Integrand size = 22, antiderivative size = 327 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {(B d-A e) \left (c d^2+a e^2\right )^3}{7 e^8 (d+e x)^7}-\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{6 e^8 (d+e x)^6}+\frac {3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{5 e^8 (d+e x)^5}+\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{4 e^8 (d+e x)^4}+\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right )}{3 e^8 (d+e x)^3}-\frac {3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right )}{2 e^8 (d+e x)^2}+\frac {c^3 (7 B d-A e)}{e^8 (d+e x)}+\frac {B c^3 \log (d+e x)}{e^8} \] Output:
1/7*(-A*e+B*d)*(a*e^2+c*d^2)^3/e^8/(e*x+d)^7-1/6*(a*e^2+c*d^2)^2*(-6*A*c*d *e+B*a*e^2+7*B*c*d^2)/e^8/(e*x+d)^6+3/5*c*(a*e^2+c*d^2)*(-A*a*e^3-5*A*c*d^ 2*e+3*B*a*d*e^2+7*B*c*d^3)/e^8/(e*x+d)^5+1/4*c*(4*A*c*d*e*(3*a*e^2+5*c*d^2 )-B*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2*d^4))/e^8/(e*x+d)^4+1/3*c^2*(-3*A*a*e ^3-15*A*c*d^2*e+15*B*a*d*e^2+35*B*c*d^3)/e^8/(e*x+d)^3-3/2*c^2*(-2*A*c*d*e +B*a*e^2+7*B*c*d^2)/e^8/(e*x+d)^2+c^3*(-A*e+7*B*d)/e^8/(e*x+d)+B*c^3*ln(e* x+d)/e^8
Time = 0.19 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.12 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {-12 A e \left (5 a^3 e^6+a^2 c e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+a c^2 e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 c^3 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )+B \left (-10 a^3 e^6 (d+7 e x)-9 a^2 c e^4 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )-30 a c^2 e^2 \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+c^3 d \left (1089 d^6+7203 d^5 e x+20139 d^4 e^2 x^2+30625 d^3 e^3 x^3+26950 d^2 e^4 x^4+13230 d e^5 x^5+2940 e^6 x^6\right )\right )+420 B c^3 (d+e x)^7 \log (d+e x)}{420 e^8 (d+e x)^7} \] Input:
Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^8,x]
Output:
(-12*A*e*(5*a^3*e^6 + a^2*c*e^4*(d^2 + 7*d*e*x + 21*e^2*x^2) + a*c^2*e^2*( d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 5*c^3*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5 *x^5 + 7*e^6*x^6)) + B*(-10*a^3*e^6*(d + 7*e*x) - 9*a^2*c*e^4*(d^3 + 7*d^2 *e*x + 21*d*e^2*x^2 + 35*e^3*x^3) - 30*a*c^2*e^2*(d^5 + 7*d^4*e*x + 21*d^3 *e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + c^3*d*(1089*d^6 + 7203*d^5*e*x + 20139*d^4*e^2*x^2 + 30625*d^3*e^3*x^3 + 26950*d^2*e^4*x^4 + 13230*d*e^5*x^5 + 2940*e^6*x^6)) + 420*B*c^3*(d + e*x)^7*Log[d + e*x])/( 420*e^8*(d + e*x)^7)
Time = 0.62 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {652, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{(d+e x)^8} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {c \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7 (d+e x)^5}-\frac {3 c^2 \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7 (d+e x)^3}+\frac {c^2 \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7 (d+e x)^4}+\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7 (d+e x)^7}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)^8}+\frac {3 c \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7 (d+e x)^6}+\frac {c^3 (A e-7 B d)}{e^7 (d+e x)^2}+\frac {B c^3}{e^7 (d+e x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{4 e^8 (d+e x)^4}-\frac {3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{2 e^8 (d+e x)^2}+\frac {c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{3 e^8 (d+e x)^3}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{6 e^8 (d+e x)^6}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{7 e^8 (d+e x)^7}+\frac {3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{5 e^8 (d+e x)^5}+\frac {c^3 (7 B d-A e)}{e^8 (d+e x)}+\frac {B c^3 \log (d+e x)}{e^8}\) |
Input:
Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^8,x]
Output:
((B*d - A*e)*(c*d^2 + a*e^2)^3)/(7*e^8*(d + e*x)^7) - ((c*d^2 + a*e^2)^2*( 7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(6*e^8*(d + e*x)^6) + (3*c*(c*d^2 + a*e^ 2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(5*e^8*(d + e*x)^5) + (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a ^2*e^4)))/(4*e^8*(d + e*x)^4) + (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d *e^2 - 3*a*A*e^3))/(3*e^8*(d + e*x)^3) - (3*c^2*(7*B*c*d^2 - 2*A*c*d*e + a *B*e^2))/(2*e^8*(d + e*x)^2) + (c^3*(7*B*d - A*e))/(e^8*(d + e*x)) + (B*c^ 3*Log[d + e*x])/e^8
