/(2*sqrt(-c*v-b))) /(2*sqrt(-c*v-b)); /* Problem 356: NEW 9-1 * Wrong simplified. */ diffcheck(integrate(%e^(d*z+b*sqrt(z))*(%e^(g+c*sqrt(z)))^v,z),z,radcan), domain:complex; (%e^(c*sqrt(z)))^v*%e^(d*z+b*sqrt(z)+g*v); /* ----- %e^(b*z^r+d*z+e)*(%e^(c*z^r+g))^v ---------------------------------- */ /* Problem 357: OK */ integrate(%e^(e+d*z+b*z^2)*(%e^(g+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(d^2/(4*(-c*v-b))+g*v+e) *erf(sqrt(-c*v-b)*z-d/(2*sqrt(-c*v-b))) /(2*sqrt(-c*v-b)); /* Problem 358: NEW 9-1 */ diffcheck(integrate(%e^(e+d*z+b*sqrt(z))*(%e^(g+c*sqrt(z)))^v,z),z,radcan), domain:complex; (%e^(c*sqrt(z)))^v*%e^(d*z+b*sqrt(z)+g*v+e); /* ----- %e^(d*z)*(%e^(c*z^r+f*z))^v ---------------------------------------- */ /* Problem 359: OK */ integrate(%e^(d*z)*(%e^(f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^-((-f*v-d)^2/(4*c*v)) *erf(sqrt(-c*v)*z+(-f*v-d)/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 360: NEW 8-1 */ diffcheck(integrate(%e^(d*z)*(%e^(f*z+c*sqrt(z)))^v,z),z,factor); %e^(d*z)*(%e^(f*z+c*sqrt(z)))^v$ /* ----- %e^(d*z+e)*(%e^(c*z^r+f*z))^v -------------------------------------- */ /* Problem 361: OK */ integrate(%e^(d*z+e)*(%e^(f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(e-(-f*v-d)^2/(4*c*v)) *erf(sqrt(-c*v)*z+(-f*v-d)/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 362: NEW 8-1 */ integrate(%e^(d*z+e)*(%e^(f*z+c*sqrt(z)))^v,z); (2*sqrt(f*v+d)*%e^((2*f*v*sqrt(z)+2*d*sqrt(z)+c*v)^2/(4*(f*v+d))) -sqrt(%pi)*c*v*erfi((2*f*v*sqrt(z)+2*d*sqrt(z)+c*v)/(2*sqrt(f*v+d)))) *%e^(d*z-(2*f*v*sqrt(z)+2*d*sqrt(z)+c*v)^2/(4*(f*v+d))+e) *(%e^(f*z+c*sqrt(z)))^v /(2*(f*v+d)^(3/2)); /* ----- %e^(b*z^r)*(%e^(c*z^r+f*z))^v -------------------------------------- */ /* Problem 363: OK */ integrate(%e^(b*z^2)*(%e^(f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(f^2*v^2/(4*(-c*v-b))) *erf(sqrt(-c*v-b)*z-f*v/(2*sqrt(-c*v-b))) /(2*sqrt(-c*v-b)); /* Problem 364: NEW 9-1 */ diffcheck(integrate(%e^(b*sqrt(z))*(%e^(f*z+c*sqrt(z)))^v,z),z,factor),domain:real; %e^(f*v*z+c*v*sqrt(z)+b*sqrt(z))$ /* ----- %e^(b*z^r+e)*(%e^(c*z^r+f*z))^v ------------------------------------ */ /* Problem 365: OK */ integrate(%e^(e+b*z^2)*(%e^(f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(f^2*v^2/(4*(-c*v-b))+e) *erf(sqrt(-c*v-b)*z-f*v/(2*sqrt(-c*v-b))) /(2*sqrt(-c*v-b)); /* Problem 366: NEW 9-1 */ diffcheck(integrate(%e^(e+b*sqrt(z))*(%e^(f*z+c*sqrt(z)))^v,z),z,factor), domain:real; %e^(f*v*z+c*v*sqrt(z)+b*sqrt(z)+e)$ /* ----- %e^(b*z^r+d*z)*(%e^(c*z^r+f*z))^v ---------------------------------- */ /* Problem 367: OK */ integrate(%e^(d*z+b*z^2)*(%e^(f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^((-f*v-d)^2/(4*(-c*v-b))) *erf(sqrt(-c*v-b)*z+(-f*v-d)/(2*sqrt(-c*v-b))) /(2*sqrt(-c*v-b)); /* Problem 368: NEW 9-1 */ diffcheck(integrate(%e^(d*x+b*sqrt(z))*(%e^(f*z+c*sqrt(z)))^v,z),z,factor), domain:real; %e^(f*v*z+c*v*sqrt(z)+b*sqrt(z)+d*x)$ /* ----- %e^(b*z^r+d*z+e)*(%e^(c*z^r+f*z))^v -------------------------------- */ /* Problem 369: OK */ integrate(%e^(e+d*z+b*z^2)*(%e^(f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^((-f*v-d)^2/(4*(-c*v-b))+e) *erf(sqrt(-c*v-b)*z+(-f*v-d)/(2*sqrt(-c*v-b))) /(2*sqrt(-c*v-b)); /* Problem 370: NEW 9-1 */ diffcheck(integrate(%e^(e+d*z+b*sqrt(z))*(%e^(f*z+c*sqrt(z)))^v,z),z,radcan), domain:complex; (%e^(c*sqrt(z)))^v*(%e^(f*z))^v*%e^(d*z+b*sqrt(z)+e); /* ----- %e^(d*z)*(%e^(c*z^r+f*z+g))^v -------------------------------------- */ /* Problem 371: OK */ integrate(%e^(d*z)*(%e^(g+f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(g*v-(-f*v-d)^2/(4*c*v)) *erf(sqrt(-c*v)*z+(-f*v-d)/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 372: NEW 8-1 */ integrate(%e^(d*z)*(%e^(g+f*z+c*sqrt(z)))^v,z); (2*sqrt(f*v+d)*%e^((2*f*v*sqrt(z)+2*d*sqrt(z)+c*v)^2/(4*(f*v+d))) -sqrt(%pi)*c*v*erfi((2*f*v*sqrt(z)+2*d*sqrt(z)+c*v)/(2*sqrt(f*v+d)))) *%e^(d*z-(2*f*v*sqrt(z)+2*d*sqrt(z)+c*v)^2/(4*(f*v+d))) *(%e^(f*z+c*sqrt(z)+g))^v /(2*(f*v+d)^(3/2)); /* ----- %e^(d*z+e)*(%e^(c*z^r+f*z+g))^v ------------------------------------ */ /* Problem 373: OK */ integrate(%e^(d*z+e)*(%e^(g+f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(-(-f*v-d)^2/(4*c*v)+g*v+e) *erf(sqrt(-c*v)*z+(-f*v-d)/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 374: NEW 8-1 */ integrate(%e^(d*z+e)*(%e^(g+f*z+c*sqrt(z)))^v,z); (2*sqrt(f*v+d)*%e^((2*f*v*sqrt(z)+2*d*sqrt(z)+c*v)^2/(4*(f*v+d))) -sqrt(%pi)*c*v*erfi((2*f*v*sqrt(z)+2*d*sqrt(z)+c*v)/(2*sqrt(f*v+d)))) *%e^(d*z-(2*f*v*sqrt(z)+2*d*sqrt(z)+c*v)^2/(4*(f*v+d))+e) *(%e^(f*z+c*sqrt(z)+g))^v /(2*(f*v+d)^(3/2)); /* ----- %e^(b*z^r)*(%e^(c*z^r+f*z+g))^v ------------------------------------ */ /* Problem 375: OK */ integrate(%e^(b*z^2)*(%e^(g+f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(f^2*v^2/(4*(-c*v-b))+g*v) *erf(sqrt(-c*v-b)*z-f*v/(2*sqrt(-c*v-b))) /(2*sqrt(-c*v-b)); /* Problem 376: NEW 9-1 */ diffcheck(integrate(%e^(b*sqrt(z))*(%e^(g+f*z+c*sqrt(z)))^v,z),z,radcan), domain:complex; (%e^(c*sqrt(z)))^v*%e^(b*sqrt(z)+g*v)*(%e^(f*z))^v; /* ----- %e^(b*z^r+e)*(%e^(c*z^r+f*z+g))^v ---------------------------------- */ /* Problem 377: OK */ integrate(%e^(e+b*z^2)*(%e^(g+f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(f^2*v^2/(4*(-c*v-b))+g*v+e) *erf(sqrt(-c*v-b)*z-f*v/(2*sqrt(-c*v-b))) /(2*sqrt(-c*v-b)); /* Problem 378: NEW 9-1 */ diffcheck(integrate(%e^(e+b*sqrt(z))*(%e^(g+f*z+c*sqrt(z)))^v,z),z,radcan), domain:complex; (%e^(c*sqrt(z)))^v*%e^(b*sqrt(z)+g*v+e)*(%e^(f*z))^v; /* ----- %e^(b*z^r+d*z)*(%e^(c*z^r+f*z+g))^v -------------------------------- */ /* Problem 379: OK */ integrate(%e^(d*z+b*z^2)*(%e^(g+f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^((-f*v-d)^2/(4*(-c*v-b))+g*v) *erf(sqrt(-c*v-b)*z+(-f*v-d)/(2*sqrt(-c*v-b))) /(2*sqrt(-c*v-b)); /* Problem 380: NEW 9-1 */ diffcheck(integrate(%e^(d*z+b*sqrt(z))*(%e^(g+f*z+c*sqrt(z)))^v,z),z,radcan), domain:complex; (%e^(c*sqrt(z)))^v*(%e^(f*z))^v*%e^(d*z+b*sqrt(z)+g*v); /* ----- %e^(b*z^r+d*z+e)*(%e^(c*z^r+f*z+g))^v ------------------------------ */ /* Problem 381: OK */ integrate(%e^(e+d*z+b*z^2)*(%e^(g+f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^((-f*v-d)^2/(4*(-c*v-b))+g*v+e) *erf(sqrt(-c*v-b)*z+(-f*v-d)/(2*sqrt(-c*v-b))) /(2*sqrt(-c*v-b)); /* Problem 382: NEW 9-1 */ diffcheck(integrate(%e^(e+d*z+b*sqrt(z))*(%e^(g+f*z+c*sqrt(z)))^v,z),z,radcan), domain:complex; (%e^(c*sqrt(z)))^v*(%e^(f*z))^v*%e^(d*z+b*sqrt(z)+g*v+e); /* ----- (%e^(c*z))^u*(%e^(a*z)^v ------------------------------------------- */ /* Problem 383: OK */ integrate((%e^(c*x))^u*(%e^(a*x))^v,x); %e^(a*v*x+c*u*x)/(a*v+c*u)$ /* ----- (%e^(c*z))^u*(%e^(a*z+b)^v ----------------------------------------- */ /* Problem 384: OK */ integrate((%e^(c*x))^u*(%e^(a*x+b))^v,x); %e^(v*(a*x+b)+c*u*x)/(a*v+c*u)$ /* ----- (%e^(c*z+d))^u*(%e^(a*z+b)^v --------------------------------------- */ /* Problem 385: OK */ integrate((%e^(c*x+d))^u*(%e^(a*x+b))^v,x); %e^(u*(c*x+d)+v*(a*x+b))/(a*v+c*u)$ /* ----- (%e^(b*z))^u*(%e^(c*z^r)^v ----------------------------------------- */ /* Problem 386: OK */ integrate((%e^(b*z))^u*(%e^(c*z^2))^v,z), domain:real; sqrt(%pi)*%e^-(b^2*u^2/(4*c*v))*erf(sqrt(-c*v)*z-b*u/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 387: NEW 9-1 */ diffcheck(integrate((%e^(b*z))^u*(%e^(c*sqrt(z)))^v,z),z,factor), domain:real; %e^(b*u*z+c*v*sqrt(z))$ /* ----- (%e^(d*z+e))^u*(%e^(c*z^r)^v --------------------------------------- */ /* Problem 388: OK */ integrate((%e^(d*z+e))^u*(%e^(c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(e*u-d^2*u^2/(4*c*v)) *erf(sqrt(-c*v)*z-d*u/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 389: NEW 9-1 */ diffcheck(integrate((%e^(d*z+e))^u*(%e^(c*sqrt(z)))^v,z),z,factor), domain:real; %e^(d*u*z+c*v*sqrt(z)+e*u)$ /* ----- (%e^(b*z^r))^u*(%e^(c*z^r+g)^v ------------------------------------- */ /* Problem 390: NEW 7-1 */ diffcheck(integrate((%e^(b*z^r))^u*(%e^(c*z^r))^v,z),z,factor); (%e^(b*z^r))^u*(%e^(c*z^r))^v; /* Problem 391: OK */ integrate((%e^(b*z^2))^u*(%e^(c*z^2))^v,z), domain:real; sqrt(%pi)*erf(sqrt(-c*v-b*u)*z)/(2*sqrt(-c*v-b*u)); /* Problem 392: OK */ integrate((%e^(b*sqrt(z)))^u*(%e^(c*sqrt(z)))^v,z), domain:real; 2*((c*v+b*u)*sqrt(z)-1)*%e^(c*v*sqrt(z)+b*u*sqrt(z)) /(c^2*v^2+2*b*c*u*v+b^2*u^2); /* ----- (%e^(d*z))^u*(%e^(c*z^r+g)^v --------------------------------------- */ /* Problem 393: OK */ integrate((%e^(d*z))^u*(%e^(g+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(g*v-d^2*u^2/(4*c*v)) *erf(sqrt(-c*v)*z-d*u/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 394: NEW 9-1 */ diffcheck(integrate((%e^(d*z))^u*(%e^(g+c*sqrt(z)))^v,z),z,factor), domain:real; %e^(d*u*z+c*v*sqrt(z)+g*v)$ /* ----- (%e^(d*z+e))^u*(%e^(c*z^r+g)^v ------------------------------------- */ /* Problem 395: OK */ integrate((%e^(d*z+e))^u*(%e^(g+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(g*v-d^2*u^2/(4*c*v)+e*u) *erf(sqrt(-c*v)*z-d*u/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 396: NEW 9-1 */ diffcheck(integrate((%e^(d*z+e))^u*(%e^(g+c*sqrt(z)))^v,z),z,factor), domain:real; %e^(d*u*z+c*v*sqrt(z)+g*v+e*u)$ /* ----- (%e^(b*z^r))^u*(%e^(c*z^r+g)^v ------------------------------------- */ /* Problem 397: NEW 7-1 */ diffcheck(integrate((%e^(b*z^r))^u*(%e^(g+c*z^r))^v,z),z,factor); (%e^(b*z^r))^u*(%e^(g+c*z^r))^v; /* Problem 398: OK */ integrate((%e^(b*z^2))^u*(%e^(g+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(g*v)*erf(sqrt(-c*v-b*u)*z)/(2*sqrt(-c*v-b*u)); /* Problem 399: OK */ integrate((%e^(b*sqrt(z)))^u*(%e^(g+c*sqrt(z)))^v,z), domain:real; 2*((c*v*%e^(g*v)+b*u*%e^(g*v))*sqrt(z)-%e^(g*v))*%e^(c*v*sqrt(z)+b*u*sqrt(z)) /(c^2*v^2+2*b*c*u*v+b^2*u^2); /* ----- (%e^(b*z^r+e))^u*(%e^(c*z^r+g)^v ----------------------------------- */ /* Problem 400: NEW 7-1 */ diffcheck(integrate((%e^(e+b*z^r))^u*(%e^(g+c*z^r))^v,z),z,factor); (%e^(e+b*z^r))^u*(%e^(g+c*z^r))^v; /* Problem 401: OK */ integrate((%e^(e+b*z^2))^u*(%e^(g+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(g*v+e*u)*erf(sqrt(-c*v-b*u)*z)/(2*sqrt(-c*v-b*u)); /* Problem 402: OK */ integrate((%e^(e+b*sqrt(z)))^u*(%e^(g+c*sqrt(z)))^v,z), domain:real; 2*((c*v*%e^(g*v+e*u)+b*u*%e^(g*v+e*u))*sqrt(z)-%e^(g*v+e*u)) *%e^(c*v*sqrt(z)+b*u*sqrt(z)) /(c^2*v^2+2*b*c*u*v+b^2*u^2); /* ----- (%e^(d*z))^u*(%e^(c*z^r+f*z)^v ------------------------------------- */ /* Problem 403: OK */ integrate((%e^(d*z))^u*(%e^(f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^-((-f*v-d*u)^2/(4*c*v)) *erf(sqrt(-c*v)*z+(-f*v-d*u)/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 404: NEW 8-1 */ integrate((%e^(d*z))^u*(%e^(f*z+c*sqrt(z)))^v,z); %e^-((2*f*v*sqrt(z)+2*d*u*sqrt(z)+c*v)^2/(4*(f*v+d*u))) *(2*sqrt(f*v+d*u)*%e^((2*f*v*sqrt(z)+2*d*u*sqrt(z)+c*v)^2/(4*(f*v+d*u))) -sqrt(%pi)*c*v*erfi((2*f*v*sqrt(z)+2*d*u*sqrt(z)+c*v)/(2*sqrt(f*v+d*u)))) *(%e^(d*z))^u*(%e^(f*z+c*sqrt(z)))^v /(2*(f*v+d*u)^(3/2)); /* ----- (%e^(d*z+e))^u*(%e^(c*z^r+f*z)^v ----------------------------------- */ /* Problem 405: OK */ integrate((%e^(e+d*z))^u*(%e^(f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(e*u-(-f*v-d*u)^2/(4*c*v)) *erf(sqrt(-c*v)*z+(-f*v-d*u)/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 406: NEW 8-1 */ diffcheck(integrate((%e^(e+d*z))^u*(%e^(f*z+c*sqrt(z)))^v,z),z,factor); (%e^(e+d*z))^u*(%e^(f*z+c*sqrt(z)))^v; /* ----- (%e^(b*z^r))^u*(%e^(c*z^r+f*z)^v ----------------------------------- */ /* Problem 407: OK */ integrate((%e^(b*z^2))^u*(%e^(f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(f^2*v^2/(4*(-c*v-b*u))) *erf(sqrt(-c*v-b*u)*z-f*v/(2*sqrt(-c*v-b*u))) /(2*sqrt(-c*v-b*u)); /* Problem 408: NEW 9-1 */ diffcheck(integrate((%e^(b*sqrt(z)))^u*(%e^(f*z+c*sqrt(z)))^v,z),z,factor), domain:real; %e^(f*v*z+c*v*sqrt(z)+b*u*sqrt(z))$ /* ----- (%e^(b*z^r+e))^u*(%e^(c*z^r+f*z)^v --------------------------------- */ /* Problem 409: OK */ integrate((%e^(e+b*z^2))^u*(%e^(f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(f^2*v^2/(4*(-c*v-b*u))+e*u) *erf(sqrt(-c*v-b*u)*z-f*v/(2*sqrt(-c*v-b*u))) /(2*sqrt(-c*v-b*u)); /* Problem 410: NEW 9-1 */ diffcheck(integrate((%e^(e+b*sqrt(z)))^u*(%e^(f*z+c*sqrt(z)))^v,z),z,factor), domain:real; %e^(f*v*z+c*v*sqrt(z)+b*u*sqrt(z)+e*u)$ /* ----- (%e^(b*z^r+d*z))^u*(%e^(c*z^r+f*z)^v ------------------------------- */ /* Problem 411: OK */ integrate((%e^(d*z+b*z^2))^u*(%e^(f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^((-f*v-d*u)^2/(4*(-c*v-b*u))) *erf(sqrt(-c*v-b*u)*z+(-f*v-d*u)/(2*sqrt(-c*v-b*u))) /(2*sqrt(-c*v-b*u)); /* Problem 412: NEW 9-1 */ diffcheck(integrate((%e^(d*z+b*sqrt(z)))^u*(%e^(f*z+c*sqrt(z)))^v,z),z,radcan), domain:complex; (%e^(b*sqrt(z)))^u*(%e^(c*sqrt(z)))^v*(%e^(d*z))^u*(%e^(f*z))^v$ /* ----- (%e^(d*z))^u*(%e^(c*z^r+f*z+g)^v ----------------------------------- */ /* Problem 413: OK */ integrate((%e^(d*z))^u*(%e^(g+f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(g*v-(-f*v-d*u)^2/(4*c*v)) *erf(sqrt(-c*v)*z+(-f*v-d*u)/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 414: NEW 8-1 */ integrate((%e^(d*z))^u*(%e^(g+f*z+c*sqrt(z)))^v,z); %e^-((2*f*v*sqrt(z)+2*d*u*sqrt(z)+c*v)^2/(4*(f*v+d*u))) *(2*sqrt(f*v+d*u)*%e^((2*f*v*sqrt(z)+2*d*u*sqrt(z)+c*v)^2/(4*(f*v+d*u))) -sqrt(%pi)*c*v*erfi((2*f*v*sqrt(z)+2*d*u*sqrt(z)+c*v)/(2*sqrt(f*v+d*u)))) *(%e^(d*z))^u*(%e^(f*z+c*sqrt(z)+g))^v /(2*(f*v+d*u)^(3/2)); /* ----- (%e^(d*z+e))^u*(%e^(c*z^r+f*z+g)^v --------------------------------- */ /* Problem 415: OK */ integrate((%e^(e+d*z))^u*(%e^(g+f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(-(-f*v-d*u)^2/(4*c*v)+g*v+e*u) *erf(sqrt(-c*v)*z+(-f*v-d*u)/(2*sqrt(-c*v))) /(2*sqrt(-c*v)); /* Problem 416: NEW 8-1 */ integrate((%e^(e+d*z))^u*(%e^(g+f*z+c*sqrt(z)))^v,z); %e^-((2*f*v*sqrt(z)+2*d*u*sqrt(z)+c*v)^2/(4*(f*v+d*u))) *(2*sqrt(f*v+d*u)*%e^((2*f*v*sqrt(z)+2*d*u*sqrt(z)+c*v)^2/(4*(f*v+d*u))) -sqrt(%pi)*c*v*erfi((2*f*v*sqrt(z)+2*d*u*sqrt(z)+c*v)/(2*sqrt(f*v+d*u)))) *(%e^(d*z+e))^u*(%e^(f*z+c*sqrt(z)+g))^v /(2*(f*v+d*u)^(3/2)); /* ----- (%e^(b*z^r))^u*(%e^(c*z^r+f*z+g)^v --------------------------------- */ /* Problem 417: OK */ integrate((%e^(b*z^2))^u*(%e^(g+f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(f^2*v^2/(4*(-c*v-b*u))+g*v) *erf(sqrt(-c*v-b*u)*z-f*v/(2*sqrt(-c*v-b*u))) /(2*sqrt(-c*v-b*u)); /* Problem 418: NEW 9-1 */ diffcheck(integrate((%e^(b*sqrt(z)))^u*(%e^(g+f*z+c*sqrt(z)))^v,z),z,factor),domain:real; %e^(f*v*z+c*v*sqrt(z)+b*u*sqrt(z)+g*v)$ /* ----- (%e^(b*z^r+e))^u*(%e^(c*z^r+f*z+g)^v ------------------------------- */ /* Problem 419: OK */ integrate((%e^(e+b*z^2))^u*(%e^(g+f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^(f^2*v^2/(4*(-c*v-b*u))+g*v+e*u) *erf(sqrt(-c*v-b*u)*z-f*v/(2*sqrt(-c*v-b*u))) /(2*sqrt(-c*v-b*u)); /* Problem 420: NEW 9-1 */ diffcheck(integrate((%e^(e+b*sqrt(z)))^u*(%e^(g+f*z+c*sqrt(z)))^v,z),z,factor),domain:real; %e^(f*v*z+c*v*sqrt(z)+b*u*sqrt(z)+g*v+e*u)$ /* ----- (%e^(b*z^r+d*z))^u*(%e^(c*z^r+f*z+g)^v ----------------------------- */ /* Problem 421: OK */ integrate((%e^(d*z+b*z^2))^u*(%e^(g+f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^((-f*v-d*u)^2/(4*(-c*v-b*u))+g*v) *erf(sqrt(-c*v-b*u)*z+(-f*v-d*u)/(2*sqrt(-c*v-b*u))) /(2*sqrt(-c*v-b*u)); /* Problem 422: NEW 9-1 */ diffcheck(integrate((%e^(d*z+b*sqrt(z)))^u*(%e^(g+f*z+c*sqrt(z)))^v,z),z,radcan), domain:complex; %e^(g*v)*(%e^(b*sqrt(z)))^u*(%e^(c*sqrt(z)))^v*(%e^(d*z))^u*(%e^(f*z))^v; /* ----- (%e^(b*z^r+d*z+e))^u*(%e^(c*z^r+f*z+g)^v --------------------------- */ /* Problem 423: OK */ integrate((%e^(e+d*z+b*z^2))^u*(%e^(g+f*z+c*z^2))^v,z), domain:real; sqrt(%pi)*%e^((-f*v-d*u)^2/(4*(-c*v-b*u))+g*v+e*u) *erf(sqrt(-c*v-b*u)*z+(-f*v-d*u)/(2*sqrt(-c*v-b*u))) /(2*sqrt(-c*v-b*u)); /* Problem 424: NEW 9-1 */ diffcheck(integrate((%e^(e+d*z+b*sqrt(z)))^u*(%e^(g+f*z+c*sqrt(z)))^v,z),z,radcan), domain:complex; %e^(g*v+e*u)*(%e^(b*sqrt(z)))^u*(%e^(c*sqrt(z)))^v*(%e^(d*z))^u*(%e^(f*z))^v; /* ----- Involving rational functions of the direct function ---------------- */ /* ----- Involving 1/(a+b*%e^(c*z)) ----------------------------------------- */ /* Problem 425: OK */ integrate(1/(b*%e^(c*z)+a),z), domain:real; z/a-log(b*%e^(c*z)+a)/(a*c); /* Problem 426: OK */ integrate(1/(%e^(c*z)+1),z), domain:real; z-log(%e^(c*z)+1)/c; /* Problem 427: OK */ integrate(1/(1-%e^(c*z)),z), domain:real; z-log(%e^(c*z)-1)/c; /* ----- Involving (a+b*%e^(c*z))^(-n)---------------------------------------- */ /* Problem 428: OK */ integrate(1/(b*%e^(c*z)+a)^2,z), domain:real; -log(b*%e^(c*z)+a)/(a^2*c)+1/(c*(a*b*%e^(c*z)+a^2))+z/a^2; /* Problem 429: OK */ integrate(1/(%e^(c*z)+1)^2,z), domain:real; -log(%e^(c*z)+1)/c+1/(c*(%e^(c*z)+1))+z; /* Problem 430: OK */ integrate(1/(1-%e^(c*z))^2,z), domain:real; -log(%e^(c*z)-1)/c-1/(c*(%e^(c*z)-1))+z; /* Problem 431: OK */ integrate(1/(b*%e^(c*z)+a)^3,z), domain:real; -log(b*%e^(c*z)+a)/(a^3*c)+(2*b*%e^(c*z)+3*a) /(c*(2*a^2*b^2*%e^(2*c*z)+4*a^3*b*%e^(c*z)+2*a^4)) +z/a^3; /* Problem 432: OK */ integrate(1/(b*%e^(c*z)+a)^4,z), domain:real; -log(b*%e^(c*z)+a)/(a^4*c)+(6*b^2*%e^(2*c*z)+15*a*b*%e^(c*z)+11*a^2) /(c*(6*a^3*b^3*%e^(3*c*z)+18*a^4*b^2*%e^(2*c*z) +18*a^5*b*%e^(c*z)+6*a^6)) +z/a^4; /* ----- Involving %e^(d*z)/(a+b*%e^(c*z)) ---------------------------------- */ /* Problem 433: NOUN */ integrate(%e^(d*z)/(b*%e^(c*z)+a),z); a*c *'integrate(%e^(d*z)/(b^2*(c-d)*%e^(2*c*z)+a*b*(2*c-2*d)*%e^(c*z)+a^2*(c-d)), z) -%e^(d*z)/(b*(c-d)*%e^(c*z)+a*(c-d)); /* Problem 435: OK */ assume(a>0,b>0); [a>0,b>0]; integrate(%e^z/(a+b*%e^(2*z)),z), domain:real; atan(sqrt(b)*%e^z/sqrt(a))/(sqrt(a)*sqrt(b)); forget(a>0,b>0); [a>0,b>0]; /* Problem 437: OK */ integrate(%e^(c*z)/(b*%e^(c*z)+a),z); log(b*%e^(c*z)+a)/(b*c); /* Problem 438: OK */ integrate((B*%e^(c*z)+A)/(b*%e^(c*z)+a),z), domain:real; (A*z)/a-((A*b-B*a)*log(b*%e^(c*z)+a))/(a*b*c); /* Problem 439: OK */ integrate(%e^(2*c*z)/(b*%e^(c*z)+a),z); %e^(c*z)/(b*c)-a*log(b*%e^(c*z)+a)/(b^2*c); /* Problem 441: OK */ assume(a>0,b>0); [a>0,b>0]; integrate(%e^(c*z)/(b*%e^(2*c*z)+a),z); atan(b*%e^(c*z)/sqrt(a*b))/(sqrt(a*b)*c); forget(a>0,b>0); [a>0,b>0]; /* Problem 444: OK */ assume(a>0,b>0); [a>0,b>0]; integrate(%e^(c*z)/(b*%e^(3*c*z)+a),z), domain:real; -log(b^(2/3)*%e^(2*c*z)-a^(1/3)*b^(1/3)*%e^(c*z)+a^(2/3)) /(6*a^(2/3)*b^(1/3)*c) +atan((2*b^(2/3)*%e^(c*z)-a^(1/3)*b^(1/3))/(sqrt(3)*a^(1/3)*b^(1/3))) /(sqrt(3)*a^(2/3)*b^(1/3)*c) +log((b^(1/3)*%e^(c*z)+a^(1/3))/b^(1/3))/(3*a^(2/3)*b^(1/3)*c); forget(a>0,b>0); [a>0,b>0]; /* ----- Involving %e^(f*z)/(a+b*%e^(c*z))^-n ------------------------------- */ /* Problem 446: NOUN */ integrate(%e^(f*z)/(b*c^(d*z)+a),z); 'integrate(%e^(f*z)/(b*c^(d*z)+a),z); /* a*log(c)*d *'integrate(%e^(f*z)/(b^2*(log(c)*d-f)*%e^(2*log(c)*d*z) +a*b*(2*log(c)*d-2*f)*%e^(log(c)*d*z)+a^2*(log(c)*d-f)), z) -%e^(f*z)/(b*(log(c)*d-f)*%e^(log(c)*d*z)+a*(log(c)*d-f)); */ /* Problem 447: NOUN */ integrate(%e^(d*z)/(b*%e^(c*z)+a)^2,z); 2*a*c *'integrate(%e^(d*z)/(b^3*(2*c-d)*%e^(3*c*z)+a*b^2*(6*c-3*d)*%e^(2*c*z) +a^2*b*(6*c-3*d)*%e^(c*z) +a^3*(2*c-d)),z) -%e^(d*z)/(b^2*(2*c-d)*%e^(2*c*z)+a*b*(4*c-2*d)*%e^(c*z)+a^2*(2*c-d)); /* Problem 448: OK */ integrate((B*%e^(c*z)+A)/(b*%e^(c*z)+a)^2,z), domain:real; (-(A*log(b*%e^(c*z)+a))/(a^2*c))+(A*b-B*a)/(c*(a^2*b+a*b^2*%e^(c*z))) +(A*z)/a^2; /* Problem 449: OK */ integrate((B*%e^(c*z)+A)/(b*%e^(c*z)+a)^3,z), domain:real; (-a^2*B+2*b^2*%e^(c*z)*A+3*a*b*A)/(c*(2*a^2*b^3*%e^(2*c*z) +4*a^3*b^2*%e^(c*z)+2*a^4*b)) -log(b*%e^(c*z)+a)*A/(a^3*c)+z*A/a^3$ /* Problem 450: OK */ integrate((C*%e^(2*c*z)+B*%e^(c*z)+A)/(b*%e^(c*z)+a)^3,z), domain:real; (-(A*log(b*%e^(c*z)+a))/(a^3*c))+((2*A*b^3-2*C*a^2*b)*%e^(c*z) +3*A*a*b^2-B*a^2*b-C*a^3) /(c*(2*a^4*b^2+4*a^3*b^3*%e^(c*z) +2*a^2*b^4*%e^(2*c*z))) +(A*z)/a^3; /* ----- Involving 1/(a*%e^(2*d*z)+b*%e^(d*z)+c)----------------------------- */ /* Problem 452: OK */ assume(4*a*c-b^2>0); [4*a*c > b^2]; integrate(1/(c+b*%e^(d*z)+a*%e^(2*d*z)),z), domain:real; -log(a*%e^(2*d*z)+b*%e^(d*z)+c)/(2*c*d)-b *atan( (2*a*%e^(d*z)+b)/sqrt(4*a*c-b^2)) /(c*sqrt(4*a*c-b^2)*d)+z/c$ forget(4*a*c-b^2>0); [4*a*c > b^2]; /* ----- Involving %e^(e*z)/(a*%e^(2*d*z)+b*%e^(d*z)+c)---------------------- */ /* Problem 454: NOUN */ integrate(%e^(e*z)/(c+b*%e^(d*z)+a*%e^(2*d*z)),z); 'integrate((b*d*%e^(d*z)+2*c*d)*%e^(e*z) /(a^2*(2*d-e)*%e^(4*d*z)+a*b*(4*d-2*e)*%e^(3*d*z) +(b^2*(2*d-e)+a*c*(4*d-2*e))*%e^(2*d*z) +b*c*(4*d-2*e)*%e^(d*z)+c^2*(2*d-e)),z) -%e^(e*z)/(a*(2*d-e)*%e^(2*d*z)+b*(2*d-e)*%e^(d*z)+c*(2*d-e)); /* Problem 456: OK */ assume(4*a*c-b^2>0); [4*a*c > b^2]; integrate(%e^(2*d*z)/(c+b*%e^(d*z)+a*%e^(2*d*z)),z),logexpand; log(b*sqrt(%e^(2*d*z))+a*%e^(2*d*z)+c)/(2*a*d) -b*atan((2*a*sqrt(%e^(2*d*z))+b)/sqrt(4*a*c-b^2))/(a*sqrt(4*a*c-b^2)*d); forget(4*a*c-b^2>0); [4*a*c > b^2]; /* ----- Other integrals ---------------------------------------------------- */ /* Problem 458: NOUN */ integrate(6^z/(9^z*b+4^z*a),z); 'integrate(6^z/(b*9^z+a*4^z),z); /* Problem 459: OK */ integrate((%e^(5*z)+%e^z)/(%e^(3*z)-%e^(2*z)+%e^z-1),z); -log(%e^(2*z)+1)/2-atan(%e^z)+log(%e^z-1)+(%e^(2*z)+2*%e^z)/2; /* Problem 460: OK */ integrate(1/(%e^(z/2)+%e^(z/3)+%e^(z/6)+1),z), domain:real; -3*log(%e^(z/3)+1)/2-3*log(%e^(z/6)+1)-3*atan(%e^(z/6))+z$ /* ----- Involving algebraic functions of the direct function --------------- */ /* ----- Involving (a+b*c^(d*z))^v ------------------------------------------ */ /* Problem 461: NOUN */ integrate((b*c^(d*z)+a)^v,z); 'integrate((b*c^(d*z)+a)^v,z); /* Problem 463: OK */ assume(a>0); [a>0]; integrate((b*c^(d*z)+a)^(1/2),z); 2*sqrt(b*c^(d*z)+a)/(log(c)*d)+sqrt(a)*log( (2*sqrt(b*c^(d*z)+a)-2*sqrt(a)) /(2*sqrt(b*c^(d*z)+a)+2*sqrt(a))) /(log(c)*d); forget(a>0); [a>0]; /* Problem 465: NOUN */ integrate((b*%e^(c*z)+a)^v,z); 'integrate((b*%e^(c*z)+a)^v,z); /* Problem 466: NOUN */ integrate((a*%e^(c*z)+a)^v,z); 'integrate((a*%e^(c*z)+a)^v,z); /* Problem 467: NOUN */ integrate((a-a*%e^(c*z))^v,z); 'integrate((a-a*%e^(c*z))^v,z); /* This assumptions is for the problems 457 - 468 */ assume(a>0); [a>0]; /* Problem 469: OK */ integrate((b*%e^(c*z)+a)^(5/2),z); a^(5/2)*log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a))/(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a))) /c +(6*(b*%e^(c*z)+a)^(5/2)+10*a*(b*%e^(c*z)+a)^(3/2)+30*a^2*sqrt(b*%e^(c*z)+a)) /(15*c); /* Problem 470: OK */ integrate((b*%e^(c*z)+a)^(3/2),z); a^(3/2)*log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a)) /(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a))) /c +(2*(b*%e^(c*z)+a)^(3/2)+6*a*sqrt(b*%e^(c*z)+a))/(3*c); /* Problem 471: OK */ integrate((b*%e^(c*z)+a)^(1/2),z); sqrt(a)*log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a)) /(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a))) /c +2*sqrt(b*%e^(c*z)+a)/c; /* Problem 472: OK */ integrate((a*%e^(c*z)+a)^(1/2),z); sqrt(a)*log((2*sqrt(a*%e^(c*z)+a)-2*sqrt(a)) /(2*sqrt(a*%e^(c*z)+a)+2*sqrt(a))) /c +2*sqrt(a*%e^(c*z)+a)/c; /* Problem 473: OK */ integrate((a-a*%e^(c*z))^(1/2),z); sqrt(a)*log((2*sqrt(a-a*%e^(c*z))-2*sqrt(a)) /(2*sqrt(a-a*%e^(c*z))+2*sqrt(a))) /c +2*sqrt(a-a*%e^(c*z))/c; /* Problem 474: OK */ integrate((b*%e^(c*z)+a)^((-1)/2),z); log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a))/(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a))) /(sqrt(a)*c); /* Problem 475: OK */ integrate((a*%e^(c*z)+a)^((-1)/2),z); log((2*sqrt(a*%e^(c*z)+a)-2*sqrt(a))/(2*sqrt(a*%e^(c*z)+a)+2*sqrt(a))) /(sqrt(a)*c); /* Problem 476: OK */ integrate((a-a*%e^(c*z))^((-1)/2),z); log((2*sqrt(a-a*%e^(c*z))-2*sqrt(a))/(2*sqrt(a-a*%e^(c*z))+2*sqrt(a))) /(sqrt(a)*c); /* Problem 477: OK */ integrate((b*%e^(c*z)+a)^((-3)/2),z); log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a))/(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a))) /(a^(3/2)*c) +2/(a*c*sqrt(b*%e^(c*z)+a)); /* Problem 478: OK */ integrate((a*%e^(c*z)+a)^((-3)/2),z); log((2*sqrt(a*%e^(c*z)+a)-2*sqrt(a))/(2*sqrt(a*%e^(c*z)+a)+2*sqrt(a))) /(a^(3/2)*c) +2/(a*c*sqrt(a*%e^(c*z)+a)); /* Problem 479: OK */ integrate((a-a*%e^(c*z))^((-3)/2),z); log((2*sqrt(a-a*%e^(c*z))-2*sqrt(a))/(2*sqrt(a-a*%e^(c*z))+2*sqrt(a))) /(a^(3/2)*c) +2/(a*c*sqrt(a-a*%e^(c*z))); /* Problem 480: OK */ integrate((b*%e^(c*z)+a)^((-5)/2),z); log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a))/(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a))) /(a^(5/2)*c) +(6*(b*%e^(c*z)+a)+2*a)/(3*a^2*c*(b*%e^(c*z)+a)^(3/2)); /* Forget assumption for Problems 457 - 468 */ forget(a>0); [a>0]; /* ----- Involving ((a+b*%e^(c*z))^v)^p ------------------------------------- */ /* Problem 482: NOUN */ integrate(((b*%e^(c*z)+a)^v)^p,z); 'integrate(((b*%e^(c*z)+a)^v)^p,z); /* Assumption for Problems 472 - 475 */ assume(a>0); [a>0]; /* Problem 484: OK */ integrate(((b*%e^(c*z)+a)^5)^(1/2),z),radexpand:all; a^(5/2)*log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a)) /(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a))) /c +(6*(b*%e^(c*z)+a)^(5/2)+10*a*(b*%e^(c*z)+a)^(3/2) +30*a^2*sqrt(b*%e^(c*z)+a)) /(15*c); /* Problem 485: OK */ integrate(((b*%e^(c*z)+a)^3)^(1/2),z),radexpand:all; a^(3/2)*log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a)) /(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a))) /c +(2*(b*%e^(c*z)+a)^(3/2)+6*a*sqrt(b*%e^(c*z)+a))/(3*c); /* Problem 486: OK */ integrate((b*%e^(c*z)+a)^(-3/2),z); log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a))/(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a))) /(a^(3/2)*c) +2/(a*c*sqrt(b*%e^(c*z)+a)); /* Problem 487: OK */ integrate((b*%e^(c*z)+a)^(-5/2),z); log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a))/(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a))) /(a^(5/2)*c) +(6*(b*%e^(c*z)+a)+2*a)/(3*a^2*c*(b*%e^(c*z)+a)^(3/2)); /* Forget assumption for Problems 472 - 475 */ forget(a>0); [a>0]; /* ----- Involving (a+b*c*%e^(d*z))^p*%e^(f*z) ------------------------------ */ /* Problem 489: NOUN */ integrate((b*c^(d*z)+a)^p*%e^(f*z),z); 'integrate((b*c^(d*z)+a)^p*%e^(f*z),z); /* Problem 490: NOUN */ integrate((b*%e^(c*z)+a)^p*%e^(f*z),z); 'integrate((b*%e^(c*z)+a)^p*%e^(f*z),z); /* Assumption for problems 480 - 490 */ assume(notequal(p+1,0)); [notequal(p,-1)]; /* Problem 492: OK */ integrate((b*%e^(c*z)+a)^p*%e^(c*z),z); (b*%e^(c*z)+a)^(p+1)/(b*c*(p+1)); /* Problem 493: OK */ integrate((a*%e^(c*z)+a)^p*%e^(c*z),z); (a*%e^(c*z)+a)^(p+1)/(a*c*(p+1)); /* Problem 494: OK */ integrate((a-a*%e^(c*z))^p*%e^(c*z),z); -(a-a*%e^(c*z))^(p+1)/(a*c*(p+1)); /* Problem 495: OK */ integrate((b*%e^(c*z)+a)^(1/2)*%e^(c*z),z); 2*(b*%e^(c*z)+a)^(3/2)/(3*b*c); /* Problem 496: OK */ integrate((a*%e^(c*z)+a)^(1/2)*%e^(c*z),z); 2*(a*%e^(c*z)+a)^(3/2)/(3*a*c); /* Problem 497: OK */ integrate((a-a*%e^(c*z))^(1/2)*%e^(c*z),z); -2*(a-a*%e^(c*z))^(3/2)/(3*a*c); /* Problem 498: OK */ integrate((b*%e^(c*z)+a)^((-1)/2)*%e^(c*z),z); 2*sqrt(b*%e^(c*z)+a)/(b*c); /* Problem 499: OK */ integrate((a*%e^(c*z)+a)^((-1)/2)*%e^(c*z),z); 2*sqrt(a*%e^(c*z)+a)/(a*c); /* Problem 500: OK */ integrate((a-a*%e^(c*z))^((-1)/2)*%e^(c*z),z); -2*sqrt(a-a*%e^(c*z))/(a*c); /* Problem 501: OK */ integrate((b*%e^(c*z)+a)^((-3)/2)*%e^(c*z),z); -2/(b*c*sqrt(b*%e^(c*z)+a)); /* Problem 502: OK */ integrate((b*%e^(c*z)+a)^((-5)/2)*%e^(c*z),z); -2/(3*b*c*(b*%e^(c*z)+a)^(3/2)); /* Forget assumption for problems 480 - 490 */ forget(notequal(p+1,0)); [notequal(p,-1)]; /* ----- Involving ((a+b*c*%e^(c*z))^v)^p*%e^(d*z) -------------------------- */ /* Problem 504: NOUN */ integrate(((b*%e^(c*z)+a)^v)^p*%e^(d*z),z); 'integrate(((b*%e^(c*z)+a)^v)^p*%e^(d*z),z); /* Problem 505: OK */ integrate(((b*%e^(c*z)+a)^5)^(1/2)*%e^(c*z),z),domain:real; 2*(b*%e^(c*z)+a)^(7/2)/(7*b*c)$ /* Problem 506: OK */ integrate(((b*%e^(c*z)+a)^3)^(1/2)*%e^(c*z),z),domain:real; 2*(b*%e^(c*z)+a)^(5/2)/(5*b*c)$ /* Problem 507: OK */ integrate((b*%e^(c*z)+a)^(-3/2)*%e^(c*z),z); -2/(b*c*sqrt(b*%e^(c*z)+a)); /* Problem 508: OK */ integrate((a*%e^(c*z)+a)^(-3/2)*%e^(c*z),z); -2/(a*c*sqrt(a*%e^(c*z)+a)); /* Problem 509: OK */ integrate((a-a*%e^(c*z))^(-3/2)*%e^(c*z),z); 2/(a*c*sqrt(a-a*%e^(c*z))); /* Problem 510: OK */ integrate((b*%e^(c*z)+a)^(-5/2)*%e^(c*z),z); -2/(3*b*c*(b*%e^(c*z)+a)^(3/2)); /* ----- Involving (a+b*%e^(c*z))^p*(%e^(d*z))^v ---------------------------- */ /* Problem 511: NOUN */ integrate((b*%e^(c*z)+a)^p*(%e^(d*z))^v,z); 'integrate((b*%e^(c*z)+a)^p*(%e^(d*z))^v,z); /* Problem 512: NOUN */ integrate((b*%e^(c*z)+a)^p*(%e^(c*z))^v,z); 'integrate((%e^(c*z))^v*(b*%e^(c*z)+a)^p,z); /* Problem 514: OK */ assume(a>0,b>0); [a>0,b>0]; integrate((%e^(c*z))^(1/2)/(b*%e^(c*z)+a),z); 2*atan(b*sqrt(%e^(c*z))/sqrt(a*b))/(sqrt(a*b)*c); forget(a>0,b>0); [a>0,b>0]; /* ----- Involving (a+b*%e^(c*z))^p and rational function of %e^(c*z) ------- */ /* Problem 516: NOUN */ integrate(e^(q*z)*(b*%e^(c*z)+a)^p/(e*%e^(c*z)+d),z); 'integrate(e^(q*z)*(b*%e^(c*z)+a)^p/(e*%e^(c*z)+d),z); /* Problem 517: NOUN */ integrate((b*%e^(c*z)+a)^p/(e*%e^(c*z)+d),z); 'integrate((b*%e^(c*z)+a)^p/(e*%e^(c*z)+d),z); assume(a>0,e>0,(a*e-b*d)>0); [a > 0,e>0,a*e>b*d]; /* Problem 519: NEW 08-2010 */ integrate((b*%e^(c*z)+a)^(1/2)/(e*%e^(c*z)+d),z); sqrt(a)*log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a))/(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a))) /(c*d) -(a*e-b*d)*log((2*e*sqrt(b*%e^(c*z)+a)-2*sqrt(e*(a*e-b*d))) /(2*e*sqrt(b*%e^(c*z)+a)+2*sqrt(e*(a*e-b*d)))) /(c*d*sqrt(e*(a*e-b*d)))$ /* Problem 520: NEW 08-2010 */ integrate((b*%e^(c*z)+a)^((-1)/2)/(e*%e^(c*z)+d),z); log((2*sqrt(b*%e^(c*z)+a)-2*sqrt(a))/(2*sqrt(b*%e^(c*z)+a)+2*sqrt(a))) /(sqrt(a)*c*d) -e*log((2*e*sqrt(b*%e^(c*z)+a)-2*sqrt(e*(a*e-b*d))) /(2*e*sqrt(b*%e^(c*z)+a)+2*sqrt(e*(a*e-b*d)))) /(c*d*sqrt(e*(a*e-b*d)))$ /* Problem 521: NEW 08-2010 */ integrate(%e^(c*z)*(b*%e^(c*z)+a)^((-1)/2)/(e*%e^(c*z)+d),z); log((2*e*sqrt(b*%e^(c*z)+a)-2*sqrt(e*(a*e-b*d))) /(2*e*sqrt(b*%e^(c*z)+a)+2*sqrt(e*(a*e-b*d)))) /(c*sqrt(e*(a*e-b*d)))$ forget(a>0,e>0,(a*e-b*d)>0); [a > 0,e>0,a*e>b*d]; /* Problem 523: NOUN */ integrate(e^(d*z)*(b*%e^(c*z)+a)^p/(e*%e^(c*z)+d),z); 'integrate(e^(d*z)*(b*%e^(c*z)+a)^p/(e*%e^(c*z)+d),z); /* ----- Involving (a+b*%e^(2*c*z))^p*%e^(c*z) ------------------------------ */ assume(a>0,b>0); [a>0,b>0]; /* Problem 525: NEW 08-2010 */ integrate((b*%e^(2*c*z)+a)^(5/2)*%e^(c*z),z); 5*a^3*asinh(2*b*%e^(c*z)/sqrt(4*a*b))/(16*sqrt(b)*c) +%e^(c*z)*(b*%e^(2*c*z)+a)^(5/2)/(6*c) +5*a*%e^(c*z)*(b*%e^(2*c*z)+a)^(3/2)/(24*c) +5*a^2*%e^(c*z)*sqrt(b*%e^(2*c*z)+a)/(16*c)$ /* Problem 526: NEW 08-2010 */ integrate((b*%e^(2*c*z)+a)^(3/2)*%e^(c*z),z); 3*a^2*asinh(2*b*%e^(c*z)/sqrt(4*a*b))/(8*sqrt(b)*c) +%e^(c*z)*(b*%e^(2*c*z)+a)^(3/2)/(4*c) +3*a*%e^(c*z)*sqrt(b*%e^(2*c*z)+a)/(8*c)$ /* Problem 527: NEW 08-2010 */ integrate((b*%e^(2*c*z)+a)^(1/2)*%e^(c*z),z); a*asinh(2*b*%e^(c*z)/sqrt(4*a*b))/(2*sqrt(b)*c) +%e^(c*z)*sqrt(b*%e^(2*c*z)+a)/(2*c)$ /* Problem 528: NEW 08-2010 */ integrate((b*%e^(2*c*z)+a)^((-1)/2)*%e^(c*z),z); asinh(2*b*%e^(c*z)/sqrt(4*a*b))/(sqrt(b)*c)$ /* Problem 529: NEW 08-2010 */ integrate((b*%e^(2*c*z)+a)^((-3)/2)*%e^(c*z),z); %e^(c*z)/(a*c*sqrt(b*%e^(2*c*z)+a)); /* Problem 530: NEW 08-2010 */ integrate((b*%e^(2*c*z)+a)^((-5)/2)*%e^(c*z),z); 2*%e^(c*z)/(3*a^2*c*sqrt(b*%e^(2*c*z)+a)) +%e^(c*z)/(3*a*c*(b*%e^(2*c*z)+a)^(3/2)); forget(a>0,b>0); [a>0,b>0]; /* ----- Involving ((a+b*%e^(2*c*z))^v)^p*%e^(c*z) -------------------------- */ /* Problem 532: NOUN */ integrate(((b*%e^(2*c*z)+a)^v)^p*%e^(c*z),z); 'integrate(%e^(c*z)*((b*%e^(2*c*z)+a)^v)^p,z); assume(a>0,b>0); [a>0,b>0]; /* Problem 534: NEW 08-2010 */ integrate(((b*%e^(2*c*z)+a)^5)^(1/2)*%e^(c*z),z),domain:real; 5*a^3*asinh(sqrt(b)*%e^(c*z)/sqrt(a))/(16*sqrt(b)*c) +%e^(c*z)*(b*%e^(2*c*z)+a)^(5/2)/(6*c) +5*a*%e^(c*z)*(b*%e^(2*c*z)+a)^(3/2)/(24*c) +5*a^2*%e^(c*z)*sqrt(b*%e^(2*c*z)+a)/(16*c)$ /* Problem 535: NEW 08-2010 */ integrate(((b*%e^(2*c*z)+a)^3)^(1/2)*%e^(c*z),z),domain:real; 3*a^2*asinh(sqrt(b)*%e^(c*z)/sqrt(a))/(8*sqrt(b)*c) +%e^(c*z)*(b*%e^(2*c*z)+a)^(3/2)/(4*c) +3*a*%e^(c*z)*sqrt(b*%e^(2*c*z)+a)/(8*c)$ forget(a>0,b>0); [a>0,b>0]; /* Problem 537: NEW 08-2010 */ integrate((b*%e^(2*c*z)+a)^(-3/2)*%e^(c*z),z); %e^(c*z)/(a*c*sqrt(b*%e^(2*c*z)+a)); /* Problem 538: NEW 08-2010 */ integrate((b*%e^(2*c*z)+a)^(-5/2)*%e^(c*z),z); 2*%e^(c*z)/(3*a^2*c*sqrt(b*%e^(2*c*z)+a)) +%e^(c*z)/(3*a*c*(b*%e^(2*c*z)+a)^(3/2)); /* ----- Involving (a+b*%e^(2*c*z))^p*(%e^(c*z))^v -------------------------- */ /* Problem 539: NOUN */ integrate((b*%e^(2*c*z)+a)^p*(%e^(c*z))^v,z); 'integrate((b*%e^(2*c*z)+a)^p*(%e^(c*z))^v,z); /* Problem 540: NOUN */ integrate((b*%e^(2*c*z)+a)^(5/2)*(%e^(c*z))^v,z); 'integrate((b*%e^(2*c*z)+a)^(5/2)*(%e^(c*z))^v,z); /* Problem 541: NOUN */ integrate((b*%e^(2*c*z)+a)^(3/2)*(%e^(c*z))^v,z); 'integrate((b*%e^(2*c*z)+a)^(3/2)*(%e^(c*z))^v,z); /* Problem 542: NOUN */ integrate((b*%e^(2*c*z)+a)^(1/2)*(%e^(c*z))^v,z); 'integrate((b*%e^(2*c*z)+a)^(1/2)*(%e^(c*z))^v,z); /* Problem 543: NOUN */ integrate((b*%e^(2*c*z)+a)^((-1)/2)*(%e^(c*z))^v,z); 'integrate((b*%e^(2*c*z)+a)^((-1)/2)*(%e^(c*z))^v,z); /* Problem 544: NOUN */ integrate((b*%e^(2*c*z)+a)^((-3)/2)*(%e^(c*z))^v,z); 'integrate((b*%e^(2*c*z)+a)^((-3)/2)*(%e^(c*z))^v,z); /* Problem 545: NOUN */ integrate((b*%e^(2*c*z)+a)^((-5)/2)*(%e^(c*z))^v,z); 'integrate((b*%e^(2*c*z)+a)^((-5)/2)*(%e^(c*z))^v,z); /* ----- Involving %e^(q*z)*(b*%e^(c*z)+a)^p*(e*%e^(c*z)+d)^v --------------- */ /* Problem 546: NOUN */ integrate(%e^(q*z)*(b*%e^(c*z)+a)^p*(e*%e^(c*z)+d)^v,z); 'integrate((b*%e^(c*z)+a)^p*(e*%e^(c*z)+d)^v*%e^(q*z),z); /* Problem 547: NOUN */ integrate((b*%e^(c*z)+a)^p*(e*%e^(c*z)+d)^v,z); 'integrate((b*%e^(c*z)+a)^p*(e*%e^(c*z)+d)^v,z); /* ----- Involvoving (a*%e^(2*d*z)+b*%e^(d*z)+c)^p -------------------------- */ /* Problem 548: NOUN */ integrate((c+b*%e^(d*z)+a*%e^(2*d*z))^p,z); 'integrate((a*%e^(2*d*z)+b*%e^(d*z)+c)^p,z); /* Problem 550: NEW 08-2010 * We get a correct result for x a real variable. */ assume(a>0,b>0,c>0,4*a*c>b^2); [a > 0,b > 0,c > 0,4*a*c > b^2]; integrate((c+b*%e^(d*x)+a*%e^(2*d*x))^(1/2),x); b*asinh((2*a*%e^(d*x)+b)/sqrt(4*a*c-b^2))/(2*sqrt(a)*d) -sqrt(c)*asinh(2*c*%e^-(d*x)/sqrt(4*a*c-b^2)+b/sqrt(4*a*c-b^2))/d +sqrt(a*%e^(2*d*x)+b*%e^(d*x)+c)/d; forget(a>0,b>0,c>0,4*a*c>b^2); [a > 0,b > 0,c > 0,4*a*c > b^2]; /* ----- Involvoving %e^(e*z)*(a*%e^(2*d*z)+b*%e^(d*z)+c)^p ----------------- */ /* Problem 552: NOUN */ integrate(%e^(e*z)*(c+b*%e^(d*z)+a*%e^(2*d*z))^p,z); 'integrate((a*%e^(2*d*z)+b*%e^(d*z)+c)^p*%e^(e*z),z); /* ----- Other integrals ---------------------------------------------------- */ assume(a>0,b>0,c>0,d>0); [a>0,b>0,c>0,d>0]; /* Problem 554: NEW 08-2010 */ integrate(sqrt((b*%e^(e*z)+a)/(d*%e^(e*z)+c)),z); a*log((2*c*sqrt((b*%e^(e*z)+a)/(d*%e^(e*z)+c))-2*sqrt(a*c)) /(2*c*sqrt((b*%e^(e*z)+a)/(d*%e^(e*z)+c))+2*sqrt(a*c))) /(sqrt(a*c)*e) -b*log((2*d*sqrt((b*%e^(e*z)+a)/(d*%e^(e*z)+c))-2*sqrt(b*d)) /(2*d*sqrt((b*%e^(e*z)+a)/(d*%e^(e*z)+c))+2*sqrt(b*d))) /(sqrt(b*d)*e)$ assume(a*d>b*c); [a*d>b*c]; /* Problem 556: NEW 08-2010 */ integrate(sqrt((%e^(c*z)-1)/(%e^(c*z)+1)),z); log(sqrt((%e^(c*z)-1)/(%e^(c*z)+1))+1)/c -2*atan(sqrt((%e^(c*z)-1)/(%e^(c*z)+1)))/c -log(sqrt((%e^(c*z)-1)/(%e^(c*z)+1))-1)/c$ forget(a>0,b>0,c>0,d>0,a*d>b*c); [a>0,b>0,c>0,d>0,a*d>b*c]; /* Problem 558: OK */ integrate(sqrt((%e^(c*z)-1)/(%e^(c*z)+1)),z); log(sqrt((%e^(c*z)-1)/(%e^(c*z)+1))+1)/c -2*atan(sqrt((%e^(c*z)-1)/(%e^(c*z)+1)))/c -log(sqrt((%e^(c*z)-1)/(%e^(c*z)+1))-1)/c; /* Problem 559: NOUN */ integrate(1/(sqrt(1-%e^z)+sqrt(1+%e^z)),z); 'integrate(1/(sqrt(1-%e^z)+sqrt(1+%e^z)),z); /* ----- Involving powers of the dircect function and a power function ------ */ /* ----- Involving z^(v-1) and arguments a*z -------------------------------- */ /* Problem 560: NEW 2-1 */ diffcheck(integrate(z^(v-1)*(%e^(a*z))^u,z),z,factor); z^(v-1)*(%e^(a*z))^u; /* Problem 561: NEW 2 * We get the most simple result for a real argument x. */ integrate((%e^(a*x))^u/x,x),gamma_expand:true,logcontract; expintegral_ei(a*u*x); /* Problem 562: NEW 2-1 */ diffcheck(integrate(z^n*(%e^(a*z))^u,z),z,factor); z^n*(%e^(a*z))^u; /* Problem 563: OK */ integrate(z*(%e^(a*z))^u,z), domain:real; (a*u*z-1)*%e^(a*u*z)/(a^2*u^2); /* Problem 564: NEW 2-1*/ integrate(z^(1/2+n)*(%e^(a*z))^u,z); -gamma_incomplete(n+3/2,-a*u*z)*z^(n+3/2)*(-a*u*z)^(-n-3/2)*(%e^(a*z))^u *%e^-(a*u*z); /* ----- Involving z^(v-1) and arguments a*z+b ------------------------------ */ /* Problem 565: NEW 2-1 */ -integrate(z^(v-1)*(%e^(a*z+b))^u,z); gamma_incomplete(v,-a*u*z)*z^v*%e^-(a*u*z)*(%e^(a*z+b))^u/(-a*u*z)^v; /* Problem 566: NEW Typ 2 * We get the most simple result only for a real argument x. */ integrate((%e^(a*x+b))^u/x,x),gamma_expand:true,logcontract; %e^(b*u)*expintegral_ei(a*u*x); /* Problem 567: NEW 2-1 */ integrate(z^n*(%e^(a*z+b))^u,z); -a^(-n-1)*(-1)^(n+1)*gamma_incomplete(n+1,-a*u*z)*u^(-n-1)*%e^-(a*u*z) *(%e^(a*z+b))^u; /* Problem 568: OK */ integrate(z*(%e^(a*z+b))^u,z), domain:real; (a*u*%e^(b*u)*z-%e^(b*u))*%e^(a*u*z)/(a^2*u^2); /* Problem 569: NEW 2-1 */ -integrate(z^(1/2+n)*(%e^(a*z+b))^u,z); gamma_incomplete(n+3/2,-a*u*z)*z^(n+3/2)*(-a*u*z)^(-n-3/2)*%e^-(a*u*z) *(%e^(a*z+b))^u; /* ----- Involving z^(v-1) and arguments a*z^r ------------------------------ */ /* Problem 570: NEW 2-1 */ integrate(z^(v-1)*(%e^(a*z^r))^u,z); -gamma_incomplete(v/r,-a*u*z^r)*z^v*(%e^(a*z^r))^u*%e^-(a*u*z^r) /(r*(-a*u*z^r)^(v/r)); /* Problem 571: NEW 2 * We get the most simple result only for a real argument x. */ integrate((%e^(a*x^r))^u/x,x),gamma_expand:true,logcontract; expintegral_ei(a*u*x^r)/r; /* Problem 572: NEW 2-1 */ integrate(z^n*(%e^(a*z^2))^u,z); -gamma_incomplete((n+1)/2,-a*u*z^2)*z^(n+1)*(-a*u*z^2)^((-n-1)/2) *(%e^(a*z^2))^u*%e^-(a*u*z^2) /2; /* Problem 573: NEW 2-1 */ integrate(z^(2*n)*(%e^(a*z^2))^u,z); -gamma_incomplete((2*n+1)/2,-a*u*z^2)*z^(2*n+1)*(-a*u*z^2)^((-2*n-1)/2) *(%e^(a*z^2))^u*%e^-(a*u*z^2) /2; /* Problem 574: NEW 2-1 */ integrate(z^(1+2*n)*(%e^(a*z^2))^u,z),factor; a^(-n-1)*(-1)^n*gamma_incomplete(n+1,-a*u*z^2)*u^(-n-1)*(%e^(a*z^2))^u *%e^-(a*u*z^2) /2; /* Problem 575: NEW 2-1 */ integrate(z^n*(%e^(a*sqrt(z)))^u,z),factor; -2*a^(-2*n-2)*gamma_incomplete(2*(n+1),-a*u*sqrt(z))*u^(-2*n-2) *(%e^(a*sqrt(z)))^u*%e^-(a*u*sqrt(z)); /* ----- Involving z^(v-1) and arguments a*z^r+b ---------------------------- */ /* Problem 576: NEW 2-1*/ integrate(z^(v-1)*(%e^(b+a*z^r))^u,z); -gamma_incomplete(v/r,-a*u*z^r)*z^v*%e^-(a*u*z^r)*(%e^(a*z^r+b))^u /(r*(-a*u*z^r)^(v/r)); /* Problem 577: NEW 2 * We get the most simple result only for a real argument x. */ integrate((%e^(b+a*x^r))^u/x,x),gamma_expand:true,logcontract; %e^(b*u)*expintegral_ei(a*u*x^r)/r; /* Problem 578: NEW 2-1 */ integrate(z^n*(%e^(b+a*z^2))^u,z),ratsimp; -gamma_incomplete((n+1)/2,-a*u*z^2)*z^(n+1)*(-a*u*z^2)^(-n/2-1/2) *%e^-(a*u*z^2)*(%e^(a*z^2+b))^u /2; /* Problem 579: NEW 2-1 */ integrate(z^(2*n)*(%e^(b+a*z^2))^u,z),ratsimp; -(-1)^n*gamma_incomplete((2*n+1)/2,-a*u*z^2)*z*%e^-(a*u*z^2)*(%e^(a*z^2+b))^u /(2*a^n*u^n*sqrt(-a*u*z^2)); /* Problem 580: NEW 2-1 */ integrate(z^(1+2*n)*(%e^(b+a*z^2))^u,z),ratsimp; a^(-n-1)*(-1)^n*gamma_incomplete(n+1,-a*u*z^2)*u^(-n-1)*%e^-(a*u*z^2) *(%e^(a*z^2+b))^u /2; /* Problem 581: NEW 2-1 */ integrate(z^n*(%e^(b+a*sqrt(z)))^u,z),ratsimp; -2*a^(-2*n-2)*gamma_incomplete(2*n+2,-a*u*sqrt(z))*u^(-2*n-2) *%e^-(a*u*sqrt(z))*(%e^(a*sqrt(z)+b))^u; /* ----- Involving z^(v-1) and arguments a*z^r+b*z -------------------------- */ /* Problem 582: NEW 5-1 */ expand(niceindices(integrate(z^n*(%e^(b*z+a*z^2))^u,z))- (-a^(-n-1)*(%e^(a*z^2+b*z))^u*%e^(-u*(a*z^2+b*z)-b^2*u/(4*a)) *('sum(b^(n-i)*gamma_incomplete((i+1)/2,-u*(2*a*z+b)^2/(4*a)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*a*z+b)^(i+1) /(-u*(2*a*z+b)^2/a)^((i+1)/2),i,0,n)) /2)); 0; /* Problem 583: NEW 6-1 */ niceindices(integrate(z^n*(%e^(b*z+a*sqrt(z)))^u,z)); 2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(a^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*b*sqrt(z)+a)^(j+i) *(a *gamma_incomplete((j+i+1)/2, -u*(2*b*sqrt(z)+a)^2 /(4*b)) *(2*b*sqrt(z)+a) +2*b *gamma_incomplete((j+i+2)/2, -u*(2*b*sqrt(z)+a)^2 /(4*b)) *sqrt(-u*(2*b*sqrt(z)+a)^2/b) /u) /(-u*(2*b*sqrt(z)+a)^2/b)^((j+i+1)/2),i,0,j),j,0, n))*(%e^(b*z+a*sqrt(z)))^u*%e^(u*(b*z+a*sqrt(z))-a^2*u/(4*b)) /b^(2*(n+1)); /* ----- Involving z^(v-1) and arguments a*z^r+b*z+c ------------------------ */ /* Problem 584: NEW 5-1 */ expand(niceindices(integrate(z^n*(%e^(c+b*z+a*z^2))^u,z))- (-a^(-n-1)*(%e^(a*z^2+b*z+c))^u*%e^(-u*(a*z^2+b*z)-b^2*u/(4*a)) *('sum(b^(n-i)*gamma_incomplete((i+1)/2,-u*(2*a*z+b)^2/(4*a)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*a*z+b)^(i+1) /(-u*(2*a*z+b)^2/a)^((i+1)/2),i,0,n)) /2)); 0; /* Problem 585: NEW 6-1 */ niceindices(integrate(z^n*(%e^(c+b*z+a*sqrt(z)))^u,z)); 2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(a^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*b*sqrt(z)+a)^(j+i) *(a *gamma_incomplete((j+i+1)/2, -u*(2*b*sqrt(z)+a)^2 /(4*b)) *(2*b*sqrt(z)+a) +2*b *gamma_incomplete((j+i+2)/2, -u*(2*b*sqrt(z)+a)^2 /(4*b)) *sqrt(-u*(2*b*sqrt(z)+a)^2/b) /u) /(-u*(2*b*sqrt(z)+a)^2/b)^((j+i+1)/2),i,0,j),j,0, n))*(%e^(b*z+a*sqrt(z)))^u *%e^(u*(b*z+a*sqrt(z)+c)-a^2*u/(4*b)) /b^(2*(n+1)); /* ----- Involving z^(v-1)*a^(d*z)*h^(c*z) ---------------------------------- */ /* Problem 586: NEW 7 */ expand(integrate(z^(v-1)*a^(d*z)*h^(c*z),z)- (-gamma_incomplete(v,-(c*log(h)+log(a)*d)*z)*z^v/(-(c*log(h)+log(a)*d)*z)^v)); 0; /* Problem 587: NEW 7 */ expand(integrate(z^n*a^(d*z)*h^(c*z),z) - (-(c*log(h)+log(a)*d)^(-n-1)*(-1)^(n+1) *gamma_incomplete(n+1,-(c*log(h)+log(a)*d)*z))); 0; /* Problem 588: NEW 7 */ expand(integrate(z^(1/2+n)*a^(d*z)*h^(c*z),z)- (-gamma_incomplete(n+3/2,-(c*log(h)+log(a)*d)*z) *z^(n+3/2)*(-(c*log(h)+log(a)*d)*z)^(-n-3/2))); 0; /* ----- Involving z^(v-1)*a^(d*z)*h^(c*z+g) -------------------------------- */ /* Problem 589: NEW 7 */ expand(integrate(z^(v-1)*a^(d*z)*h^(g+c*z),z)- -h^g*gamma_incomplete(v,-(c*log(h)+log(a)*d)*z)*z^v/(-(c*log(h)+log(a)*d)*z)^v); 0; /* Problem 590: NEW 7 */ expand(integrate(z^n*a^(d*z)*h^(g+c*z),z)- -h^g*(c*log(h)+log(a)*d)^(-n-1)*(-1)^(n+1) *gamma_incomplete(n+1,-(c*log(h)+log(a)*d)*z)); 0; /* Problem 591: NEW 7 */ expand(integrate(z^(1/2+n)*a^(d*z)*h^(g+c*z),z)- -h^g*gamma_incomplete(n+3/2,-(c*log(h)+log(a)*d)*z)*z^(n+3/2) *(-(c*log(h)+log(a)*d)*z)^(-n-3/2)); 0; /* ----- Involving z^(v-1)*a^(d*z+e)*h^(c*z+g) ------------------------------ */ /* Problem 592: NEW 7 */ expand(integrate(z^(v-1)*a^(e+d*z)*h^(g+c*z),z)- -a^e*h^g*gamma_incomplete(v,-(c*log(h)+log(a)*d)*z)*z^v /(-(c*log(h)+log(a)*d)*z)^v); 0; /* Problem 593: NEW 7 */ expand(integrate(z^n*a^(e+d*z)*h^(g+c*z),z)- -a^e*h^g*(c*log(h)+log(a)*d)^(-n-1)*(-1)^(n+1) *gamma_incomplete(n+1,-(c*log(h)+log(a)*d)*z)); 0; /* Problem 594: NEW 7 */ expand(integrate(z^(1/2+n)*a^(e+d*z)*h^(g+c*z),z)- -a^e*h^g*gamma_incomplete(n+3/2,-(c*log(h)+log(a)*d)*z)*z^(n+3/2) *(-(c*log(h)+log(a)*d)*z)^(-n-3/2)); 0; /* ----- Involving z^n*a^(b*z)*h^(c*z^r) ------------------------------------ */ /* Problem 595: NEW 9 */ expand(niceindices(integrate(z^n*a^(b*z)*h^(c*z^2),z))- (-c^(-n-1)*%e^-(log(a)^2*b^2/(4*c*log(h)))*log(h)^(-n-1) *('sum(log(a)^(n-i)*b^(n-i) *gamma_incomplete((i+1)/2, -(2*c*log(h)*z+log(a)*b)^2 /(4*c*log(h)))*(-1)^(i-n) *2^(i-n)*binomial(n,i) *(2*c*log(h)*z+log(a)*b)^(i+1) /(-(2*c*log(h)*z+log(a)*b)^2/(c*log(h)))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 596: NEW 10 */ expand(niceindices(integrate(z^n*a^(b*z)*h^(c*sqrt(z)),z))- (%e^-(c^2*log(h)^2/(4*log(a)*b))*2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum( c^(2*n-j-i)*log(h)^(2*n-j-i) *(-1)^(i-j)*binomial(j,i) *(2*log(a)*b*sqrt(z) +c*log(h)) ^(j+i) *(c*log(h) *gamma_incomplete( (j+i+1)/2, -(2*log(a)*b*sqrt(z) +c*log(h)) ^2 /(4*log(a)*b)) *(2*log(a)*b*sqrt(z) +c*log(h)) +2*log(a)*b *gamma_incomplete( (j+i+2)/2, -(2*log(a)*b*sqrt(z) +c*log(h)) ^2 /(4*log(a)*b)) *sqrt( -(2*log(a)*b*sqrt(z) +c*log(h)) ^2 /(log(a)*b))) /(-(2*log(a)*b*sqrt(z)+c*log(h))^2 /(log(a)*b)) ^((j+i+1)/2),i,0,j),j,0,n)) /(log(a)^(2*(n+1))*b^(2*(n+1))))); 0; /* ----- Involving z^n*a^(d*z+e)*h^(c*z^r) ---------------------------------- */ /* Problem 597: NEW 9 */ expand(niceindices(integrate(z^n*a^(e+d*z)*h^(c*z^2),z))- (-a^e*c^(-n-1)*%e^-(log(a)^2*d^2/(4*c*log(h)))*log(h)^(-n-1) *('sum(log(a)^(n-i)*d^(n-i) *gamma_incomplete((i+1)/2, -(2*c*log(h)*z+log(a)*d)^2 /(4*c*log(h)))*(-1)^(i-n)*2^(i-n) *binomial(n,i)*(2*c*log(h)*z+log(a)*d)^(i+1) /(-(2*c*log(h)*z+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 598: NEW 10 */ expand(niceindices(integrate(z^n*a^(e+d*z)*h^(c*sqrt(z)),z))- (a^e*%e^-(c^2*log(h)^2/(4*log(a)*d))*2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*log(h)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*log(a)*d*sqrt(z)+c*log(h))^(j+i) *(c*log(h) *gamma_incomplete((j+i+1)/2, -(2*log(a)*d*sqrt(z) +c*log(h)) ^2 /(4*log(a)*d)) *(2*log(a)*d*sqrt(z)+c*log(h)) +2*log(a)*d *gamma_incomplete((j+i+2)/2, -(2*log(a)*d*sqrt(z) +c*log(h)) ^2 /(4*log(a)*d)) *sqrt(-(2*log(a)*d*sqrt(z)+c*log(h))^2 /(log(a)*d))) /(-(2*log(a)*d*sqrt(z)+c*log(h))^2/(log(a)*d)) ^((j+i+1)/2),i,0,j),j,0,n)) /(log(a)^(2*(n+1))*d^(2*(n+1))))); 0; /* ----- Involving z^(v-1)*a^(b*z^r)*h^(c*z^r) ------------------------------ */ /* Problem 599: NEW 7 */ expand(integrate(z^(v-1)*a^(b*z^r)*h^(c*z^r),z)- -gamma_incomplete(v/r,-(c*log(h)+log(a)*b)*z^r)*z^v /(r*(-(c*log(h)+log(a)*b)*z^r)^(v/r))); 0; /* Problem 600: NEW 7 */ expand(integrate(z^n*a^(b*z^2)*h^(c*z^2),z)- (-gamma_incomplete((n+1)/2,-(c*log(h)+log(a)*b)*z^2) *z^(n+1)*(-(c*log(h)+log(a)*b)*z^2)^((-n-1)/2) /2)); 0; /* Problem 601: NEW 7 */ expand(integrate(z^(2*n)*a^(b*z^2)*h^(c*z^2),z)- (-gamma_incomplete((2*n+1)/2,-(c*log(h)+log(a)*b)*z^2) *z^(2*n+1)*(-(c*log(h)+log(a)*b)*z^2)^((-2*n-1)/2) /2)); 0; /* Problem 602: NEW 7 */ expand(integrate(z^(1+2*n)*a^(b*z^2)*h^(c*z^2),z)- -(c*log(h)+log(a)*b)^((-2*n-2)/2)*(-1)^((-2*n-2)/2) *gamma_incomplete((2*n+2)/2, -(c*log(h)+log(a)*b)*z^2) /2); 0; /* Problem 603: NEW 7 */ expand(integrate(z^n*a^(b*sqrt(z))*h^(c*sqrt(z)),z)- (-2*(c*log(h)+log(a)*b)^(2*(-n-1)) *gamma_incomplete(2*(n+1),-(c*log(h)+log(a)*b)*sqrt(z)))); 0; /* ----- Involving z^n*a^(d*z)*h^(c*z^r+g) ---------------------------------- */ /* Problem 604: NEW 9 */ expand(niceindices(integrate(z^n*a^(d*z)*h^(g+c*z^2),z))- (-c^(-n-1)*h^g*%e^-(log(a)^2*d^2/(4*c*log(h)))*log(h)^(-n-1) *('sum(log(a)^(n-i)*d^(n-i) *gamma_incomplete((i+1)/2, -(2*c*log(h)*z+log(a)*d)^2 /(4*c*log(h)))*(-1)^(i-n) *2^(i-n)*binomial(n,i) *(2*c*log(h)*z+log(a)*d)^(i+1) /(-(2*c*log(h)*z+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 605: NEW 10 */ expand(niceindices(integrate(z^n*a^(d*z)*h^(g+c*sqrt(z)),z))- (h^g*%e^-(c^2*log(h)^2/(4*log(a)*d))*2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*log(h)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*log(a)*d*sqrt(z)+c*log(h))^(j+i) *(c*log(h) *gamma_incomplete((j+i+1)/2, -(2*log(a)*d*sqrt(z) +c*log(h)) ^2 /(4*log(a)*d)) *(2*log(a)*d*sqrt(z)+c*log(h)) +2*log(a)*d *gamma_incomplete((j+i+2)/2, -(2*log(a)*d*sqrt(z) +c*log(h)) ^2 /(4*log(a)*d)) *sqrt(-(2*log(a)*d*sqrt(z)+c*log(h))^2 /(log(a)*d))) /(-(2*log(a)*d*sqrt(z)+c*log(h))^2/(log(a)*d)) ^((j+i+1)/2),i,0,j),j,0,n)) /(log(a)^(2*(n+1))*d^(2*(n+1))))); 0; /* ----- Involving z^n*a^(d*z+e)*h^(c*z^r+g) -------------------------------- */ /* Problem 606: NEW 9 */ expand(niceindices(integrate(z^n*a^(e+d*z)*h^(g+c*z^2),z))- (-a^e*c^(-n-1)*h^g*%e^-(log(a)^2*d^2/(4*c*log(h)))*log(h)^(-n-1) *('sum(log(a)^(n-i)*d^(n-i) *gamma_incomplete((i+1)/2, -(2*c*log(h)*z+log(a)*d)^2 /(4*c*log(h)))*(-1)^(i-n)*2^(i-n) *binomial(n,i)*(2*c*log(h)*z+log(a)*d)^(i+1) /(-(2*c*log(h)*z+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 607: NEW 10 */ expand(niceindices(integrate(z^n*a^(e+d*z)*h^(g+c*sqrt(z)),z))- (a^e*h^g*%e^-(c^2*log(h)^2/(4*log(a)*d))*2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*log(h)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*log(a)*d*sqrt(z)+c*log(h))^(j+i) *(c*log(h) *gamma_incomplete((j+i+1)/2, -(2*log(a)*d*sqrt(z) +c*log(h)) ^2 /(4*log(a)*d)) *(2*log(a)*d*sqrt(z)+c*log(h)) +2*log(a)*d *gamma_incomplete((j+i+2)/2, -(2*log(a)*d*sqrt(z) +c*log(h)) ^2 /(4*log(a)*d)) *sqrt(-(2*log(a)*d*sqrt(z)+c*log(h))^2 /(log(a)*d))) /(-(2*log(a)*d*sqrt(z)+c*log(h))^2/(log(a)*d)) ^((j+i+1)/2),i,0,j),j,0,n)) /(log(a)^(2*(n+1))*d^(2*(n+1))))); 0; /* ----- Involving z^(v-1)*a^(b*z^r)*h^(c*z^r+g) ---------------------------- */ /* Problem 608: NEW 7 */ expand(integrate(z^(v-1)*a^(b*z^r)*h^(g+c*z^r),z)- -h^g*gamma_incomplete(v/r,-(c*log(h)+log(a)*b)*z^r)*z^v /(r*(-(c*log(h)+log(a)*b)*z^r)^(v/r))); 0; /* Problem 609: NEW 7 */ expand(integrate(z^n*a^(b*z^2)*h^(g+c*z^2),z)- (-h^g*gamma_incomplete((n+1)/2,-(c*log(h)+log(a)*b)*z^2)*z^(n+1) *(-(c*log(h)+log(a)*b)*z^2)^((-n-1)/2) /2)); 0; /* Problem 610: NEW 7 */ expand(integrate(z^(2*n)*a^(b*z^2)*h^(g+c*z^2),z)- (-h^g*gamma_incomplete((2*n+1)/2,-(c*log(h)+log(a)*b)*z^2)*z^(2*n+1) *(-(c*log(h)+log(a)*b)*z^2)^((-2*n-1)/2) /2)); 0; /* Problem 611: NEW 7 */ expand(integrate(z^(1+2*n)*a^(b*z^2)*h^(g+c*z^2),z)- -h^g*(c*log(h)+log(a)*b)^((-2*n-2)/2)*(-1)^((-2*n-2)/2) *gamma_incomplete((2*n+2)/2,-(c*log(h)+log(a)*b)*z^2) /2); 0; /* Problem 612: NEW 7 */ expand(integrate(z^(1+2*n)*a^(b*sqrt(z))*h^(g+c*sqrt(z)),z)- -2*h^g*(c*log(h)+log(a)*b)^(2*(-2*n-2)) *gamma_incomplete(2*(2*n+2),-(c*log(h)+log(a)*b)*sqrt(z))); 0; /* ----- Involving z^(v-1)*a^(b*z^r+e)*h^(c*z^r+g) -------------------------- */ /* Problem 613: NEW 7 */ expand(integrate(z^(v-1)*a^(e+b*z^r)*h^(g+c*z^r),z)- -a^e*h^g*gamma_incomplete(v/r,-(c*log(h)+log(a)*b)*z^r)*z^v /(r*(-(c*log(h)+log(a)*b)*z^r)^(v/r))); 0; /* Problem 614: NEW 7 */ expand(integrate(z^n*a^(e+b*z^2)*h^(g+c*z^2),z)- (-a^e*h^g*gamma_incomplete((n+1)/2,-(c*log(h)+log(a)*b)*z^2)*z^(n+1) *(-(c*log(h)+log(a)*b)*z^2)^((-n-1)/2) /2)); 0; /* Problem 615: NEW 7 */ expand(integrate(z^(2*n)*a^(e+b*z^2)*h^(g+c*z^2),z)- (-a^e*h^g*gamma_incomplete((2*n+1)/2,-(c*log(h)+log(a)*b)*z^2)*z^(2*n+1) *(-(c*log(h)+log(a)*b)*z^2)^((-2*n-1)/2) /2)); 0; /* Problem 616: NEW 7 */ expand(integrate(z^(1+2*n)*a^(e+b*z^2)*h^(g+c*z^2),z)- -a^e*h^g*(c*log(h)+log(a)*b)^((-2*n-2)/2)*(-1)^((-2*n-2)/2) *gamma_incomplete((2*n+2)/2,-(c*log(h)+log(a)*b)*z^2) /2); 0; /* Problem 617: NEW 7 */ expand(integrate(z^(1+2*n)*a^(e+b*sqrt(z))*h^(g+c*sqrt(z)),z)- -2*a^e*h^g*(c*log(h)+log(a)*b)^(2*(-2*n-2)) *gamma_incomplete(2*(2*n+2),-(c*log(h)+log(a)*b)*sqrt(z))); 0; /* ----- Involving z^n*a^(d*z)*h^(c*z^r+f*z) -------------------------------- */ /* Problem 618: NEW 9 */ expand(niceindices(integrate(z^n*a^(d*z)*h^(f*z+c*z^2),z))- (-c^(-n-1)*%e^-((f*log(h)+log(a)*d)^2/(4*c*log(h)))*log(h)^(-n-1) *('sum((-f*log(h)-log(a)*d)^(n-i) *gamma_incomplete((i+1)/2, -(log(h)*(2*c*z+f)+log(a)*d)^2 /(4*c*log(h)))*2^(i-n)*binomial(n,i) *(log(h)*(2*c*z+f)+log(a)*d)^(i+1) /(-(log(h)*(2*c*z+f)+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0, n)) /2)); 0; /* Problem 619: NEW 10 */ expand(niceindices(integrate(z^n*a^(d*z)*h^(f*z+c*sqrt(z)),z)) - (%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d))) *2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*log(h)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h))^(j+i) *(c*log(h) *gamma_incomplete((j+i+1)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)) ^2 /(4*(f*log(h)+log(a)*d))) *(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h)) +2*(f*log(h)+log(a)*d) *gamma_incomplete((j+i+2)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)) ^2 /(4*(f*log(h)+log(a)*d))) *sqrt(-(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h)) ^2 /(f*log(h)+log(a)*d))) /(-(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h))^2 /(f*log(h)+log(a)*d)) ^((j+i+1)/2),i,0,j),j,0,n)) /(f*log(h)+log(a)*d)^(2*(n+1)))); 0; /* ----- Involving z^n*a^(d*z+e)*h^(c*z^r+f*z) ------------------------------ */ /* Problem 620: NEW 9 */ expand(niceindices(integrate(z^n*a^(e+d*z)*h^(f*z+c*z^2),z))- (-a^e*c^(-n-1)*%e^-((f*log(h)+log(a)*d)^2/(4*c*log(h)))*log(h)^(-n-1) *('sum((-f*log(h)-log(a)*d)^(n-i)*gamma_incomplete((i+1)/2, -(log(h)*(2*c*z+f) +log(a)*d) ^2 /(4*c*log(h)))*2^(i-n) *binomial(n,i) *(log(h)*(2*c*z+f)+log(a)*d)^(i+1) /(-(log(h)*(2*c*z+f)+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 621: NEW 10 */ expand(niceindices(integrate(z^n*a^(e+d*z)*h^(f*z+c*sqrt(z)),z))- (a^e*%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*log(h)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h))^(j+i) *(c*log(h) *gamma_incomplete((j+i+1)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)) ^2 /(4*(f*log(h)+log(a)*d))) *(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h)) +2*(f*log(h)+log(a)*d) *gamma_incomplete((j+i+2)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)) ^2 /(4*(f*log(h)+log(a)*d))) *sqrt(-(2*(f*log(h)+log(a)*d)*sqrt(z) +c*log(h)) ^2 /(f*log(h)+log(a)*d))) /(-(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h))^2 /(f*log(h)+log(a)*d)) ^((j+i+1)/2),i,0,j),j,0,n)) /(f*log(h)+log(a)*d)^(2*(n+1)))); 0; /* ----- Involving z^n*a^(b*z^r)*h^(c*z^r+f*z) ------------------------------ */ /* Problem 622: NEW 9 */ expand(niceindices(integrate(z^n*a^(b*z^2)*h^(f*z+c*z^2),z))- (-%e^-(f^2*log(h)^2/(4*(c*log(h)+log(a)*b))) *(c*log(h)+log(a)*b)^(-n-1) *('sum(f^(n-i)*log(h)^(n-i) *gamma_incomplete((i+1)/2, -(log(h)*(2*c*z+f)+2*log(a)*b*z)^2 /(4*(c*log(h)+log(a)*b)))*(-1)^(i-n)*2^(i-n) *binomial(n,i)*(log(h)*(2*c*z+f)+2*log(a)*b*z)^(i+1) /(-(log(h)*(2*c*z+f)+2*log(a)*b*z)^2/(c*log(h)+log(a)*b))^((i+1)/2), i,0,n)) /2)); 0; /* Problem 623: NEW 10 */ expand(niceindices(integrate(z^n*a^(b*sqrt(z))*h^(f*z+c*sqrt(z)),z))- (%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h))) *2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum((c*log(h)+log(a)*b)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^(j+i) *((c*log(h)+log(a)*b) *gamma_incomplete( (j+i+1)/2, -(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*f*log(h))) *(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) +2*f*log(h) *gamma_incomplete( (j+i+2)/2, -(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*f*log(h))) *sqrt( -(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^2 /(f*log(h)))) /(-(2*f*log(h)*sqrt(z)+c*log(h)+log(a)*b)^2/(f*log(h))) ^((j+i+1)/2),i,0,j),j,0,n)) /(f^(2*(n+1))*log(h)^(2*(n+1))))); 0; /* ----- Involving z^n*a^(b*z^r+e)*h^(c*z^r+f*z) ---------------------------- */ /* Problem 624: NEW 9 */ expand(niceindices(integrate(z^n*a^(e+b*z^2)*h^(f*z+c*z^2),z))- (-a^e*%e^-(f^2*log(h)^2/(4*(c*log(h)+log(a)*b)))*(c*log(h)+log(a)*b)^(-n-1) *('sum(f^(n-i)*log(h)^(n-i) *gamma_incomplete((i+1)/2, -(log(h)*(2*c*z+f)+2*log(a)*b*z)^2 /(4*(c*log(h)+log(a)*b)))*(-1)^(i-n) *2^(i-n)*binomial(n,i)*(log(h)*(2*c*z+f)+2*log(a)*b*z)^(i+1) /(-(log(h)*(2*c*z+f)+2*log(a)*b*z)^2/(c*log(h)+log(a)*b)) ^((i+1)/2),i,0,n)) /2)); 0; /* Problem 625: NEW 10 */ expand(niceindices(integrate(z^n*a^(e+b*sqrt(z))*h^(f*z+c*sqrt(z)),z))- (a^e*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))*2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum((c*log(h)+log(a)*b)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^(j+i) *((c*log(h)+log(a)*b) *gamma_incomplete( (j+i+1)/2, -(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*f*log(h))) *(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) +2*f*log(h) *gamma_incomplete( (j+i+2)/2, -(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*f*log(h))) *sqrt( -(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^2 /(f*log(h)))) /(-(2*f*log(h)*sqrt(z)+c*log(h)+log(a)*b)^2/(f*log(h))) ^((j+i+1)/2),i,0,j),j,0,n)) /(f^(2*(n+1))*log(h)^(2*(n+1))))); 0; /* ----- Involving z^n*a^(b*z^r+d*z)*h^(c*z^r+f*z) -------------------------- */ /* Problem 626: NEW 9 */ expand(niceindices(integrate(z^n*a^(d*z+b*z^2)*h^(f*z+c*z^2),z))- (-%e^-((f*log(h)+log(a)*d)^2/(4*(c*log(h)+log(a)*b))) *(c*log(h)+log(a)*b)^(-n-1) *('sum((-f*log(h)-log(a)*d)^(n-i)*gamma_incomplete((i+1)/2, -(log(h)*(2*c*z+f) +log(a)*(2*b*z+d)) ^2 /(4*(c*log(h)+log(a)*b))) *2^(i-n)*binomial(n,i) *(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))^(i+1) /(-(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))^2/(c*log(h)+log(a)*b)) ^((i+1)/2),i,0,n)) /2)); 0; /* Problem 627: NEW 10 */ expand(niceindices(integrate(z^n*a^(d*z+b*sqrt(z))*h^(f*z+c*sqrt(z)),z))- (%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d))) *2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum((c*log(h)+log(a)*b)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*(f*log(h)+log(a)*d)*sqrt(z) +c*log(h)+log(a)*b) ^(j+i) *((c*log(h)+log(a)*b) *gamma_incomplete( (j+i+1)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*(f*log(h)+log(a)*d))) *(2*(f*log(h)+log(a)*d)*sqrt(z) +c*log(h)+log(a)*b) +2*(f*log(h)+log(a)*d) *gamma_incomplete( (j+i+2)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*(f*log(h)+log(a)*d))) *sqrt( -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)+log(a)*b) ^2 /(f*log(h)+log(a)*d))) /(-(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h)+log(a)*b)^2 /(f*log(h)+log(a)*d)) ^((j+i+1)/2),i,0,j),j,0,n)) /(f*log(h)+log(a)*d)^(2*(n+1)))); 0; /* ----- Involving z^n*a^(d*z)*h^(c*z^r+f*z+g) ------------------------------ */ /* Problem 628: NEW 9 */ expand(niceindices(integrate(z^n*a^(d*z)*h^(g+f*z+c*z^2),z))- (-c^(-n-1)*h^g*%e^-((f*log(h)+log(a)*d)^2/(4*c*log(h)))*log(h)^(-n-1) *('sum((-f*log(h)-log(a)*d)^(n-i) *gamma_incomplete((i+1)/2, -(log(h)*(2*c*z+f)+log(a)*d)^2 /(4*c*log(h)))*2^(i-n)*binomial(n,i) *(log(h)*(2*c*z+f)+log(a)*d)^(i+1) /(-(log(h)*(2*c*z+f)+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0, n)) /2)); 0; /* Problem 629: NEW 10 */ expand(niceindices(integrate(z^n*a^(d*z)*h^(g+f*z+c*sqrt(z)),z))- (h^g*%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*log(h)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h))^(j+i) *(c*log(h) *gamma_incomplete((j+i+1)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)) ^2 /(4*(f*log(h)+log(a)*d))) *(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h)) +2*(f*log(h)+log(a)*d) *gamma_incomplete((j+i+2)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)) ^2 /(4*(f*log(h)+log(a)*d))) *sqrt(-(2*(f*log(h)+log(a)*d)*sqrt(z) +c*log(h)) ^2 /(f*log(h)+log(a)*d))) /(-(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h))^2 /(f*log(h)+log(a)*d)) ^((j+i+1)/2),i,0,j),j,0,n)) /(f*log(h)+log(a)*d)^(2*(n+1)))); 0; /* ----- Involving z^n*a^(d*z+e)*h^(c*z^r+f*z+g) ---------------------------- */ /* Problem 630: NEW 9 */ expand(niceindices(integrate(z^n*a^(e+d*z)*h^(g+f*z+c*z^2),z))- (-a^e*c^(-n-1)*h^g*%e^-((f*log(h)+log(a)*d)^2/(4*c*log(h)))*log(h)^(-n-1) *('sum((-f*log(h)-log(a)*d)^(n-i)*gamma_incomplete((i+1)/2, -(log(h)*(2*c*z+f) +log(a)*d) ^2 /(4*c*log(h)))*2^(i-n) *binomial(n,i) *(log(h)*(2*c*z+f)+log(a)*d)^(i+1) /(-(log(h)*(2*c*z+f)+log(a)*d)^2/(c*log(h)))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 631: NEW 10 */ expand(niceindices(integrate(z^n*a^(e+d*z)*h^(g+f*z+c*sqrt(z)),z))- (a^e*h^g*%e^-(c^2*log(h)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*log(h)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h))^(j+i) *(c*log(h) *gamma_incomplete((j+i+1)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)) ^2 /(4*(f*log(h)+log(a)*d))) *(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h)) +2*(f*log(h)+log(a)*d) *gamma_incomplete((j+i+2)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)) ^2 /(4*(f*log(h)+log(a)*d))) *sqrt(-(2*(f*log(h)+log(a)*d)*sqrt(z) +c*log(h)) ^2 /(f*log(h)+log(a)*d))) /(-(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h))^2 /(f*log(h)+log(a)*d)) ^((j+i+1)/2),i,0,j),j,0,n)) /(f*log(h)+log(a)*d)^(2*(n+1)))); 0; /* ----- Involving z^n*a^(b*z^r)*h^(c*z^r+f*z+g) ---------------------------- */ /* Problem 632: NEW 9 */ expand(niceindices(integrate(z^n*a^(b*z^2)*h^(g+f*z+c*z^2),z))- (-h^g*%e^-(f^2*log(h)^2/(4*(c*log(h)+log(a)*b)))*(c*log(h)+log(a)*b)^(-n-1) *('sum(f^(n-i)*log(h)^(n-i) *gamma_incomplete((i+1)/2, -(log(h)*(2*c*z+f)+2*log(a)*b*z)^2 /(4*(c*log(h)+log(a)*b)))*(-1)^(i-n) *2^(i-n)*binomial(n,i)*(log(h)*(2*c*z+f)+2*log(a)*b*z)^(i+1) /(-(log(h)*(2*c*z+f)+2*log(a)*b*z)^2/(c*log(h)+log(a)*b)) ^((i+1)/2),i,0,n)) /2)); 0; /* Problem 633: NEW 10 */ expand(niceindices(integrate(z^n*a^(b*sqrt(z))*h^(g+f*z+c*sqrt(z)),z))- (h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))*2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum((c*log(h)+log(a)*b)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^(j+i) *((c*log(h)+log(a)*b) *gamma_incomplete( (j+i+1)/2, -(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*f*log(h))) *(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) +2*f*log(h) *gamma_incomplete( (j+i+2)/2, -(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*f*log(h))) *sqrt( -(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^2 /(f*log(h)))) /(-(2*f*log(h)*sqrt(z)+c*log(h)+log(a)*b)^2/(f*log(h))) ^((j+i+1)/2),i,0,j),j,0,n)) /(f^(2*(n+1))*log(h)^(2*(n+1))))); 0; /* ----- Involving z^n*a^(b*z^r+e)*h^(c*z^r+f*z+g) -------------------------- */ /* Problem 634: NEW 9 */ expand(niceindices(integrate(z^n*a^(e+b*z^2)*h^(g+f*z+c*z^2),z))- (-a^e*h^g*%e^-(f^2*log(h)^2/(4*(c*log(h)+log(a)*b)))*(c*log(h)+log(a)*b)^(-n-1) *('sum(f^(n-i)*log(h)^(n-i) *gamma_incomplete((i+1)/2, -(log(h)*(2*c*z+f)+2*log(a)*b*z)^2 /(4*(c*log(h)+log(a)*b)))*(-1)^(i-n) *2^(i-n)*binomial(n,i)*(log(h)*(2*c*z+f)+2*log(a)*b*z)^(i+1) /(-(log(h)*(2*c*z+f)+2*log(a)*b*z)^2/(c*log(h)+log(a)*b)) ^((i+1)/2),i,0,n)) /2)); 0; /* Problem 635: NEW 10 */ expand(niceindices(integrate(z^n*a^(e+b*sqrt(z))*h^(g+f*z+c*sqrt(z)),z))- (a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*f*log(h)))*2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum((c*log(h)+log(a)*b)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^(j+i) *((c*log(h)+log(a)*b) *gamma_incomplete( (j+i+1)/2, -(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*f*log(h))) *(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) +2*f*log(h) *gamma_incomplete( (j+i+2)/2, -(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*f*log(h))) *sqrt( -(2*f*log(h)*sqrt(z) +c*log(h)+log(a)*b) ^2 /(f*log(h)))) /(-(2*f*log(h)*sqrt(z)+c*log(h)+log(a)*b)^2/(f*log(h))) ^((j+i+1)/2),i,0,j),j,0,n)) /(f^(2*(n+1))*log(h)^(2*(n+1))))); 0; /* ----- Involving z^n*a^(b*z^r+d*z)*h^(c*z^r+f*z+g) ------------------------ */ /* Problem 636: NEW 9 */ expand(niceindices(integrate(z^n*a^(d*z+b*z^2)*h^(g+f*z+c*z^2),z))- (-h^g*%e^-((f*log(h)+log(a)*d)^2/(4*(c*log(h)+log(a)*b))) *(c*log(h)+log(a)*b)^(-n-1) *('sum((-f*log(h)-log(a)*d)^(n-i)*gamma_incomplete((i+1)/2, -(log(h)*(2*c*z+f) +log(a)*(2*b*z+d)) ^2 /(4 *(c*log(h) +log(a)*b)))*2^(i-n) *binomial(n,i) *(log(h)*(2*c*z+f)+log(a)*(2*b*z+d)) ^(i+1) /(-(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))^2/(c*log(h)+log(a)*b)) ^((i+1)/2),i,0,n))/2)); 0; /* Problem 637: NEW 10 */ expand(niceindices(integrate(z^n*a^(d*z+b*sqrt(z))*h^(g+f*z+c*sqrt(z)),z))- (h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum((c*log(h)+log(a)*b)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)+log(a)*b) ^(j+i) *((c*log(h)+log(a)*b) *gamma_incomplete( (j+i+1)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*(f*log(h)+log(a)*d))) *(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)+log(a)*b) +2*(f*log(h)+log(a)*d) *gamma_incomplete( (j+i+2)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*(f*log(h)+log(a)*d))) *sqrt( -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)+log(a)*b) ^2 /(f*log(h)+log(a)*d))) /(-(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h)+log(a)*b)^2 /(f*log(h)+log(a)*d)) ^((j+i+1)/2),i,0,j),j,0,n)) /(f*log(h)+log(a)*d)^(2*(n+1)))); 0; /* ----- Involving z^n*a^(b*z^r+d*z+e)*h^(c*z^r+f*z+g) ---------------------- */ /* Problem 638: NEW 9 */ expand(niceindices(integrate(z^n*a^(e+d*z+b*z^2)*h^(g+f*z+c*z^2),z))- (-a^e*h^g*%e^-((f*log(h)+log(a)*d)^2/(4*(c*log(h)+log(a)*b))) *(c*log(h)+log(a)*b)^(-n-1) *('sum((-f*log(h)-log(a)*d)^(n-i)*gamma_incomplete((i+1)/2, -(log(h)*(2*c*z+f) +log(a)*(2*b*z+d)) ^2 /(4 *(c*log(h) +log(a)*b)))*2^(i-n) *binomial(n,i) *(log(h)*(2*c*z+f)+log(a)*(2*b*z+d)) ^(i+1) /(-(log(h)*(2*c*z+f)+log(a)*(2*b*z+d))^2/(c*log(h)+log(a)*b)) ^((i+1)/2),i,0,n)) /2)); 0; /* Problem 639: NEW 10 */ expand(niceindices(integrate(z^n*a^(e+d*z+b*sqrt(z))*h^(g+f*z+c*sqrt(z)),z))- (a^e*h^g*%e^-((c*log(h)+log(a)*b)^2/(4*(f*log(h)+log(a)*d)))*2^(-2*n-1) *('sum(4^j*binomial(n,j) *'sum((c*log(h)+log(a)*b)^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)+log(a)*b) ^(j+i) *((c*log(h)+log(a)*b) *gamma_incomplete( (j+i+1)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*(f*log(h)+log(a)*d))) *(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)+log(a)*b) +2*(f*log(h)+log(a)*d) *gamma_incomplete( (j+i+2)/2, -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)+log(a)*b) ^2 /(4*(f*log(h)+log(a)*d))) *sqrt( -(2*(f*log(h)+log(a)*d) *sqrt(z) +c*log(h)+log(a)*b) ^2 /(f*log(h)+log(a)*d))) /(-(2*(f*log(h)+log(a)*d)*sqrt(z)+c*log(h)+log(a)*b)^2 /(f*log(h)+log(a)*d)) ^((j+i+1)/2),i,0,j),j,0,n)) /(f*log(h)+log(a)*d)^(2*(n+1)))); 0; /* ----- Products of powers of the direct function and a power function ----- */ /* ----- Involving z^(v-1)*%e^(c*z)*(%e^(a*z))^u ---------------------------- */ /* Problem 640: NEW 7-1*/ expand(integrate(z^(v-1)*%e^(c*z)*(%e^(a*z))^u,z)- -gamma_incomplete(v,-(a*u+c)*z)*z^v*(%e^(a*z))^u*%e^(c*z-(a*u+c)*z) /(-(a*u+c)*z)^v)$ 0; /* Problem 641: NEW 7-1 */ expand(integrate(z^n*%e^(c*z)*(%e^(a*z))^u,z)- -(-1)^(n+1)*gamma_incomplete(n+1,-(a*u+c)*z)*(a*u+c)^(-n-1)*(%e^(a*z))^u *%e^(c*z-(a*u+c)*z))$ 0; /* Problem 642: NEW 7-1 */ expand(integrate(z^(1/2+n)*%e^(c*z)*(%e^(a*z))^u,z)- -gamma_incomplete(n+3/2,-(a*u+c)*z)*z^(n+3/2)*(-(a*u+c)*z)^(-n-3/2) *(%e^(a*z))^u*%e^(c*z-(a*u+c)*z))$ 0; /* ----- Involving z^(v-1)*%e^(c*z+d)*(%e^(a*z))^u -------------------------- */ /* Problem 643: NEW 7-1 */ expand(integrate(z^(v-1)*%e^(c*z+d)*(%e^(a*z))^u,z)- -gamma_incomplete(v,-(a*u+c)*z)*z^v*(%e^(a*z))^u*%e^(-(a*u+c)*z+c*z+d) /(-(a*u+c)*z)^v)$ 0; /* Problem 644: NEW 7-1 */ expand(integrate(z^n*%e^(c*z+d)*(%e^(a*z))^u,z)- -(-1)^(n+1)*gamma_incomplete(n+1,-(a*u+c)*z)*(a*u+c)^(-n-1)*(%e^(a*z))^u *%e^(-(a*u+c)*z+c*z+d)); 0; /* Problem 645: NEW 7-1 */ expand(integrate(z^(1/2+n)*%e^(c*z+d)*(%e^(a*z))^u,z)- -gamma_incomplete(n+3/2,-(a*u+c)*z)*z^(n+3/2)*(-(a*u+c)*z)^(-n-3/2) *(%e^(a*z))^u*%e^(-(a*u+c)*z+c*z+d))$ 0; /* ----- Involving z^(v-1)*%e^(c*z)*(%e^(a*z+b))^u -------------------------- */ /* Problem 646: NEW 7-1 */ expand(integrate(z^(v-1)*%e^(c*z)*(%e^(a*z+b))^u,z)- -gamma_incomplete(v,-(a*u+c)*z)*z^v*(%e^(a*z+b))^u*%e^(c*z-(a*u+c)*z) /(-(a*u+c)*z)^v); 0; /* Problem 647: NEW 7-1 */ expand(integrate(z^n*%e^(c*z)*(%e^(a*z+b))^u,z)- -(-1)^(n+1)*gamma_incomplete(n+1,-(a*u+c)*z)*(a*u+c)^(-n-1)*(%e^(a*z+b))^u *%e^(c*z-(a*u+c)*z))$ 0; /* Problem 648: NEW 7-1 */ expand(integrate(z^(1/2+n)*%e^(c*z)*(%e^(a*z+b))^u,z)- -gamma_incomplete(n+3/2,-(a*u+c)*z)*z^(n+3/2)*(-(a*u+c)*z)^(-n-3/2) *(%e^(a*z+b))^u*%e^(c*z-(a*u+c)*z))$ 0; /* ----- Involving z^(v-1)*%e^(c*z+d)*(%e^(a*z+b))^u ------------------------ */ /* Problem 649: NEW 7-1 */ expand(integrate(z^(v-1)*%e^(c*z+d)*(%e^(a*z+b))^u,z)- -gamma_incomplete(v,-(a*u+c)*z)*z^v*(%e^(a*z+b))^u*%e^(-(a*u+c)*z+c*z+d) /(-(a*u+c)*z)^v)$ 0; /* Problem 650: NEW 7-1 */ expand(integrate(z^n*%e^(c*z+d)*(%e^(a*z+b))^u,z)- -(-1)^(n+1)*gamma_incomplete(n+1,-(a*u+c)*z)*(a*u+c)^(-n-1)*(%e^(a*z+b))^u *%e^(-(a*u+c)*z+c*z+d))$ 0; /* Problem 651: NEW 7-1 */ expand(integrate(z^(1/2+n)*%e^(c*z+d)*(%e^(a*z+b))^u,z)- -gamma_incomplete(n+3/2,-(a*u+c)*z)*z^(n+3/2)*(-(a*u+c)*z)^(-n-3/2) *(%e^(a*z+b))^u*%e^(-(a*u+c)*z+c*z+d))$ 0; /* ----- Involving z^n*%e^(b*z^r)*(%e^(c*z))^u ------------------------------ */ /* Problem 652: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(b*z^2)*(%e^(c*z))^u,z))- (-b^(-n-1)*(%e^(c*z))^u*%e^(b*z^2-z*(b*z+c*u)-c^2*u^2/(4*b)) *('sum(c^(n-i)*gamma_incomplete((i+1)/2,-(2*b*z+c*u)^2/(4*b)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*u^(n-i) *(2*b*z+c*u)^(i+1) /(-(2*b*z+c*u)^2/b)^((i+1)/2),i,0,n)) /2)); 0; /* Problem 653: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(b*sqrt(z))*(%e^(c*z))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*c*u*sqrt(z)+b)^(j+i) *(b *gamma_incomplete((j+i+1)/2, -(2*c*u*sqrt(z)+b)^2 /(4*c*u)) *(2*c*u*sqrt(z)+b) +2*c *gamma_incomplete((j+i+2)/2, -(2*c*u*sqrt(z)+b)^2 /(4*c*u))*u *sqrt((2*c*u*sqrt(z)+b)^2/(c*u))) /(-(2*c*u*sqrt(z)+b)^2/(c*u))^((j+i+1)/2),i,0,j),j, 0,n))*(%e^(c*z))^u*%e^(-c*u*z-b^2/(4*c*u)) /(c^(2*(n+1))*u^(2*(n+1))))); 0; /* ----- Involving z^n*%e^(b*z^r+e)*(%e^(c*z))^u ---------------------------- */ /* Problem 654: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+b*z^2)*(%e^(c*z))^u,z))- (-b^(-n-1)*(%e^(c*z))^u*%e^(b*z^2-z*(b*z+c*u)-c^2*u^2/(4*b)+e) *('sum(c^(n-i)*gamma_incomplete((i+1)/2,-(2*b*z+c*u)^2/(4*b)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*u^(n-i) *(2*b*z+c*u)^(i+1) /(-(2*b*z+c*u)^2/b)^((i+1)/2),i,0,n)) /2)); 0; /* Problem 655: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+b*sqrt(z))*(%e^(c*z))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*c*u*sqrt(z)+b)^(j+i) *(b *gamma_incomplete((j+i+1)/2, -(2*c*u*sqrt(z)+b)^2 /(4*c*u)) *(2*c*u*sqrt(z)+b) +2*c *gamma_incomplete((j+i+2)/2, -(2*c*u*sqrt(z)+b)^2 /(4*c*u))*u *sqrt((2*c*u*sqrt(z)+b)^2/(c*u))) /(-(2*c*u*sqrt(z)+b)^2/(c*u))^((j+i+1)/2),i,0,j),j, 0,n))*(%e^(c*z))^u*%e^(-c*u*z-b^2/(4*c*u)+e) /(c^(2*(n+1))*u^(2*(n+1))))); 0; /* ----- Involving z^n*%e^(b*z^r+d*z)*(%e^(c*z))^u -------------------------- */ /* Problem 656: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(d*z+b*z^2)*(%e^(c*z))^u,z))- (-b^(-n-1)*(%e^(c*z))^u*%e^(-z*(b*z+c*u+d)+z*(b*z+d)-(c*u+d)^2/(4*b)) *('sum(gamma_incomplete((i+1)/2,-(2*b*z+c*u+d)^2/(4*b)) *2^(i-n)*binomial(n,i)*(-c*u-d)^(n-i)*(2*b*z+c*u+d)^(i+1) /(-(2*b*z+c*u+d)^2/b)^((i+1)/2),i,0,n)) /2)); 0; /* Problem 657: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(d*z+b*sqrt(z))*(%e^(c*z))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*(c*u+d)*sqrt(z)+b)^(j+i) *(b *gamma_incomplete((j+i+1)/2, -(2*(c*u+d)*sqrt(z)+b) ^2 /(4*(c*u+d))) *(2*(c*u+d)*sqrt(z)+b) +2 *gamma_incomplete((j+i+2)/2, -(2*(c*u+d)*sqrt(z) +b) ^2 /(4*(c*u+d))) *(c*u+d) *sqrt((2*(c*u+d)*sqrt(z)+b)^2/(c*u+d))) /((2*(c*u+d)*sqrt(z)+b)^2/(-c*u-d))^((j+i+1)/2),i, 0,j),j,0,n))*(%e^(c*z))^u *%e^(-c*u*z-b^2/(4*(c*u+d))) /(c*u+d)^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(b*z^r+d*z+e)*(%e^(c*z^r))^u ---------------------- */ /* Problem 658: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z+b*z^2)*(%e^(c*z))^u,z))- (-b^(-n-1)*(%e^(c*z))^u*%e^(-z*(b*z+c*u+d)+z*(b*z+d)-(c*u+d)^2/(4*b)+e) *('sum(gamma_incomplete((i+1)/2,-(2*b*z+c*u+d)^2/(4*b)) *2^(i-n)*binomial(n,i)*(-c*u-d)^(n-i)*(2*b*z+c*u+d)^(i+1) /(-(2*b*z+c*u+d)^2/b)^((i+1)/2),i,0,n)) /2)); 0; /* Problem 659: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z+b*sqrt(z))*(%e^(c*z))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*(c*u+d)*sqrt(z)+b)^(j+i) *(b *gamma_incomplete((j+i+1)/2, -(2*(c*u+d)*sqrt(z)+b) ^2 /(4*(c*u+d))) *(2*(c*u+d)*sqrt(z)+b) +2 *gamma_incomplete((j+i+2)/2, -(2*(c*u+d)*sqrt(z) +b) ^2 /(4*(c*u+d))) *(c*u+d) *sqrt((2*(c*u+d)*sqrt(z)+b)^2/(c*u+d))) /((2*(c*u+d)*sqrt(z)+b)^2/(-c*u-d))^((j+i+1)/2),i, 0,j),j,0,n))*(%e^(c*z))^u *%e^(-c*u*z-b^2/(4*(c*u+d))+e) /(c*u+d)^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(b*z^r)*(%e^(f*z+g))^u ---------------------------- */ /* Problem 660: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(b*z^2)*(%e^(g+f*z))^u,z))- (-b^(-n-1)*(%e^(f*z+g))^u*%e^(b*z^2-z*(b*z+f*u)-f^2*u^2/(4*b)) *('sum(f^(n-i)*gamma_incomplete((i+1)/2,-(2*b*z+f*u)^2/(4*b)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*u^(n-i) *(2*b*z+f*u)^(i+1) /(-(2*b*z+f*u)^2/b)^((i+1)/2),i,0,n)) /2)); 0; /* Problem 661: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(b*sqrt(z))*(%e^(g+f*z))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*f*u*sqrt(z)+b)^(j+i) *(b *gamma_incomplete((j+i+1)/2, -(2*f*u*sqrt(z)+b)^2 /(4*f*u)) *(2*f*u*sqrt(z)+b) +2*f *gamma_incomplete((j+i+2)/2, -(2*f*u*sqrt(z)+b)^2 /(4*f*u))*u *sqrt((2*f*u*sqrt(z)+b)^2/(f*u))) /(-(2*f*u*sqrt(z)+b)^2/(f*u))^((j+i+1)/2),i,0,j),j, 0,n))*(%e^(f*z+g))^u*%e^(-f*u*z-b^2/(4*f*u)) /(f^(2*(n+1))*u^(2*(n+1))))); 0; /* ----- Involving z^n*%e^(b*z^r+e)*(%e^(f*z+g))^u -------------------------- */ /* Problem 662: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+b*z^2)*(%e^(g+f*z))^u,z))- (-b^(-n-1)*(%e^(f*z+g))^u*%e^(b*z^2-z*(b*z+f*u)-f^2*u^2/(4*b)+e) *('sum(f^(n-i)*gamma_incomplete((i+1)/2,-(2*b*z+f*u)^2/(4*b)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*u^(n-i) *(2*b*z+f*u)^(i+1) /(-(2*b*z+f*u)^2/b)^((i+1)/2),i,0,n)) /2)); 0; /* Problem 663: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+b*sqrt(z))*(%e^(g+f*z))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*f*u*sqrt(z)+b)^(j+i) *(b *gamma_incomplete((j+i+1)/2, -(2*f*u*sqrt(z)+b)^2 /(4*f*u)) *(2*f*u*sqrt(z)+b) +2*f *gamma_incomplete((j+i+2)/2, -(2*f*u*sqrt(z)+b)^2 /(4*f*u))*u *sqrt((2*f*u*sqrt(z)+b)^2/(f*u))) /(-(2*f*u*sqrt(z)+b)^2/(f*u))^((j+i+1)/2),i,0,j),j, 0,n))*(%e^(f*z+g))^u*%e^(-f*u*z-b^2/(4*f*u)+e) /(f^(2*(n+1))*u^(2*(n+1))))); 0; /* ----- Involving z^n*%e^(b*z^r+d*z)*(%e^(f*z+g))^u ------------------------ */ /* Problem 664: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(d*z+b*z^2)*(%e^(g+f*z))^u,z))- (-b^(-n-1)*(%e^(f*z+g))^u*%e^(-z*(b*z+f*u+d)+z*(b*z+d)-(f*u+d)^2/(4*b)) *('sum(gamma_incomplete((i+1)/2,-(2*b*z+f*u+d)^2/(4*b)) *2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i)*(2*b*z+f*u+d)^(i+1) /(-(2*b*z+f*u+d)^2/b)^((i+1)/2),i,0,n)) /2)); 0; /* Problem 665: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(d*z+b*sqrt(z))*(%e^(g+f*z))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*(f*u+d)*sqrt(z)+b)^(j+i) *(b *gamma_incomplete((j+i+1)/2, -(2*(f*u+d)*sqrt(z)+b) ^2 /(4*(f*u+d))) *(2*(f*u+d)*sqrt(z)+b) +2 *gamma_incomplete((j+i+2)/2, -(2*(f*u+d)*sqrt(z) +b) ^2 /(4*(f*u+d))) *(f*u+d) *sqrt((2*(f*u+d)*sqrt(z)+b)^2/(f*u+d))) /((2*(f*u+d)*sqrt(z)+b)^2/(-f*u-d))^((j+i+1)/2),i, 0,j),j,0,n))*(%e^(f*z+g))^u *%e^(-f*u*z-b^2/(4*(f*u+d))) /(f*u+d)^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(b*z^r+d*z+e)*(%e^(f*z+g))^u ---------------------- */ /* Problem 666: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z+b*z^2)*(%e^(g+f*z))^u,z))- (-b^(-n-1)*(%e^(f*z+g))^u*%e^(-z*(b*z+f*u+d)+z*(b*z+d)-(f*u+d)^2/(4*b)+e) *('sum(gamma_incomplete((i+1)/2,-(2*b*z+f*u+d)^2/(4*b)) *2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i)*(2*b*z+f*u+d)^(i+1) /(-(2*b*z+f*u+d)^2/b)^((i+1)/2),i,0,n)) /2)); 0; /* Problem 667: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z+b*sqrt(z))*(%e^(g+f*z))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(b^(2*n-j-i)*(-1)^(i-j)*binomial(j,i) *(2*(f*u+d)*sqrt(z)+b)^(j+i) *(b *gamma_incomplete((j+i+1)/2, -(2*(f*u+d)*sqrt(z)+b) ^2 /(4*(f*u+d))) *(2*(f*u+d)*sqrt(z)+b) +2 *gamma_incomplete((j+i+2)/2, -(2*(f*u+d)*sqrt(z) +b) ^2 /(4*(f*u+d))) *(f*u+d) *sqrt((2*(f*u+d)*sqrt(z)+b)^2/(f*u+d))) /((2*(f*u+d)*sqrt(z)+b)^2/(-f*u-d))^((j+i+1)/2),i, 0,j),j,0,n))*(%e^(f*z+g))^u *%e^(-f*u*z-b^2/(4*(f*u+d))+e) /(f*u+d)^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(d*z)*(%e^(c*z^r))^u ------------------------------ */ /* Problem 668: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(d*z)*(%e^(c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*%e^(-z*(c*u*z+d)+d*z-d^2/(4*c*u))*(%e^(c*z^2))^u *('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d)^2/(4*c*u)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*c*u*z+d)^(i+1) /(-(2*c*u*z+d)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 669: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(d*z)*(%e^(c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*d*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*d*sqrt(z)+c*u)^2 /(4*d))*u *(2*d*sqrt(z)+c*u) +2*d *gamma_incomplete((j+i+2)/2, -(2*d*sqrt(z)+c*u)^2 /(4*d)) *sqrt((2*d*sqrt(z)+c*u)^2/d)) /(-(2*d*sqrt(z)+c*u)^2/d)^((j+i+1)/2),i,0,j),j,0, n))*(%e^(c*sqrt(z)))^u*%e^(-c*u*sqrt(z)-c^2*u^2/(4*d)) /d^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(d*z+e)*(%e^(c*z^r))^u ---------------------------- */ /* Problem 670: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z)*(%e^(c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*%e^(-z*(c*u*z+d)+d*z-d^2/(4*c*u)+e)*(%e^(c*z^2))^u *('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d)^2/(4*c*u)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*c*u*z+d)^(i+1) /(-(2*c*u*z+d)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 671: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z)*(%e^(c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*d*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*d*sqrt(z)+c*u)^2 /(4*d))*u *(2*d*sqrt(z)+c*u) +2*d *gamma_incomplete((j+i+2)/2, -(2*d*sqrt(z)+c*u)^2 /(4*d)) *sqrt((2*d*sqrt(z)+c*u)^2/d)) /(-(2*d*sqrt(z)+c*u)^2/d)^((j+i+1)/2),i,0,j),j,0, n))*(%e^(c*sqrt(z)))^u*%e^(-c*u*sqrt(z)-c^2*u^2/(4*d)+e) /d^(2*(n+1)))); 0; /* ----- Involving z^(v-1)*%e^(b*z^r)*(%e^(c*z^r))^u ------------------------ */ /* Problem 672: NEW 7-1*/ diffcheck(integrate(z^(v-1)*%e^(b*z^r)*(%e^(c*z^r))^u,z),z,factor); z^(v-1)*%e^(b*z^r)*(%e^(c*z^r))^u; /* Problem 673: NEW 7-1 */ diffcheck(integrate(z^n*%e^(b*z^2)*(%e^(c*z^2))^u,z),z,factor); z^n*%e^(b*z^2)*(%e^(c*z^2))^u; /* Problem 674: NEW 7-1 */ diffcheck(integrate(z^(2*n)*%e^(b*z^2)*(%e^(c*z^2))^u,z),z,factor); z^(2*n)*%e^(b*z^2)*(%e^(c*z^2))^u; /* Problem 675: NEW 7-1 */ diffcheck(integrate(z^(1+2*n)*%e^(b*z^2)*(%e^(c*z^2))^u,z),z,factor); z^(1+2*n)*%e^(b*z^2)*(%e^(c*z^2))^u; /* Problem 676: NEW 7-1 */ diffcheck(integrate(z^n*%e^(b*sqrt(z))*(%e^(c*sqrt(z)))^u,z),z,factor); z^n*%e^(b*sqrt(z))*(%e^(c*sqrt(z)))^u; /* ----- Involving z^(v-1)*%e^(b*z^r+e)*(%e^(c*z^r))^u ---------------------- */ /* Problem 677: NEW 7-1 */ diffcheck(integrate(z^(v-1)*%e^(e+b*z^r)*(%e^(c*z^r))^u,z),z,factor); z^(v-1)*%e^(e+b*z^r)*(%e^(c*z^r))^u; /* Problem 678: NEW 7-1 */ diffcheck(integrate(z^n*%e^(e+b*z^2)*(%e^(c*z^2))^u,z),z,factor); z^n*%e^(e+b*z^2)*(%e^(c*z^2))^u; /* Problem 679: NEW 7-1 */ diffcheck(integrate(z^(2*n)*%e^(e+b*z^2)*(%e^(c*z^2))^u,z),z,factor); z^(2*n)*%e^(e+b*z^2)*(%e^(c*z^2))^u; /* Problem 680: NEW 7-1 */ diffcheck(integrate(z^(1+2*n)*%e^(e+b*z^2)*(%e^(c*z^2))^u,z),z,factor); z^(1+2*n)*%e^(e+b*z^2)*(%e^(c*z^2))^u; /* Problem 681: NEW 7-1 */ diffcheck(integrate(z^n*%e^(e+b*sqrt(z))*(%e^(c*sqrt(z)))^u,z),z,factor); z^n*%e^(e+b*sqrt(z))*(%e^(c*sqrt(z)))^u; /* ----- Involving z^n*%e^(b*z^r+d*z)*(%e^(c*z^r))^u ------------------------ */ /* Problem 682: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(d*z+b*z^2)*(%e^(c*z^2))^u,z))- (-(c*u+b)^(-n-1)*%e^(-z*(c*u*z+b*z+d)+z*(b*z+d)-d^2/(4*(c*u+b)))*(%e^(c*z^2))^u *('sum(d^(n-i)*gamma_incomplete((i+1)/2, -(2*(c*u+b)*z+d)^2/(4*(c*u+b))) *(-1)^(i-n)*2^(i-n)*binomial(n,i) *(2*(c*u+b)*z+d)^(i+1) /((2*(c*u+b)*z+d)^2/(-c*u-b))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 683: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(d*z+b*sqrt(z))*(%e^(c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i) *(2*d*sqrt(z)+c*u+b)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*d*sqrt(z)+c*u+b)^2 /(4*d)) *(c*u+b)*(2*d*sqrt(z)+c*u+b) +2*d *gamma_incomplete((j+i+2)/2, -(2*d*sqrt(z)+c*u+b) ^2 /(4*d)) *sqrt((2*d*sqrt(z)+c*u+b)^2/d)) /(-(2*d*sqrt(z)+c*u+b)^2/d)^((j+i+1)/2),i,0,j),j,0, n))*(%e^(c*sqrt(z)))^u*%e^(-c*u*sqrt(z)-(c*u+b)^2/(4*d)) /d^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(b*z^r+d*z+e)*(%e^(c*z^r))^u ---------------------- */ /* Problem 684: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z+b*z^2)*(%e^(c*z^2))^u,z))- (-(c*u+b)^(-n-1)*%e^(-z*(c*u*z+b*z+d)+z*(b*z+d)-d^2/(4*(c*u+b))+e) *(%e^(c*z^2))^u *('sum(d^(n-i)*gamma_incomplete((i+1)/2, -(2*(c*u+b)*z+d)^2/(4*(c*u+b))) *(-1)^(i-n)*2^(i-n)*binomial(n,i) *(2*(c*u+b)*z+d)^(i+1) /((2*(c*u+b)*z+d)^2/(-c*u-b))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 685: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z+b*sqrt(z))*(%e^(c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i) *(2*d*sqrt(z)+c*u+b)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*d*sqrt(z)+c*u+b)^2 /(4*d)) *(c*u+b)*(2*d*sqrt(z)+c*u+b) +2*d *gamma_incomplete((j+i+2)/2, -(2*d*sqrt(z)+c*u+b) ^2 /(4*d)) *sqrt((2*d*sqrt(z)+c*u+b)^2/d)) /(-(2*d*sqrt(z)+c*u+b)^2/d)^((j+i+1)/2),i,0,j),j,0, n))*(%e^(c*sqrt(z)))^u*%e^(-c*u*sqrt(z)-(c*u+b)^2/(4*d)+e) /d^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(d*z)*(%e^(c*z^r+g))^u ---------------------------- */ /* Problem 686: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(d*z)*(%e^(g+c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*%e^(-z*(c*u*z+d)+d*z-d^2/(4*c*u))*(%e^(c*z^2+g))^u *('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d)^2/(4*c*u)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*c*u*z+d)^(i+1) /(-(2*c*u*z+d)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 687: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(d*z)*(%e^(g+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*d*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*d*sqrt(z)+c*u)^2 /(4*d))*u *(2*d*sqrt(z)+c*u) +2*d *gamma_incomplete((j+i+2)/2, -(2*d*sqrt(z)+c*u)^2 /(4*d)) *sqrt((2*d*sqrt(z)+c*u)^2/d)) /(-(2*d*sqrt(z)+c*u)^2/d)^((j+i+1)/2),i,0,j),j,0, n))*(%e^(c*sqrt(z)+g))^u*%e^(-c*u*sqrt(z)-c^2*u^2/(4*d)) /d^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(d*z+e)*(%e^(c*z^r+g))^u -------------------------- */ /* Problem 688: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z)*(%e^(g+c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*%e^(-z*(c*u*z+d)+d*z-d^2/(4*c*u)+e)*(%e^(c*z^2+g))^u *('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d)^2/(4*c*u)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*(2*c*u*z+d)^(i+1) /(-(2*c*u*z+d)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 689: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z)*(%e^(g+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*d*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*d*sqrt(z)+c*u)^2 /(4*d))*u *(2*d*sqrt(z)+c*u) +2*d *gamma_incomplete((j+i+2)/2, -(2*d*sqrt(z)+c*u)^2 /(4*d)) *sqrt((2*d*sqrt(z)+c*u)^2/d)) /(-(2*d*sqrt(z)+c*u)^2/d)^((j+i+1)/2),i,0,j),j,0, n))*(%e^(c*sqrt(z)+g))^u*%e^(-c*u*sqrt(z)-c^2*u^2/(4*d)+e) /d^(2*(n+1)))); 0; /* ----- Involving z^(v-1)*%e^(b*z^r)*(%e^(c*z^r+g))^u ---------------------- */ /* Problem 690: NEW 7-1 */ diffcheck(integrate(z^(v-1)*%e^(b*z^r)*(%e^(g+c*z^r))^u,z),z,factor); z^(v-1)*%e^(b*z^r)*(%e^(g+c*z^r))^u; /* Problem 691: NEW 7-1 */ diffcheck(integrate(z^n*%e^(b*z^2)*(%e^(g+c*z^2))^u,z),z,factor); z^n*%e^(b*z^2)*(%e^(g+c*z^2))^u; /* Problem 692: NEW 7-1 */ diffcheck(integrate(z^(2*n)*%e^(b*z^2)*(%e^(g+c*z^2))^u,z),z,factor); z^(2*n)*%e^(b*z^2)*(%e^(g+c*z^2))^u; /* Problem 693: NEW 7-1 */ diffcheck(integrate(z^(1+2*n)*%e^(b*z^2)*(%e^(g+c*z^2))^u,z),z,factor); z^(1+2*n)*%e^(b*z^2)*(%e^(g+c*z^2))^u; /* Problem 694: NEW 7-1*/ diffcheck(integrate(z^n*%e^(b*sqrt(z))*(%e^(g+c*sqrt(z)))^u,z),z,factor); z^n*%e^(b*sqrt(z))*(%e^(g+c*sqrt(z)))^u; /* ----- Involving z^(v-1)*%e^(b*z^r+e)*(%e^(c*z^r+g))^u -------------------- */ /* Problem 695: NEW 7-1 */ diffcheck(integrate(z^(v-1)*%e^(e+b*z^r)*(%e^(g+c*z^r))^u,z),z,factor); z^(v-1)*%e^(e+b*z^r)*(%e^(g+c*z^r))^u; /* Problem 696: NEW 7-1 */ diffcheck(integrate(z^n*%e^(e+b*z^2)*(%e^(g+c*z^2))^u,z),z,factor); z^n*%e^(e+b*z^2)*(%e^(g+c*z^2))^u; /* Problem 697: NEW 7-1 */ diffcheck(integrate(z^(2*n)*%e^(e+b*z^2)*(%e^(g+c*z^2))^u,z),z,factor); z^(2*n)*%e^(e+b*z^2)*(%e^(g+c*z^2))^u; /* Problem 698: NEW 7-1 */ diffcheck(integrate(z^(1+2*n)*%e^(e+b*z^2)*(%e^(g+c*z^2))^u,z),z,factor); z^(1+2*n)*%e^(e+b*z^2)*(%e^(g+c*z^2))^u; /* Problem 699: NEW 7-1 */ diffcheck(integrate(z^n*%e^(e+b*sqrt(z))*(%e^(g+c*sqrt(z)))^u,z),z,factor); z^n*%e^(e+b*sqrt(z))*(%e^(g+c*sqrt(z)))^u; /* ----- Involving z^n*%e^(b*z^r+d*z)*(%e^(c*z^r+g))^u ---------------------- */ /* Problem 700: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(d*z+b*z^2)*(%e^(g+c*z^2))^u,z))- (-(c*u+b)^(-n-1)*%e^(-z*(c*u*z+b*z+d)+z*(b*z+d)-d^2/(4*(c*u+b))) *(%e^(c*z^2+g))^u *('sum(d^(n-i)*gamma_incomplete((i+1)/2, -(2*(c*u+b)*z+d)^2/(4*(c*u+b))) *(-1)^(i-n)*2^(i-n)*binomial(n,i) *(2*(c*u+b)*z+d)^(i+1) /((2*(c*u+b)*z+d)^2/(-c*u-b))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 701: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(d*z+b*sqrt(z))*(%e^(g+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i) *(2*d*sqrt(z)+c*u+b)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*d*sqrt(z)+c*u+b)^2 /(4*d)) *(c*u+b)*(2*d*sqrt(z)+c*u+b) +2*d *gamma_incomplete((j+i+2)/2, -(2*d*sqrt(z)+c*u+b) ^2 /(4*d)) *sqrt((2*d*sqrt(z)+c*u+b)^2/d)) /(-(2*d*sqrt(z)+c*u+b)^2/d)^((j+i+1)/2),i,0,j),j,0, n))*(%e^(c*sqrt(z)+g))^u*%e^(-c*u*sqrt(z)-(c*u+b)^2/(4*d)) /d^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(b*z^r+d*z+e)*(%e^(c*z^r+g))^u -------------------- */ /* Problem 702: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z+b*z^2)*(%e^(g+c*z^2))^u,z))- (-(c*u+b)^(-n-1)*%e^(-z*(c*u*z+b*z+d)+z*(b*z+d)-d^2/(4*(c*u+b))+e) *(%e^(c*z^2+g))^u *('sum(d^(n-i)*gamma_incomplete((i+1)/2, -(2*(c*u+b)*z+d)^2/(4*(c*u+b))) *(-1)^(i-n)*2^(i-n)*binomial(n,i) *(2*(c*u+b)*z+d)^(i+1) /((2*(c*u+b)*z+d)^2/(-c*u-b))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 703: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z+b*sqrt(z))*(%e^(g+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i) *(2*d*sqrt(z)+c*u+b)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*d*sqrt(z)+c*u+b)^2 /(4*d)) *(c*u+b)*(2*d*sqrt(z)+c*u+b) +2*d *gamma_incomplete((j+i+2)/2, -(2*d*sqrt(z)+c*u+b) ^2 /(4*d)) *sqrt((2*d*sqrt(z)+c*u+b)^2/d)) /(-(2*d*sqrt(z)+c*u+b)^2/d)^((j+i+1)/2),i,0,j),j,0, n))*(%e^(c*sqrt(z)+g))^u*%e^(-c*u*sqrt(z)-(c*u+b)^2/(4*d)+e) /d^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(d*z)*(%e^(c*z^r+f*z))^u -------------------------- */ /* Problem 704: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(d*z)*(%e^(f*z+c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*(%e^(z*(c*z+f)))^u *%e^(-z*(c*u*z+f*u+d)+d*z-(f*u+d)^2/(4*c*u)) *('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d)^2/(4*c*u)) *2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i)*(2*c*u*z+f*u+d)^(i+1) /(-(2*c*u*z+f*u+d)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 705: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(d*z)*(%e^(f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*(f*u+d)*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*(f*u+d)*sqrt(z) +c*u) ^2 /(4*(f*u+d)))*u *(2*(f*u+d)*sqrt(z)+c*u) +2 *gamma_incomplete((j+i+2)/2, -(2*(f*u+d)*sqrt(z) +c*u) ^2 /(4*(f*u+d))) *(f*u+d) *sqrt( (2*(f*u+d)*sqrt(z)+c*u)^2/(f*u+d))) /((2*(f*u+d)*sqrt(z)+c*u)^2/(-f*u-d))^((j+i+1)/2), i,0,j),j,0,n))*(%e^(f*z+c*sqrt(z)))^u *%e^(-f*u*z-c*u*sqrt(z)-c^2*u^2/(4*(f*u+d))) /(f*u+d)^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(d*z+e)*(%e^(c*z^r+f*z))^u ------------------------ */ /* Problem 706: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z)*(%e^(f*z+c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*(%e^(z*(c*z+f)))^u *%e^(-z*(c*u*z+f*u+d)+d*z-(f*u+d)^2/(4*c*u)+e) *('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d)^2/(4*c*u)) *2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i)*(2*c*u*z+f*u+d)^(i+1) /(-(2*c*u*z+f*u+d)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 707: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z)*(%e^(f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*(f*u+d)*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*(f*u+d)*sqrt(z) +c*u) ^2 /(4*(f*u+d)))*u *(2*(f*u+d)*sqrt(z)+c*u) +2 *gamma_incomplete((j+i+2)/2, -(2*(f*u+d)*sqrt(z) +c*u) ^2 /(4*(f*u+d))) *(f*u+d) *sqrt( (2*(f*u+d)*sqrt(z)+c*u)^2/(f*u+d))) /((2*(f*u+d)*sqrt(z)+c*u)^2/(-f*u-d))^((j+i+1)/2), i,0,j),j,0,n))*(%e^(f*z+c*sqrt(z)))^u *%e^(-f*u*z-c*u*sqrt(z)-c^2*u^2/(4*(f*u+d))+e) /(f*u+d)^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(b*z^r)*(%e^(c*z^r+f*z))^u ------------------------ */ /* Problem 708: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(b*z^2)*(%e^(f*z+c*z^2))^u,z))- (-(c*u+b)^(-n-1)*(%e^(z*(c*z+f)))^u *%e^(b*z^2-z*(c*u*z+b*z+f*u)-f^2*u^2/(4*(c*u+b))) *('sum(f^(n-i)*gamma_incomplete((i+1)/2, -(2*(c*u+b)*z+f*u)^2 /(4*(c*u+b)))*(-1)^(i-n) *2^(i-n)*binomial(n,i)*u^(n-i) *(2*(c*u+b)*z+f*u)^(i+1) /((2*(c*u+b)*z+f*u)^2/(-c*u-b))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 709: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(b*sqrt(z))*(%e^(f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i) *(2*f*u*sqrt(z)+c*u+b)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*f*u*sqrt(z)+c*u+b) ^2 /(4*f*u)) *(c*u+b)*(2*f*u*sqrt(z)+c*u+b) +2*f *gamma_incomplete((j+i+2)/2, -(2*f*u*sqrt(z) +c*u+b) ^2 /(4*f*u))*u *sqrt((2*f*u*sqrt(z)+c*u+b)^2/(f*u))) /(-(2*f*u*sqrt(z)+c*u+b)^2/(f*u))^((j+i+1)/2),i,0, j),j,0,n))*(%e^(f*z+c*sqrt(z)))^u *%e^(-f*u*z-c*u*sqrt(z)-(c*u+b)^2/(4*f*u)) /(f^(2*(n+1))*u^(2*(n+1))))); 0; /* ----- Involving z^n*%e^(b*z^r+e)*(%e^(c*z^r+f*z))^u ---------------------- */ /* Problem 710: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+b*z^2)*(%e^(f*z+c*z^2))^u,z))- (-(c*u+b)^(-n-1)*(%e^(z*(c*z+f)))^u *%e^(b*z^2-z*(c*u*z+b*z+f*u)-f^2*u^2/(4*(c*u+b))+e) *('sum(f^(n-i)*gamma_incomplete((i+1)/2, -(2*(c*u+b)*z+f*u)^2 /(4*(c*u+b)))*(-1)^(i-n) *2^(i-n)*binomial(n,i)*u^(n-i) *(2*(c*u+b)*z+f*u)^(i+1) /((2*(c*u+b)*z+f*u)^2/(-c*u-b))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 711: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+b*sqrt(z))*(%e^(f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i) *(2*f*u*sqrt(z)+c*u+b)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*f*u*sqrt(z)+c*u+b) ^2 /(4*f*u)) *(c*u+b)*(2*f*u*sqrt(z)+c*u+b) +2*f *gamma_incomplete((j+i+2)/2, -(2*f*u*sqrt(z) +c*u+b) ^2 /(4*f*u))*u *sqrt((2*f*u*sqrt(z)+c*u+b)^2/(f*u))) /(-(2*f*u*sqrt(z)+c*u+b)^2/(f*u))^((j+i+1)/2),i,0, j),j,0,n))*(%e^(f*z+c*sqrt(z)))^u *%e^(-f*u*z-c*u*sqrt(z)-(c*u+b)^2/(4*f*u)+e) /(f^(2*(n+1))*u^(2*(n+1))))); 0; /* ----- Involving z^n*%e^(b*z^r+d*z)*(%e^(c*z^r+f*z))^u -------------------- */ /* Problem 712: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(d*z+b*z^2)*(%e^(f*z+c*z^2))^u,z))- (-(c*u+b)^(-n-1)*(%e^(z*(c*z+f)))^u *%e^(-z*(c*u*z+b*z+f*u+d)+z*(b*z+d)-(f*u+d)^2/(4*(c*u+b))) *('sum(gamma_incomplete((i+1)/2, -(2*(c*u+b)*z+f*u+d)^2/(4*(c*u+b))) *2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i) *(2*(c*u+b)*z+f*u+d)^(i+1) /((2*(c*u+b)*z+f*u+d)^2/(-c*u-b))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 713: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(d*z+b*sqrt(z))*(%e^(f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i) *(2*(f*u+d)*sqrt(z)+c*u+b)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*(f*u+d)*sqrt(z) +c*u+b) ^2 /(4*(f*u+d))) *(c*u+b)*(2*(f*u+d)*sqrt(z)+c*u+b) +2*gamma_incomplete((j+i+2)/2, -(2*(f*u+d)*sqrt(z) +c*u+b) ^2 /(4*(f*u+d))) *(f*u+d) *sqrt( (2*(f*u+d)*sqrt(z)+c*u+b)^2/(f*u+d))) /((2*(f*u+d)*sqrt(z)+c*u+b)^2/(-f*u-d)) ^((j+i+1)/2),i,0,j),j,0,n))*(%e^(f*z+c*sqrt(z)))^u *%e^(-f*u*z-c*u*sqrt(z)-(c*u+b)^2/(4*(f*u+d))) /(f*u+d)^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(b*z^r+d*z+e)*(%e^(c*z^r+f*z))^u ------------------ */ /* Problem 714: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z+b*z^2)*(%e^(f*z+c*z^2))^u,z))- (-(c*u+b)^(-n-1)*(%e^(z*(c*z+f)))^u *%e^(-z*(c*u*z+b*z+f*u+d)+z*(b*z+d)-(f*u+d)^2/(4*(c*u+b))+e) *('sum(gamma_incomplete((i+1)/2, -(2*(c*u+b)*z+f*u+d)^2/(4*(c*u+b))) *2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i) *(2*(c*u+b)*z+f*u+d)^(i+1) /((2*(c*u+b)*z+f*u+d)^2/(-c*u-b))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 715: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z+b*sqrt(z))*(%e^(f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i) *(2*(f*u+d)*sqrt(z)+c*u+b)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*(f*u+d)*sqrt(z) +c*u+b) ^2 /(4*(f*u+d))) *(c*u+b)*(2*(f*u+d)*sqrt(z)+c*u+b) +2*gamma_incomplete((j+i+2)/2, -(2*(f*u+d)*sqrt(z) +c*u+b) ^2 /(4*(f*u+d))) *(f*u+d) *sqrt( (2*(f*u+d)*sqrt(z)+c*u+b)^2/(f*u+d))) /((2*(f*u+d)*sqrt(z)+c*u+b)^2/(-f*u-d)) ^((j+i+1)/2),i,0,j),j,0,n))*(%e^(f*z+c*sqrt(z)))^u *%e^(-f*u*z-c*u*sqrt(z)-(c*u+b)^2/(4*(f*u+d))+e) /(f*u+d)^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(d*z)*(%e^(c*z^r+f*z+g))^u ------------------------ */ /* Problem 716: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(d*z)*(%e^(g+f*z+c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*(%e^(z*(c*z+f)+g))^u *%e^(-z*(c*u*z+f*u+d)+d*z-(f*u+d)^2/(4*c*u)) *('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d)^2/(4*c*u)) *2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i)*(2*c*u*z+f*u+d)^(i+1) /(-(2*c*u*z+f*u+d)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 717: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(d*z)*(%e^(g+f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*(f*u+d)*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*(f*u+d)*sqrt(z) +c*u) ^2 /(4*(f*u+d)))*u *(2*(f*u+d)*sqrt(z)+c*u) +2 *gamma_incomplete((j+i+2)/2, -(2*(f*u+d)*sqrt(z) +c*u) ^2 /(4*(f*u+d))) *(f*u+d) *sqrt( (2*(f*u+d)*sqrt(z)+c*u)^2/(f*u+d))) /((2*(f*u+d)*sqrt(z)+c*u)^2/(-f*u-d))^((j+i+1)/2), i,0,j),j,0,n))*(%e^(f*z+c*sqrt(z)+g))^u *%e^(-f*u*z-c*u*sqrt(z)-c^2*u^2/(4*(f*u+d))) /(f*u+d)^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(d*z+e)*(%e^(c*z^r+f*z+g))^u ---------------------- */ /* Problem 718: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z)*(%e^(g+f*z+c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*(%e^(z*(c*z+f)+g))^u *%e^(-z*(c*u*z+f*u+d)+d*z-(f*u+d)^2/(4*c*u)+e) *('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d)^2/(4*c*u)) *2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i)*(2*c*u*z+f*u+d)^(i+1) /(-(2*c*u*z+f*u+d)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 719: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z)*(%e^(g+f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*(f*u+d)*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*(f*u+d)*sqrt(z) +c*u) ^2 /(4*(f*u+d)))*u *(2*(f*u+d)*sqrt(z)+c*u) +2 *gamma_incomplete((j+i+2)/2, -(2*(f*u+d)*sqrt(z) +c*u) ^2 /(4*(f*u+d))) *(f*u+d) *sqrt( (2*(f*u+d)*sqrt(z)+c*u)^2/(f*u+d))) /((2*(f*u+d)*sqrt(z)+c*u)^2/(-f*u-d))^((j+i+1)/2), i,0,j),j,0,n))*(%e^(f*z+c*sqrt(z)+g))^u *%e^(-f*u*z-c*u*sqrt(z)-c^2*u^2/(4*(f*u+d))+e) /(f*u+d)^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(b*z^r)*(%e^(c*z^r+f*z+g))^u ---------------------- */ /* Problem 720: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(b*z^2)*(%e^(g+f*z+c*z^2))^u,z))- (-(c*u+b)^(-n-1)*(%e^(z*(c*z+f)+g))^u *%e^(b*z^2-z*(c*u*z+b*z+f*u)-f^2*u^2/(4*(c*u+b))) *('sum(f^(n-i)*gamma_incomplete((i+1)/2, -(2*(c*u+b)*z+f*u)^2 /(4*(c*u+b)))*(-1)^(i-n) *2^(i-n)*binomial(n,i)*u^(n-i) *(2*(c*u+b)*z+f*u)^(i+1) /((2*(c*u+b)*z+f*u)^2/(-c*u-b))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 721: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(b*sqrt(z))*(%e^(g+f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i) *(2*f*u*sqrt(z)+c*u+b)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*f*u*sqrt(z)+c*u+b) ^2 /(4*f*u)) *(c*u+b)*(2*f*u*sqrt(z)+c*u+b) +2*f *gamma_incomplete((j+i+2)/2, -(2*f*u*sqrt(z) +c*u+b) ^2 /(4*f*u))*u *sqrt((2*f*u*sqrt(z)+c*u+b)^2/(f*u))) /(-(2*f*u*sqrt(z)+c*u+b)^2/(f*u))^((j+i+1)/2),i,0, j),j,0,n))*(%e^(f*z+c*sqrt(z)+g))^u *%e^(-f*u*z-c*u*sqrt(z)-(c*u+b)^2/(4*f*u)) /(f^(2*(n+1))*u^(2*(n+1))))); 0; /* ----- Involving z^n*%e^(b*z^r+e)*(%e^(c*z^r+f*z+g))^u -------------------- */ /* Problem 722: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+b*z^2)*(%e^(g+f*z+c*z^2))^u,z))- (-(c*u+b)^(-n-1)*(%e^(z*(c*z+f)+g))^u *%e^(b*z^2-z*(c*u*z+b*z+f*u)-f^2*u^2/(4*(c*u+b))+e) *('sum(f^(n-i)*gamma_incomplete((i+1)/2, -(2*(c*u+b)*z+f*u)^2 /(4*(c*u+b)))*(-1)^(i-n) *2^(i-n)*binomial(n,i)*u^(n-i) *(2*(c*u+b)*z+f*u)^(i+1) /((2*(c*u+b)*z+f*u)^2/(-c*u-b))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 723: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+b*sqrt(z))*(%e^(g+f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i) *(2*f*u*sqrt(z)+c*u+b)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*f*u*sqrt(z)+c*u+b) ^2 /(4*f*u)) *(c*u+b)*(2*f*u*sqrt(z)+c*u+b) +2*f *gamma_incomplete((j+i+2)/2, -(2*f*u*sqrt(z) +c*u+b) ^2 /(4*f*u))*u *sqrt((2*f*u*sqrt(z)+c*u+b)^2/(f*u))) /(-(2*f*u*sqrt(z)+c*u+b)^2/(f*u))^((j+i+1)/2),i,0, j),j,0,n))*(%e^(f*z+c*sqrt(z)+g))^u *%e^(-f*u*z-c*u*sqrt(z)-(c*u+b)^2/(4*f*u)+e) /(f^(2*(n+1))*u^(2*(n+1))))); 0; /* ----- Involving z^n*%e^(b*z^r+d*z)*(%e^(c*z^r+f*z+g))^u ------------------ */ /* Problem 724: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(d*z+b*z^2)*(%e^(g+f*z+c*z^2))^u,z))- (-(c*u+b)^(-n-1)*(%e^(z*(c*z+f)+g))^u *%e^(-z*(c*u*z+b*z+f*u+d)+z*(b*z+d)-(f*u+d)^2/(4*(c*u+b))) *('sum(gamma_incomplete((i+1)/2, -(2*(c*u+b)*z+f*u+d)^2/(4*(c*u+b))) *2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i) *(2*(c*u+b)*z+f*u+d)^(i+1) /((2*(c*u+b)*z+f*u+d)^2/(-c*u-b))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 725: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(d*z+b*sqrt(z))*(%e^(g+f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i) *(2*(f*u+d)*sqrt(z)+c*u+b)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*(f*u+d)*sqrt(z) +c*u+b) ^2 /(4*(f*u+d))) *(c*u+b)*(2*(f*u+d)*sqrt(z)+c*u+b) +2*gamma_incomplete((j+i+2)/2, -(2*(f*u+d)*sqrt(z) +c*u+b) ^2 /(4*(f*u+d))) *(f*u+d) *sqrt( (2*(f*u+d)*sqrt(z)+c*u+b)^2/(f*u+d))) /((2*(f*u+d)*sqrt(z)+c*u+b)^2/(-f*u-d)) ^((j+i+1)/2),i,0,j),j,0,n)) *(%e^(f*z+c*sqrt(z)+g))^u *%e^(-f*u*z-c*u*sqrt(z)-(c*u+b)^2/(4*(f*u+d))) /(f*u+d)^(2*(n+1)))); 0; /* ----- Involving z^n*%e^(b*z^r+d*z+e)*(%e^(c*z^r+f*z+g))^u ---------------- */ /* Problem 726: NEW 9-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z+b*z^2)*(%e^(g+f*z+c*z^2))^u,z))- (-(c*u+b)^(-n-1)*(%e^(z*(c*z+f)+g))^u *%e^(-z*(c*u*z+b*z+f*u+d)+z*(b*z+d)-(f*u+d)^2/(4*(c*u+b))+e) *('sum(gamma_incomplete((i+1)/2, -(2*(c*u+b)*z+f*u+d)^2/(4*(c*u+b))) *2^(i-n)*binomial(n,i)*(-f*u-d)^(n-i) *(2*(c*u+b)*z+f*u+d)^(i+1) /((2*(c*u+b)*z+f*u+d)^2/(-c*u-b))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 727: NEW 10-1 */ expand(niceindices(integrate(z^n*%e^(e+d*z+b*sqrt(z))*(%e^(g+f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b)^(2*n-j-i) *(2*(f*u+d)*sqrt(z)+c*u+b)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*(f*u+d)*sqrt(z) +c*u+b) ^2 /(4*(f*u+d))) *(c*u+b)*(2*(f*u+d)*sqrt(z)+c*u+b) +2*gamma_incomplete((j+i+2)/2, -(2*(f*u+d)*sqrt(z) +c*u+b) ^2 /(4*(f*u+d))) *(f*u+d) *sqrt( (2*(f*u+d)*sqrt(z)+c*u+b)^2/(f*u+d))) /((2*(f*u+d)*sqrt(z)+c*u+b)^2/(-f*u-d)) ^((j+i+1)/2),i,0,j),j,0,n)) *(%e^(f*z+c*sqrt(z)+g))^u *%e^(-f*u*z-c*u*sqrt(z)-(c*u+b)^2/(4*(f*u+d))+e) /(f*u+d)^(2*(n+1)))); 0; /* ----- Involving z^(v-1)*(%e^(c*z))^q*(%e^(a*z))^u ------------------------ */ /* Problem 728: NEW 7-1 */ diffcheck(integrate(z^(v-1)*(%e^(d*z))^q*(%e^(f*z))^u,z),z,factor); z^(v-1)*(%e^(d*z))^q*(%e^(f*z))^u; /* Problem 729: NEW 7-1 */ diffcheck(integrate(z^n*(%e^(d*z))^q*(%e^(f*z))^u,z),z,factor); z^n*(%e^(d*z))^q*(%e^(f*z))^u; /* Problem 730: NEW 7-1 */ diffcheck(integrate(z^(1/2+n)*(%e^(d*z))^q*(%e^(f*z))^u,z),z,factor); z^(1/2+n)*(%e^(d*z))^q*(%e^(f*z))^u; /* ----- Involving z^(v-1)*(%e^(c*z))^q*(%e^(a*z+b))^u ---------------------- */ /* Problem 731: NEW 7-1 */ diffcheck(integrate(z^(v-1)*(%e^(d*z))^q*(%e^(g+f*z))^u,z),z,factor); z^(v-1)*(%e^(d*z))^q*(%e^(g+f*z))^u; /* Problem 732: NEW 7-1 */ diffcheck(integrate(z^n*(%e^(d*z))^q*(%e^(g+f*z))^u,z),z,factor); z^n*(%e^(d*z))^q*(%e^(g+f*z))^u; /* Problem 733: NEW 7-1 */ diffcheck(integrate(z^(1/2+n)*(%e^(d*z))^q*(%e^(g+f*z))^u,z),z,factor); z^(1/2+n)*(%e^(d*z))^q*(%e^(g+f*z))^u; /* ----- Involving z^(v-1)*(%e^(c*z+d))^q*(%e^(a*z+b))^u -------------------- */ /* Problem 734: NEW 7-1 */ diffcheck(integrate(z^(v-1)*(%e^(e+d*z))^q*(%e^(g+f*z))^u,z),z,factor); z^(v-1)*(%e^(e+d*z))^q*(%e^(g+f*z))^u; /* Problem 735: NEW 7-1 */ diffcheck(integrate(z^n*(%e^(e+d*z))^q*(%e^(g+f*z))^u,z),z,factor); z^n*(%e^(e+d*z))^q*(%e^(g+f*z))^u; /* Problem 736: NEW 7-1 */ diffcheck(integrate(z^(1/2+n)*(%e^(e+d*z))^q*(%e^(g+f*z))^u,z),z,factor); z^(1/2+n)*(%e^(e+d*z))^q*(%e^(g+f*z))^u; /* ----- Involving z^n*(%e^(d*z))^q*(%e^(c*z^r))^u -------------------------- */ /* Problem 737: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(d*z))^q*(%e^(c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*(%e^(d*z))^q*%e^(-z*(c*u*z+d*q)-d^2*q^2/(4*c*u)) *(%e^(c*z^2))^u *('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d*q)^2/(4*c*u)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*q^(n-i) *(2*c*u*z+d*q)^(i+1) /(-(2*c*u*z+d*q)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 738: NEW 10-1 */ expand(niceindices(integrate(z^n*(%e^(d*z))^q*(%e^(c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*d*q*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*d*q*sqrt(z)+c*u)^2 /(4*d*q))*u *(2*d*q*sqrt(z)+c*u) +2*d *gamma_incomplete((j+i+2)/2, -(2*d*q*sqrt(z)+c*u) ^2 /(4*d*q))*q *sqrt((2*d*q*sqrt(z)+c*u)^2/(d*q))) /(-(2*d*q*sqrt(z)+c*u)^2/(d*q))^((j+i+1)/2),i,0,j), j,0,n))*(%e^(c*sqrt(z)))^u*(%e^(d*z))^q *%e^(-d*q*z-c*u*sqrt(z)-c^2*u^2/(4*d*q)) /(d^(2*(n+1))*q^(2*(n+1))))); 0; /* ----- Involving z^n*(%e^(d*z+e))^q*(%e^(c*z^r))^u ------------------------ */ /* Problem 739: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(e+d*z))^q*(%e^(c*z^2))^u,z)) - (-c^(-n-1)*u^(-n-1)*(%e^(d*z+e))^q*%e^(-z*(c*u*z+d*q)-d^2*q^2/(4*c*u)) *(%e^(c*z^2))^u *('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d*q)^2/(4*c*u)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*q^(n-i) *(2*c*u*z+d*q)^(i+1) /(-(2*c*u*z+d*q)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 740: NEW 10-1 */ expand(niceindices(integrate(z^n*(%e^(e+d*z))^q*(%e^(c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*d*q*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*d*q*sqrt(z)+c*u)^2 /(4*d*q))*u *(2*d*q*sqrt(z)+c*u) +2*d *gamma_incomplete((j+i+2)/2, -(2*d*q*sqrt(z)+c*u) ^2 /(4*d*q))*q *sqrt((2*d*q*sqrt(z)+c*u)^2/(d*q))) /(-(2*d*q*sqrt(z)+c*u)^2/(d*q))^((j+i+1)/2),i,0,j), j,0,n))*(%e^(c*sqrt(z)))^u*(%e^(d*z+e))^q *%e^(-d*q*z-c*u*sqrt(z)-c^2*u^2/(4*d*q)) /(d^(2*(n+1))*q^(2*(n+1))))); 0; /* ----- Involving z^(v-1)*(%e^(b*z^r))^q*(%e^(c*z^r))^u -------------------- */ /* Problem 741: NEW 7-1 */ diffcheck(integrate(z^(v-1)*(%e^(b*z^r))^q*(%e^(c*z^r))^u,z),z,factor); z^(v-1)*(%e^(b*z^r))^q*(%e^(c*z^r))^u; /* Problem 742: NEW 7-1 */ diffcheck(integrate(z^n*(%e^(b*z^2))^q*(%e^(c*z^2))^u,z),z,factor); z^n*(%e^(b*z^2))^q*(%e^(c*z^2))^u; /* Problem 743: NEW 7-1 */ diffcheck(integrate(z^(2*n)*(%e^(b*z^2))^q*(%e^(c*z^2))^u,z),z,factor); z^(2*n)*(%e^(b*z^2))^q*(%e^(c*z^2))^u; /* Problem 744: NEW 7-1 */ diffcheck(integrate(z^(1+2*n)*(%e^(b*z^2))^q*(%e^(c*z^2))^u,z),z,factor); z^(1+2*n)*(%e^(b*z^2))^q*(%e^(c*z^2))^u; /* Problem 745: NEW 7-1 */ diffcheck(integrate(z^n*(%e^(b*sqrt(z)))^q*(%e^(c*sqrt(z)))^u,z),z,factor); z^n*(%e^(b*sqrt(z)))^q*(%e^(c*sqrt(z)))^u; /* ----- Involving z^n*(%e^(d*z)^q*(%e^(c*z^r+g))^u ------------------------- */ /* Problem 746: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(d*z))^q*(%e^(g+c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*(%e^(d*z))^q*%e^(-z*(c*u*z+d*q)-d^2*q^2/(4*c*u)) *(%e^(c*z^2+g))^u *('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d*q)^2/(4*c*u)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*q^(n-i) *(2*c*u*z+d*q)^(i+1) /(-(2*c*u*z+d*q)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 747: NEW 10-1 */ expand(niceindices(integrate(z^n*(%e^(d*z))^q*(%e^(g+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*d*q*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*d*q*sqrt(z)+c*u)^2 /(4*d*q))*u *(2*d*q*sqrt(z)+c*u) +2*d *gamma_incomplete((j+i+2)/2, -(2*d*q*sqrt(z)+c*u) ^2 /(4*d*q))*q *sqrt((2*d*q*sqrt(z)+c*u)^2/(d*q))) /(-(2*d*q*sqrt(z)+c*u)^2/(d*q))^((j+i+1)/2),i,0,j), j,0,n))*(%e^(c*sqrt(z)+g))^u*(%e^(d*z))^q *%e^(-d*q*z-c*u*sqrt(z)-c^2*u^2/(4*d*q)) /(d^(2*(n+1))*q^(2*(n+1))))); 0; /* ----- Involving z^n*(%e^(d*z+e)^q*(%e^(c*z^r+g))^u ----------------------- */ /* Problem 748: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(e+d*z))^q*(%e^(g+c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*(%e^(d*z+e))^q*%e^(-z*(c*u*z+d*q)-d^2*q^2/(4*c*u)) *(%e^(c*z^2+g))^u *('sum(d^(n-i)*gamma_incomplete((i+1)/2,-(2*c*u*z+d*q)^2/(4*c*u)) *(-1)^(i-n)*2^(i-n)*binomial(n,i)*q^(n-i) *(2*c*u*z+d*q)^(i+1) /(-(2*c*u*z+d*q)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 749: NEW 10-1 */ expand(niceindices(integrate(z^n*(%e^(e+d*z))^q*(%e^(g+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*d*q*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*d*q*sqrt(z)+c*u)^2 /(4*d*q))*u *(2*d*q*sqrt(z)+c*u) +2*d *gamma_incomplete((j+i+2)/2, -(2*d*q*sqrt(z)+c*u) ^2 /(4*d*q))*q *sqrt((2*d*q*sqrt(z)+c*u)^2/(d*q))) /(-(2*d*q*sqrt(z)+c*u)^2/(d*q))^((j+i+1)/2),i,0,j), j,0,n))*(%e^(c*sqrt(z)+g))^u*(%e^(d*z+e))^q *%e^(-d*q*z-c*u*sqrt(z)-c^2*u^2/(4*d*q)) /(d^(2*(n+1))*q^(2*(n+1))))); 0; /* ----- Involving z^(v-1)*(%e^(b*z^r)^q*(%e^(c*z^r+g))^u ------------------- */ /* Problem 750: NEW 7-1 */ diffcheck(integrate(z^(v-1)*(%e^(b*z^r))^q*(%e^(g+c*z^r))^u,z),z,factor); z^(v-1)*(%e^(b*z^r))^q*(%e^(g+c*z^r))^u; /* Problem 751: NEW 7-1 */ diffcheck(integrate(z^n*(%e^(b*z^2))^q*(%e^(g+c*z^2))^u,z),z,factor); z^n*(%e^(b*z^2))^q*(%e^(g+c*z^2))^u; /* Problem 752: NEW 7-1 */ diffcheck(integrate(z^(2*n)*(%e^(b*z^2))^q*(%e^(g+c*z^2))^u,z),z,factor); z^(2*n)*(%e^(b*z^2))^q*(%e^(g+c*z^2))^u; /* Problem 753: NEW 7-1 */ diffcheck(integrate(z^(1+2*n)*(%e^(b*z^2))^q*(%e^(g+c*z^2))^u,z),z,factor); z^(1+2*n)*(%e^(b*z^2))^q*(%e^(g+c*z^2))^u; /* Problem 754: NEW 7-1 */ diffcheck(integrate(z^n*(%e^(b*sqrt(z)))^q*(%e^(g+c*sqrt(z)))^u,z),z,factor); z^n*(%e^(b*sqrt(z)))^q*(%e^(g+c*sqrt(z)))^u; /* ----- Involving z^(v-1)*(%e^(b*z^r+e)^q*(%e^(c*z^r+g))^u ----------------- */ /* Problem 755: NEW 7-1 */ diffcheck(integrate(z^(v-1)*(%e^(e+b*z^r))^q*(%e^(g+c*z^r))^u,z),z,factor); z^(v-1)*(%e^(e+b*z^r))^q*(%e^(g+c*z^r))^u; /* Problem 756: NEW 7-1 */ diffcheck(integrate(z^n*(%e^(e+b*z^2))^q*(%e^(g+c*z^2))^u,z),z,factor); z^n*(%e^(e+b*z^2))^q*(%e^(g+c*z^2))^u; /* Problem 757: NEW 7-1 */ diffcheck(integrate(z^(2*n)*(%e^(e+b*z^2))^q*(%e^(g+c*z^2))^u,z),z,factor); z^(2*n)*(%e^(e+b*z^2))^q*(%e^(g+c*z^2))^u; /* Problem 758: NEW 7-1 */ diffcheck(integrate(z^(1+2*n)*(%e^(e+b*z^2))^q*(%e^(g+c*z^2))^u,z),z,factor); z^(1+2*n)*(%e^(e+b*z^2))^q*(%e^(g+c*z^2))^u; /* Problem 759: NEW 7-1 */ diffcheck(integrate(z^n*(%e^(e+b*sqrt(z)))^q*(%e^(g+c*sqrt(z)))^u,z),z,factor); z^n*(%e^(e+b*sqrt(z)))^q*(%e^(g+c*sqrt(z)))^u; /* ----- Involving z^n*(%e^(d*z)^q*(%e^(c*z^r+f*z))^u ----------------------- */ /* Problem 760: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(d*z))^q*(%e^(f*z+c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*(%e^(d*z))^q*(%e^(z*(c*z+f)))^u *%e^(-z*(c*u*z+f*u+d*q)-(f*u+d*q)^2/(4*c*u)) *('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d*q)^2/(4*c*u)) *2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i) *(2*c*u*z+f*u+d*q)^(i+1) /(-(2*c*u*z+f*u+d*q)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 761: NEW 10-1 */ expand(niceindices(integrate(z^n*(%e^(d*z))^q*(%e^(f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*(f*u+d*q)*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*(f*u+d*q)*sqrt(z) +c*u) ^2 /(4*(f*u+d*q)))*u *(2*(f*u+d*q)*sqrt(z)+c*u) +2 *gamma_incomplete((j+i+2)/2, -(2*(f*u+d*q)*sqrt(z) +c*u) ^2 /(4*(f*u+d*q))) *(f*u+d*q) *sqrt( (2*(f*u+d*q)*sqrt(z)+c*u)^2 /(f*u+d*q))) /((2*(f*u+d*q)*sqrt(z)+c*u)^2/(-f*u-d*q)) ^((j+i+1)/2),i,0,j),j,0,n))*(%e^(d*z))^q *(%e^(f*z+c*sqrt(z)))^u *%e^(-f*u*z-d*q*z-c*u*sqrt(z)-c^2*u^2/(4*(f*u+d*q))) /(f*u+d*q)^(2*(n+1)))); 0; /* ----- Involving z^n*(%e^(d*z+e)^q*(%e^(c*z^r+f*z))^u --------------------- */ /* Problem 762: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(e+d*z))^q*(%e^(f*z+c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*(%e^(z*(c*z+f)))^u*(%e^(d*z+e))^q *%e^(-z*(c*u*z+f*u+d*q)-(f*u+d*q)^2/(4*c*u)) *('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d*q)^2/(4*c*u)) *2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i) *(2*c*u*z+f*u+d*q)^(i+1) /(-(2*c*u*z+f*u+d*q)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 763: NEW 10-1 */ expand(niceindices(integrate(z^n*(%e^(e+d*z))^q*(%e^(f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*(f*u+d*q)*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*(f*u+d*q)*sqrt(z) +c*u) ^2 /(4*(f*u+d*q)))*u *(2*(f*u+d*q)*sqrt(z)+c*u) +2 *gamma_incomplete((j+i+2)/2, -(2*(f*u+d*q)*sqrt(z) +c*u) ^2 /(4*(f*u+d*q))) *(f*u+d*q) *sqrt( (2*(f*u+d*q)*sqrt(z)+c*u)^2 /(f*u+d*q))) /((2*(f*u+d*q)*sqrt(z)+c*u)^2/(-f*u-d*q)) ^((j+i+1)/2),i,0,j),j,0,n))*(%e^(d*z+e))^q *(%e^(f*z+c*sqrt(z)))^u *%e^(-f*u*z-d*q*z-c*u*sqrt(z)-c^2*u^2/(4*(f*u+d*q))) /(f*u+d*q)^(2*(n+1)))); 0; /* ----- Involving z^n*(%e^(b*z^r)^q*(%e^(c*z^r+f*z))^u --------------------- */ /* Problem 764: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(b*z^2))^q*(%e^(f*z+c*z^2))^u,z))- (-(c*u+b*q)^(-n-1)*(%e^(z*(c*z+f)))^u *%e^(-z*(c*u*z+b*q*z+f*u)-f^2*u^2/(4*(c*u+b*q))) *(%e^(b*z^2))^q *('sum(f^(n-i)*gamma_incomplete((i+1)/2, -(2*(c*u+b*q)*z+f*u)^2 /(4*(c*u+b*q)))*(-1)^(i-n) *2^(i-n)*binomial(n,i)*u^(n-i) *(2*(c*u+b*q)*z+f*u)^(i+1) /((2*(c*u+b*q)*z+f*u)^2/(-c*u-b*q))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 765: NEW 10-1 */ expand(niceindices(integrate(z^n*(%e^(b*sqrt(z)))^q*(%e^(f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b*q)^(2*n-j-i) *(2*f*u*sqrt(z)+c*u+b*q)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*f*u*sqrt(z) +c*u+b*q) ^2 /(4*f*u)) *(c*u+b*q)*(2*f*u*sqrt(z)+c*u+b*q) +2*f *gamma_incomplete((j+i+2)/2, -(2*f*u*sqrt(z) +c*u+b*q) ^2 /(4*f*u))*u *sqrt((2*f*u*sqrt(z)+c*u+b*q)^2/(f*u))) /(-(2*f*u*sqrt(z)+c*u+b*q)^2/(f*u))^((j+i+1)/2),i, 0,j),j,0,n))*(%e^(b*sqrt(z)))^q *(%e^(f*z+c*sqrt(z)))^u *%e^(-f*u*z-c*u*sqrt(z)-b*q*sqrt(z)-(c*u+b*q)^2/(4*f*u)) /(f^(2*(n+1))*u^(2*(n+1))))); 0; /* ----- Involving z^n*(%e^(b*z^r+e)^q*(%e^(c*z^r+f*z))^u ------------------- */ /* Problem 766: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(e+b*z^2))^q*(%e^(f*z+c*z^2))^u,z))- (-(c*u+b*q)^(-n-1)*(%e^(z*(c*z+f)))^u *%e^(-z*(c*u*z+b*q*z+f*u)-f^2*u^2/(4*(c*u+b*q))) *(%e^(b*z^2+e))^q *('sum(f^(n-i)*gamma_incomplete((i+1)/2, -(2*(c*u+b*q)*z+f*u)^2 /(4*(c*u+b*q)))*(-1)^(i-n) *2^(i-n)*binomial(n,i)*u^(n-i) *(2*(c*u+b*q)*z+f*u)^(i+1) /((2*(c*u+b*q)*z+f*u)^2/(-c*u-b*q))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 767: NEW 10-1 */ expand(niceindices(integrate(z^n*(%e^(e+b*sqrt(z)))^q*(%e^(f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b*q)^(2*n-j-i) *(2*f*u*sqrt(z)+c*u+b*q)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*f*u*sqrt(z) +c*u+b*q) ^2 /(4*f*u)) *(c*u+b*q)*(2*f*u*sqrt(z)+c*u+b*q) +2*f *gamma_incomplete((j+i+2)/2, -(2*f*u*sqrt(z) +c*u+b*q) ^2 /(4*f*u))*u *sqrt((2*f*u*sqrt(z)+c*u+b*q)^2/(f*u))) /(-(2*f*u*sqrt(z)+c*u+b*q)^2/(f*u))^((j+i+1)/2),i, 0,j),j,0,n))*(%e^(b*sqrt(z)+e))^q *(%e^(f*z+c*sqrt(z)))^u *%e^(-f*u*z-c*u*sqrt(z)-b*q*sqrt(z)-(c*u+b*q)^2/(4*f*u)) /(f^(2*(n+1))*u^(2*(n+1))))); 0; /* ----- Involving z^n*(%e^(b*z^r+d*z)^q*(%e^(c*z^r+f*z))^u ----------------- */ /* Problem 768: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(d*z+b*z^2))^q*(%e^(f*z+c*z^2))^u,z))- (-(c*u+b*q)^(-n-1)*(%e^(z*(b*z+d)))^q*(%e^(z*(c*z+f)))^u *%e^(-z*(c*u*z+b*q*z+f*u+d*q)-(f*u+d*q)^2/(4*(c*u+b*q))) *('sum(gamma_incomplete((i+1)/2, -(2*(c*u+b*q)*z+f*u+d*q)^2 /(4*(c*u+b*q))) *2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i) *(2*(c*u+b*q)*z+f*u+d*q)^(i+1) /((2*(c*u+b*q)*z+f*u+d*q)^2/(-c*u-b*q))^((i+1)/2),i, 0,n)) /2)); 0; /* Problem 769: NEW 10-1 */ expand(niceindices(integrate(z^n*(%e^(d*z+b*sqrt(z)))^q*(%e^(f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b*q)^(2*n-j-i) *(2*(f*u+d*q)*sqrt(z)+c*u+b*q)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*(f*u+d*q)*sqrt(z) +c*u+b*q) ^2 /(4*(f*u+d*q))) *(c*u+b*q)*(2*(f*u+d*q)*sqrt(z)+c*u+b*q) +2*gamma_incomplete((j+i+2)/2, -(2*(f*u+d*q)*sqrt(z) +c*u+b*q) ^2 /(4*(f*u+d*q))) *(f*u+d*q) *sqrt( (2*(f*u+d*q)*sqrt(z)+c*u+b*q)^2 /(f*u+d*q))) /((2*(f*u+d*q)*sqrt(z)+c*u+b*q)^2/(-f*u-d*q)) ^((j+i+1)/2),i,0,j),j,0,n))*(%e^(d*z+b*sqrt(z)))^q *(%e^(f*z+c*sqrt(z)))^u *%e^(-f*u*z-d*q*z-c*u*sqrt(z)-b*q*sqrt(z)-(c*u+b*q)^2/(4*(f*u+d*q))) /(f*u+d*q)^(2*(n+1)))); 0; /* ----- Involving z^n*(%e^(d*z)^q*(%e^(c*z^r+f*z+g))^u --------------------- */ /* Problem 770: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(d*z))^q*(%e^(g+f*z+c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*(%e^(d*z))^q*(%e^(z*(c*z+f)+g))^u *%e^(-z*(c*u*z+f*u+d*q)-(f*u+d*q)^2/(4*c*u)) *('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d*q)^2/(4*c*u)) *2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i) *(2*c*u*z+f*u+d*q)^(i+1) /(-(2*c*u*z+f*u+d*q)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 771: NEW 10-1 */ expand(niceindices(integrate(z^n*(%e^(d*z))^q*(%e^(g+f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*(f*u+d*q)*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*(f*u+d*q)*sqrt(z) +c*u) ^2 /(4*(f*u+d*q)))*u *(2*(f*u+d*q)*sqrt(z)+c*u) +2 *gamma_incomplete((j+i+2)/2, -(2*(f*u+d*q)*sqrt(z) +c*u) ^2 /(4*(f*u+d*q))) *(f*u+d*q) *sqrt( (2*(f*u+d*q)*sqrt(z)+c*u)^2 /(f*u+d*q))) /((2*(f*u+d*q)*sqrt(z)+c*u)^2/(-f*u-d*q)) ^((j+i+1)/2),i,0,j),j,0,n))*(%e^(d*z))^q *(%e^(f*z+c*sqrt(z)+g))^u *%e^(-f*u*z-d*q*z-c*u*sqrt(z)-c^2*u^2/(4*(f*u+d*q))) /(f*u+d*q)^(2*(n+1)))); 0; /* ----- Involving z^n*(%e^(d*z+e)^q*(%e^(c*z^r+f*z+g))^u ------------------- */ /* Problem 772: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(e+d*z))^q*(%e^(g+f*z+c*z^2))^u,z))- (-c^(-n-1)*u^(-n-1)*(%e^(d*z+e))^q*(%e^(z*(c*z+f)+g))^u *%e^(-z*(c*u*z+f*u+d*q)-(f*u+d*q)^2/(4*c*u)) *('sum(gamma_incomplete((i+1)/2,-(2*c*u*z+f*u+d*q)^2/(4*c*u)) *2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i) *(2*c*u*z+f*u+d*q)^(i+1) /(-(2*c*u*z+f*u+d*q)^2/(c*u))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 773: NEW 10-1 */ expand(niceindices(integrate(z^n*(%e^(e+d*z))^q*(%e^(g+f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum(c^(2*n-j-i)*(-1)^(i-j)*binomial(j,i)*u^(2*n-j-i) *(2*(f*u+d*q)*sqrt(z)+c*u)^(j+i) *(c *gamma_incomplete((j+i+1)/2, -(2*(f*u+d*q)*sqrt(z) +c*u) ^2 /(4*(f*u+d*q)))*u *(2*(f*u+d*q)*sqrt(z)+c*u) +2 *gamma_incomplete((j+i+2)/2, -(2*(f*u+d*q)*sqrt(z) +c*u) ^2 /(4*(f*u+d*q))) *(f*u+d*q) *sqrt( (2*(f*u+d*q)*sqrt(z)+c*u)^2 /(f*u+d*q))) /((2*(f*u+d*q)*sqrt(z)+c*u)^2/(-f*u-d*q)) ^((j+i+1)/2),i,0,j),j,0,n))*(%e^(d*z+e))^q *(%e^(f*z+c*sqrt(z)+g))^u *%e^(-f*u*z-d*q*z-c*u*sqrt(z)-c^2*u^2/(4*(f*u+d*q))) /(f*u+d*q)^(2*(n+1)))); 0; /* ----- Involving z^n*(%e^(b*z^r)^q*(%e^(c*z^r+f*z+g))^u ------------------- */ /* Problem 774: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(b*z^2))^q*(%e^(g+f*z+c*z^2))^u,z))- (-(c*u+b*q)^(-n-1)*(%e^(z*(c*z+f)+g))^u *%e^(-z*(c*u*z+b*q*z+f*u)-f^2*u^2/(4*(c*u+b*q))) *(%e^(b*z^2))^q *('sum(f^(n-i)*gamma_incomplete((i+1)/2, -(2*(c*u+b*q)*z+f*u)^2 /(4*(c*u+b*q)))*(-1)^(i-n) *2^(i-n)*binomial(n,i)*u^(n-i) *(2*(c*u+b*q)*z+f*u)^(i+1) /((2*(c*u+b*q)*z+f*u)^2/(-c*u-b*q))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 775: NEW 10-1 */ expand(niceindices(integrate(z^n*(%e^(b*sqrt(z)))^q*(%e^(g+f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b*q)^(2*n-j-i) *(2*f*u*sqrt(z)+c*u+b*q)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*f*u*sqrt(z) +c*u+b*q) ^2 /(4*f*u)) *(c*u+b*q)*(2*f*u*sqrt(z)+c*u+b*q) +2*f *gamma_incomplete((j+i+2)/2, -(2*f*u*sqrt(z) +c*u+b*q) ^2 /(4*f*u))*u *sqrt((2*f*u*sqrt(z)+c*u+b*q)^2/(f*u))) /(-(2*f*u*sqrt(z)+c*u+b*q)^2/(f*u))^((j+i+1)/2),i, 0,j),j,0,n))*(%e^(b*sqrt(z)))^q *(%e^(f*z+c*sqrt(z)+g))^u *%e^(-f*u*z-c*u*sqrt(z)-b*q*sqrt(z)-(c*u+b*q)^2/(4*f*u)) /(f^(2*(n+1))*u^(2*(n+1))))); 0; /* ----- Involving z^n*(%e^(b*z^r+e)^q*(%e^(c*z^r+f*z+g))^u ----------------- */ /* Problem 776: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(e+b*z^2))^q*(%e^(g+f*z+c*z^2))^u,z))- (-(c*u+b*q)^(-n-1)*(%e^(z*(c*z+f)+g))^u *%e^(-z*(c*u*z+b*q*z+f*u)-f^2*u^2/(4*(c*u+b*q))) *(%e^(b*z^2+e))^q *('sum(f^(n-i)*gamma_incomplete((i+1)/2, -(2*(c*u+b*q)*z+f*u)^2 /(4*(c*u+b*q)))*(-1)^(i-n) *2^(i-n)*binomial(n,i)*u^(n-i) *(2*(c*u+b*q)*z+f*u)^(i+1) /((2*(c*u+b*q)*z+f*u)^2/(-c*u-b*q))^((i+1)/2),i,0,n)) /2)); 0; /* Problem 777: NEW 10-1 */ expand(niceindices(integrate(z^n*(%e^(e+b*sqrt(z)))^q*(%e^(g+f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b*q)^(2*n-j-i) *(2*f*u*sqrt(z)+c*u+b*q)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*f*u*sqrt(z) +c*u+b*q) ^2 /(4*f*u)) *(c*u+b*q)*(2*f*u*sqrt(z)+c*u+b*q) +2*f *gamma_incomplete((j+i+2)/2, -(2*f*u*sqrt(z) +c*u+b*q) ^2 /(4*f*u))*u *sqrt((2*f*u*sqrt(z)+c*u+b*q)^2/(f*u))) /(-(2*f*u*sqrt(z)+c*u+b*q)^2/(f*u))^((j+i+1)/2),i, 0,j),j,0,n))*(%e^(b*sqrt(z)+e))^q *(%e^(f*z+c*sqrt(z)+g))^u *%e^(-f*u*z-c*u*sqrt(z)-b*q*sqrt(z)-(c*u+b*q)^2/(4*f*u)) /(f^(2*(n+1))*u^(2*(n+1))))); 0; /* ----- Involving z^n*(%e^(b*z^r+d*z)^q*(%e^(c*z^r+f*z+g))^u --------------- */ /* Problem 778: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(d*z+b*z^2))^q*(%e^(g+f*z+c*z^2))^u,z))- (-(c*u+b*q)^(-n-1)*(%e^(z*(b*z+d)))^q*(%e^(z*(c*z+f)+g))^u *%e^(-z*(c*u*z+b*q*z+f*u+d*q)-(f*u+d*q)^2/(4*(c*u+b*q))) *('sum(gamma_incomplete((i+1)/2, -(2*(c*u+b*q)*z+f*u+d*q)^2 /(4*(c*u+b*q))) *2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i) *(2*(c*u+b*q)*z+f*u+d*q)^(i+1) /((2*(c*u+b*q)*z+f*u+d*q)^2/(-c*u-b*q))^((i+1)/2),i, 0,n)) /2)); 0; /* Problem 779: NEW 10-1 */ expand( niceindices(integrate(z^n*(%e^(d*z+b*sqrt(z)))^q*(%e^(g+f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b*q)^(2*n-j-i) *(2*(f*u+d*q)*sqrt(z)+c*u+b*q)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*(f*u+d*q)*sqrt(z) +c*u+b*q) ^2 /(4*(f*u+d*q))) *(c*u+b*q)*(2*(f*u+d*q)*sqrt(z)+c*u+b*q) +2*gamma_incomplete((j+i+2)/2, -(2*(f*u+d*q)*sqrt(z) +c*u+b*q) ^2 /(4*(f*u+d*q))) *(f*u+d*q) *sqrt( (2*(f*u+d*q)*sqrt(z)+c*u+b*q)^2 /(f*u+d*q))) /((2*(f*u+d*q)*sqrt(z)+c*u+b*q)^2/(-f*u-d*q)) ^((j+i+1)/2),i,0,j),j,0,n))*(%e^(d*z+b*sqrt(z)))^q *(%e^(f*z+c*sqrt(z)+g))^u *%e^(-f*u*z-d*q*z-c*u*sqrt(z)-b*q*sqrt(z)-(c*u+b*q)^2/(4*(f*u+d*q))) /(f*u+d*q)^(2*(n+1))))$ 0; /* ----- Involving z^n*(%e^(b*z^r+d*z+e)^q*(%e^(c*z^r+f*z+g))^u ------------- */ /* Problem 780: NEW 9-1 */ expand(niceindices(integrate(z^n*(%e^(e+d*z+b*z^2))^q*(%e^(g+f*z+c*z^2))^u,z))- (-(c*u+b*q)^(-n-1)*(%e^(z*(b*z+d)+e))^q*(%e^(z*(c*z+f)+g))^u *%e^(-z*(c*u*z+b*q*z+f*u+d*q)-(f*u+d*q)^2/(4*(c*u+b*q))) *('sum(gamma_incomplete((i+1)/2, -(2*(c*u+b*q)*z+f*u+d*q)^2 /(4*(c*u+b*q))) *2^(i-n)*binomial(n,i)*(-f*u-d*q)^(n-i) *(2*(c*u+b*q)*z+f*u+d*q)^(i+1) /((2*(c*u+b*q)*z+f*u+d*q)^2/(-c*u-b*q))^((i+1)/2),i, 0,n)) /2)); 0; /* Problem 781: NEW 10-1 * This result must be checked more carefully. */ expand( niceindices(integrate(z^n*(%e^(e+d*z+b*sqrt(z)))^q*(%e^(g+f*z+c*sqrt(z)))^u,z))- (2^(-2*n-1)*('sum(4^j*binomial(n,j) *'sum((-1)^(i-j)*binomial(j,i)*(c*u+b*q)^(2*n-j-i) *(2*(f*u+d*q)*sqrt(z)+c*u+b*q)^(j+i) *(gamma_incomplete((j+i+1)/2, -(2*(f*u+d*q)*sqrt(z) +c*u+b*q) ^2 /(4*(f*u+d*q))) *(c*u+b*q)*(2*(f*u+d*q)*sqrt(z)+c*u+b*q) +2*gamma_incomplete((j+i+2)/2, -(2*(f*u+d*q)*sqrt(z) +c*u+b*q) ^2 /(4*(f*u+d*q))) *(f*u+d*q) *sqrt( (2*(f*u+d*q)*sqrt(z)+c*u+b*q)^2 /(f*u+d*q))) /((2*(f*u+d*q)*sqrt(z)+c*u+b*q)^2/(-f*u-d*q)) ^((j+i+1)/2),i,0,j),j,0,n)) *(%e^(d*z+b*sqrt(z)+e))^q*(%e^(f*z+c*sqrt(z)+g))^u *%e^(-f*u*z-d*q*z-c*u*sqrt(z)-b*q*sqrt(z)-(c*u+b*q)^2/(4*(f*u+d*q))) /(f*u+d*q)^(2*(n+1))))$ 0; /* Involving rational functions of the direct functions and a power function */ /* ----- Involving z^n/(a+b*%e^(c*z)) --------------------------------------- */ /* Problem 782: OK */ integrate(z/(b*%e^(c*z)+a),z); log(%e^(c*z))^2/(2*a*c^2)-(log(%e^(c*z))*log(b*%e^(c*z)/a+1) +li[2](-b*%e^(c*z)/a)) /(a*c^2)$ /* Problem 783: OK */ integrate(z^2/(b*%e^(c*z)+a),z); 2*(-log(%e^(c*z))^2*log(b*%e^(c*z)/a+1)/2 +li[3](-b*%e^(c*z)/a)-log(%e^(c*z))*li[2](-b*%e^(c*z)/a)) /(a*c^3) +log(%e^(c*z))^3/(3*a*c^3)$ /* Problem 784: OK */ integrate(z^3/(b*%e^(c*z)+a),z); log(%e^(c*z))^4/(4*a*c^4)-6*(log(%e^(c*z))^3*log(b*%e^(c*z)/a+1)/6 +li[4](-b*%e^(c*z)/a) -log(%e^(c*z))*li[3](-b*%e^(c*z)/a) +log(%e^(c*z))^2*li[2](-b*%e^(c*z)/a)/2) /(a*c^4)$ /* Problem 785: OK */ integrate(z^4/(b*%e^(c*z)+a),z); 24*(-log(%e^(c*z))^4*log(b*%e^(c*z)/a+1)/24 +li[5](-b*%e^(c*z)/a)-log(%e^(c*z))*li[4](-b*%e^(c*z)/a) +log(%e^(c*z))^2*li[3](-b*%e^(c*z)/a)/2 -log(%e^(c*z))^3*li[2](-b*%e^(c*z)/a)/6) /(a*c^5) +log(%e^(c*z))^5/(5*a*c^5)$ /* Problem 786: NOUN */ integrate(z^n/(1-%e^(c*z)),z); 'integrate(z^n/(1-%e^(c*z)),z); /* ----- Involving z^n*(a+b*c^(d*z))^-m ------------------------------------- */ /* Problem 787: OK */ expand(integrate(z/(b*c^(d*z)+a)^2,z)); -log(%e^(log(c)*d*z))*log(b*%e^(log(c)*d*z)/a+1)/(a^2*log(c)^2*d^2) +log(b*%e^(log(c)*d*z)+a)/(a^2*log(c)^2*d^2) -li[2](-b*%e^(log(c)*d*z)/a)/(a^2*log(c)^2*d^2) +log(%e^(log(c)*d*z))^2/(2*a^2*log(c)^2*d^2) -log(%e^(log(c)*d*z))/(a^2*log(c)^2*d^2) +z/(a*b*log(c)*d*%e^(log(c)*d*z)+a^2*log(c)*d)$ /* Problem 788: OK */ integrate(z^2/(b*c^(d*z)+a)^2,z); 2*(-log(%e^(log(c)*d*z))^2*log(b*%e^(log(c)*d*z)/a+1)/2 +li[3](-b*%e^(log(c)*d*z)/a) -log(%e^(log(c)*d*z))*li[2](-b*%e^(log(c)*d*z)/a)) /(a^2*log(c)^3*d^3) +2*(log(%e^(log(c)*d*z))*log(b*%e^(log(c)*d*z)/a+1) +li[2](-b*%e^(log(c)*d*z)/a)) /(a^2*log(c)^3*d^3) +(log(%e^(log(c)*d*z))^3-3*log(%e^(log(c)*d*z))^2)/(3*a^2*log(c)^3*d^3) +z^2/(a*b*log(c)*d*%e^(log(c)*d*z)+a^2*log(c)*d)$ /* Problem 789: OK */ integrate(z^3/(b*c^(d*z)+a)^2,z); -6*(log(%e^(log(c)*d*z))^3*log(b*%e^(log(c)*d*z)/a+1)/6 +li[4](-b*%e^(log(c)*d*z)/a) -log(%e^(log(c)*d*z))*li[3](-b*%e^(log(c)*d*z)/a) +log(%e^(log(c)*d*z))^2*li[2](-b*%e^(log(c)*d*z)/a)/2) /(a^2*log(c)^4*d^4) -6*(-log(%e^(log(c)*d*z))^2*log(b*%e^(log(c)*d*z)/a+1)/2 +li[3](-b*%e^(log(c)*d*z)/a) -log(%e^(log(c)*d*z))*li[2](-b*%e^(log(c)*d*z)/a)) /(a^2*log(c)^4*d^4) +(log(%e^(log(c)*d*z))^4-4*log(%e^(log(c)*d*z))^3)/(4*a^2*log(c)^4*d^4) +z^3/(a*b*log(c)*d*%e^(log(c)*d*z)+a^2*log(c)*d)$ /* Problem 790: OK */ integrate(z^4/(b*c^(d*z)+a)^2,z); 24*(-log(%e^(log(c)*d*z))^4*log(b*%e^(log(c)*d*z)/a+1)/24 +li[5](-b*%e^(log(c)*d*z)/a) -log(%e^(log(c)*d*z))*li[4](-b*%e^(log(c)*d*z)/a) +log(%e^(log(c)*d*z))^2*li[3](-b*%e^(log(c)*d*z)/a)/2 -log(%e^(log(c)*d*z))^3*li[2](-b*%e^(log(c)*d*z)/a)/6) /(a^2*log(c)^5*d^5) +24*(log(%e^(log(c)*d*z))^3*log(b*%e^(log(c)*d*z)/a+1)/6 +li[4](-b*%e^(log(c)*d*z)/a) -log(%e^(log(c)*d*z))*li[3](-b*%e^(log(c)*d*z)/a) +log(%e^(log(c)*d*z))^2*li[2](-b*%e^(log(c)*d*z)/a)/2) /(a^2*log(c)^5*d^5) +(log(%e^(log(c)*d*z))^5-5*log(%e^(log(c)*d*z))^4)/(5*a^2*log(c)^5*d^5) +z^4/(a*b*log(c)*d*%e^(log(c)*d*z)+a^2*log(c)*d)$ /* ----- Involving z^n*%e^(c*z)/(a+b*%e^(c*z)) ------------------------------ */ /* Problem 791: NOUN */ integrate(z*%e^(f*z)/(b*c^(d*z)+a),z); 'integrate(z*%e^(f*z)/(b*c^(d*z)+a),z); /* 'integrate((a*(log(c)^2*d^2-log(c)*d*f)*z+a*log(c)*d)*%e^(f*z) /(b^2*(f^2-2*log(c)*d*f+log(c)^2*d^2)*%e^(2*log(c)*d*z) +a*b*(2*f^2-4*log(c)*d*f+2*log(c)^2*d^2)*%e^(log(c)*d*z) +a^2*(f^2-2*log(c)*d*f+log(c)^2*d^2)),z) -((log(c)*d-f)*z+1)*%e^(f*z)/(b*(f^2-2*log(c)*d*f+log(c)^2*d^2) *%e^(log(c)*d*z) +a*(f^2-2*log(c)*d*f+log(c)^2*d^2)); */ /* Problem 792: NOUN */ integrate(z*%e^(d*z)/(b*%e^(c*z)+a),z); 'integrate((a*(c^2-c*d)*z+a*c)*%e^(d*z)/(b^2*(d^2-2*c*d+c^2)*%e^(2*c*z) +a*b*(2*d^2-4*c*d+2*c^2)*%e^(c*z) +a^2*(d^2-2*c*d+c^2)),z) -((c-d)*z+1)*%e^(d*z)/(b*(d^2-2*c*d+c^2)*%e^(c*z)+a*(d^2-2*c*d+c^2)); /* Problem 793: OK */ integrate(z*%e^(2*c*z)/(b*%e^(c*z)+a),z); -a*(log(%e^(c*z))*log(b*%e^(c*z)/a+1)+li[2](-b*%e^(c*z)/a))/(b^2*c^2) +a*log(%e^(c*z))^2/(2*b^2*c^2)+((2*b*c*z-2*b)*%e^(c*z)-a*c^2*z^2)/(2*b^2*c^2)$ /* Problem 794: OK */ integrate(z*%e^-(2*c*z)/(b*%e^(c*z)+a),z); -b^2*(log(%e^(c*z))*log(b*%e^(c*z)/a+1)+li[2](-b*%e^(c*z)/a))/(a^3*c^2) +b^2*log(%e^(c*z))^2/(2*a^3*c^2) +((4*b*c*z+4*b)*%e^-(c*z)+(-2*a*c*z-a)*%e^-(2*c*z))/(4*a^2*c^2)$ /* Problem 795: OK */ integrate(z*%e^(c*z)/(b*%e^(c*z)+a),z); (log(%e^(c*z))*log(b*%e^(c*z)/a+1)+li[2](-b*%e^(c*z)/a))/(b*c^2) -log(%e^(c*z))^2/(2*b*c^2)+z^2/(2*b)$ /* Problem 796: NOUN */ integrate(z^2*%e^(f*z)/(b*c^(d*z)+a),z); 'integrate(z^2*%e^(f*z)/(b*c^(d*z)+a),z); /* 'integrate((a*(log(c)*d*f^2-2*log(c)^2*d^2*f+log(c)^3*d^3)*z^2 +a*(2*log(c)^2*d^2-2*log(c)*d*f)*z+2*a*log(c)*d) *%e^(f*z) /(b^2*(-f^3+3*log(c)*d*f^2-3*log(c)^2*d^2*f+log(c)^3*d^3) *%e^(2*log(c)*d*z) +a*b*(-2*f^3+6*log(c)*d*f^2-6*log(c)^2*d^2*f+2*log(c)^3*d^3) *%e^(log(c)*d*z) +a^2*(-f^3+3*log(c)*d*f^2-3*log(c)^2*d^2*f+log(c)^3*d^3)),z) -((f^2-2*log(c)*d*f+log(c)^2*d^2)*z^2+(2*log(c)*d-2*f)*z+2)*%e^(f*z) /(b*(-f^3+3*log(c)*d*f^2-3*log(c)^2*d^2*f+log(c)^3*d^3)*%e^(log(c)*d*z) +a*(-f^3+3*log(c)*d*f^2-3*log(c)^2*d^2*f+log(c)^3*d^3)); */ /* Problem 797: OK */ integrate(z^2*%e^(c*z)/(b*%e^(c*z)+a),z); -2*(-log(%e^(c*z))^2*log(b*%e^(c*z)/a+1)/2 +li[3](-b*%e^(c*z)/a)-log(%e^(c*z))*li[2](-b*%e^(c*z)/a)) /(b*c^3) -log(%e^(c*z))^3/(3*b*c^3)+z^3/(3*b)$ /* Problem 798: NOUN */ integrate(z^3*%e^(f*z)/(b*c^(d*z)+a),z); 'integrate(z^3*%e^(f*z)/(b*c^(d*z)+a),z); /* 'integrate((a*(-log(c)*d*f^3+3*log(c)^2*d^2*f^2-3*log(c)^3*d^3*f+log(c)^4*d^4) *z^3 +a*(3*log(c)*d*f^2-6*log(c)^2*d^2*f+3*log(c)^3*d^3)*z^2 +a*(6*log(c)^2*d^2-6*log(c)*d*f)*z+6*a*log(c)*d) *%e^(f*z) /(b^2*(f^4-4*log(c)*d*f^3+6*log(c)^2*d^2*f^2-4*log(c)^3*d^3*f +log(c)^4*d^4)*%e^(2*log(c)*d*z) +a*b *(2*f^4-8*log(c)*d*f^3+12*log(c)^2*d^2*f^2-8*log(c)^3*d^3*f +2*log(c)^4*d^4)*%e^(log(c)*d*z) +a^2*(f^4-4*log(c)*d*f^3+6*log(c)^2*d^2*f^2-4*log(c)^3*d^3*f +log(c)^4*d^4)),z) -((-f^3+3*log(c)*d*f^2-3*log(c)^2*d^2*f+log(c)^3*d^3)*z^3 +(3*f^2-6*log(c)*d*f+3*log(c)^2*d^2)*z^2+(6*log(c)*d-6*f)*z+6) *%e^(f*z) /(b*(f^4-4*log(c)*d*f^3+6*log(c)^2*d^2*f^2-4*log(c)^3*d^3*f+log(c)^4*d^4) *%e^(log(c)*d*z) +a*(f^4-4*log(c)*d*f^3+6*log(c)^2*d^2*f^2-4*log(c)^3*d^3*f+log(c)^4*d^4)); */ /* Problem 799: OK */ integrate(z^3*%e^(c*z)/(b*%e^(c*z)+a),z); 6*(log(%e^(c*z))^3*log(b*%e^(c*z)/a+1)/6 +li[4](-b*%e^(c*z)/a)-log(%e^(c*z))*li[3](-b*%e^(c*z)/a) +log(%e^(c*z))^2*li[2](-b*%e^(c*z)/a)/2) /(b*c^4) -log(%e^(c*z))^4/(4*b*c^4)+z^4/(4*b)$ /* Problem 800: OK */ integrate(z^4*%e^(c*z)/(b*%e^(c*z)+a),z); -24*(-log(%e^(c*z))^4*log(b*%e^(c*z)/a+1)/24 +li[5](-b*%e^(c*z)/a)-log(%e^(c*z))*li[4](-b*%e^(c*z)/a) +log(%e^(c*z))^2*li[3](-b*%e^(c*z)/a)/2 -log(%e^(c*z))^3*li[2](-b*%e^(c*z)/a)/6) /(b*c^5) -log(%e^(c*z))^5/(5*b*c^5)+z^5/(5*b)$ /* ---- Involving z^n/(a*%e^(2*d*z)+b*%e^(d*z)+c) --------------------------- */ assume(4*a*c>b^2); [4*a*c>b^2]; /* Problem 802: NOUN */ integrate(z/(c+b*%e^(d*z)+a*%e^(2*d*z)),z); 'integrate(z/(a*%e^(2*d*z)+b*%e^(d*z)+c),z); /* Problem 803: NOUN */ integrate(z^2/(c+b*%e^(d*z)+a*%e^(2*d*z)),z); 'integrate(z^2/(a*%e^(2*d*z)+b*%e^(d*z)+c),z); /* Problem 804: NOUN */ integrate(z^3/(c+b*%e^(d*z)+a*%e^(2*d*z)),z); 'integrate(z^3/(a*%e^(2*d*z)+b*%e^(d*z)+c),z); forget(4*a*c>b^2); [4*a*c>b^2]; /* ----- Involing z^n*(a+b*c^(d*z)^p ---------------------------------------- */ /* Problem 806: NOUN */ integrate(z*(b*c^(d*z)+a)^p,z); 'integrate((b*c^(d*z)+a)^p*z,z); /* Problem 807: NOUN */ integrate(z^2*(b*c^(d*z)+a)^p,z); 'integrate((b*c^(d*z)+a)^p*z^2,z); /* Problem 808: NOUN */ integrate(z^3*(b*c^(d*z)+a)^p,z); 'integrate((b*c^(d*z)+a)^p*z^3,z); /* ----- Involing z^n*%e^(f*z)*(a+b*c^(d*z)^p ------------------------------- */ /* Problem 809: NOUN */ integrate(z*%e^(f*z)*(b*c^(d*z)+a)^p,z); 'integrate((b*c^(d*z)+a)^p*z*%e^(f*z),z); /* Problem 810: NOUN */ integrate(z^2*%e^(f*z)*(b*c^(d*z)+a)^p,z); 'integrate((b*c^(d*z)+a)^p*z^2*%e^(f*z),z); /* Problem 811: NOUN */ integrate(z^3*%e^(f*z)*(b*c^(d*z)+a)^p,z); 'integrate((b*c^(d*z)+a)^p*z^3*%e^(f*z),z); /* Problem 812: NOUN */ integrate(z^4*%e^(f*z)*(b*c^(d*z)+a)^p,z); 'integrate((b*c^(d*z)+a)^p*z^4*%e^(f*z),z); /* ----- End of file -------------------------------------------------------- */