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Time = 0.86 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.30
| method | result | size |
| risch | \(\frac {-\frac {c^{3} \left (A e -7 B d \right ) x^{6}}{e^{2}}-\frac {3 c^{2} \left (2 A c d e +B a \,e^{2}-21 B c \,d^{2}\right ) x^{5}}{2 e^{3}}-\frac {c^{2} \left (6 A a \,e^{3}+30 A c \,d^{2} e +15 B a d \,e^{2}-385 B c \,d^{3}\right ) x^{4}}{6 e^{4}}-\frac {c \left (12 A a c d \,e^{3}+60 A \,c^{2} d^{3} e +9 B \,e^{4} a^{2}+30 B a c \,d^{2} e^{2}-875 B \,c^{2} d^{4}\right ) x^{3}}{12 e^{5}}-\frac {c \left (12 A \,a^{2} e^{5}+12 A a c \,d^{2} e^{3}+60 A \,c^{2} d^{4} e +9 B \,a^{2} d \,e^{4}+30 B a c \,d^{3} e^{2}-959 B \,c^{2} d^{5}\right ) x^{2}}{20 e^{6}}-\frac {\left (12 A \,a^{2} c d \,e^{5}+12 A a \,c^{2} d^{3} e^{3}+60 A \,c^{3} d^{5} e +10 B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+30 B a \,c^{2} d^{4} e^{2}-1029 B \,c^{3} d^{6}\right ) x}{60 e^{7}}-\frac {60 A \,a^{3} e^{7}+12 A \,a^{2} c \,d^{2} e^{5}+12 A a \,c^{2} d^{4} e^{3}+60 A \,c^{3} d^{6} e +10 B \,a^{3} d \,e^{6}+9 B \,a^{2} c \,d^{3} e^{4}+30 B a \,c^{2} d^{5} e^{2}-1089 B \,c^{3} d^{7}}{420 e^{8}}}{\left (e x +d \right )^{7}}+\frac {B \,c^{3} \ln \left (e x +d \right )}{e^{8}}\) | \(426\) |
| default | \(-\frac {c^{2} \left (3 A a \,e^{3}+15 A c \,d^{2} e -15 B a d \,e^{2}-35 B c \,d^{3}\right )}{3 e^{8} \left (e x +d \right )^{3}}+\frac {c \left (12 A a c d \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-30 B a c \,d^{2} e^{2}-35 B \,c^{2} d^{4}\right )}{4 e^{8} \left (e x +d \right )^{4}}-\frac {A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}}{7 e^{8} \left (e x +d \right )^{7}}+\frac {B \,c^{3} \ln \left (e x +d \right )}{e^{8}}-\frac {3 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{5 e^{8} \left (e x +d \right )^{5}}-\frac {c^{3} \left (A e -7 B d \right )}{e^{8} \left (e x +d \right )}+\frac {3 c^{2} \left (2 A c d e -B a \,e^{2}-7 B c \,d^{2}\right )}{2 e^{8} \left (e x +d \right )^{2}}-\frac {-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}}{6 e^{8} \left (e x +d \right )^{6}}\) | \(447\) |
| norman | \(\frac {-\frac {60 A \,a^{3} e^{7}+12 A \,a^{2} c \,d^{2} e^{5}+12 A a \,c^{2} d^{4} e^{3}+60 A \,c^{3} d^{6} e +10 B \,a^{3} d \,e^{6}+9 B \,a^{2} c \,d^{3} e^{4}+30 B a \,c^{2} d^{5} e^{2}-1089 B \,c^{3} d^{7}}{420 e^{8}}-\frac {\left (A \,c^{3} e -7 B \,c^{3} d \right ) x^{6}}{e^{2}}-\frac {3 \left (2 A \,c^{3} d e +B \,e^{2} a \,c^{2}-21 B \,c^{3} d^{2}\right ) x^{5}}{2 e^{3}}-\frac {\left (6 A a \,c^{2} e^{3}+30 A \,c^{3} d^{2} e +15 B a \,c^{2} d \,e^{2}-385 B \,c^{3} d^{3}\right ) x^{4}}{6 e^{4}}-\frac {\left (12 A a \,c^{2} d \,e^{3}+60 A \,c^{3} d^{3} e +9 B \,e^{4} a^{2} c +30 B a \,c^{2} d^{2} e^{2}-875 B \,c^{3} d^{4}\right ) x^{3}}{12 e^{5}}-\frac {\left (12 A \,a^{2} c \,e^{5}+12 A a \,c^{2} d^{2} e^{3}+60 A \,c^{3} d^{4} e +9 B \,a^{2} c d \,e^{4}+30 B a \,c^{2} d^{3} e^{2}-959 B \,c^{3} d^{5}\right ) x^{2}}{20 e^{6}}-\frac {\left (12 A \,a^{2} c d \,e^{5}+12 A a \,c^{2} d^{3} e^{3}+60 A \,c^{3} d^{5} e +10 B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+30 B a \,c^{2} d^{4} e^{2}-1029 B \,c^{3} d^{6}\right ) x}{60 e^{7}}}{\left (e x +d \right )^{7}}+\frac {B \,c^{3} \ln \left (e x +d \right )}{e^{8}}\) | \(449\) |
| parallelrisch | \(-\frac {9 B \,a^{2} c \,d^{3} e^{4}+12 A \,a^{2} c \,d^{2} e^{5}+12 A a \,c^{2} d^{4} e^{3}+420 A \,x^{6} c^{3} e^{7}-420 B \ln \left (e x +d \right ) c^{3} d^{7}+70 B x \,a^{3} e^{7}-8820 B \ln \left (e x +d \right ) x^{5} c^{3} d^{2} e^{5}+210 B x a \,c^{2} d^{4} e^{3}+30 B a \,c^{2} d^{5} e^{2}-8820 B \ln \left (e x +d \right ) x^{2} c^{3} d^{5} e^{2}+2100 A \,x^{4} c^{3} d^{2} e^{5}-26950 B \,x^{4} c^{3} d^{3} e^{4}+2100 A \,x^{3} c^{3} d^{3} e^{4}-1089 B \,c^{3} d^{7}-420 B \ln \left (e x +d \right ) x^{7} c^{3} e^{7}-14700 B \ln \left (e x +d \right ) x^{3} c^{3} d^{4} e^{3}-14700 B \ln \left (e x +d \right ) x^{4} c^{3} d^{3} e^{4}-2940 B \ln \left (e x +d \right ) x^{6} c^{3} d \,e^{6}+1260 A \,x^{5} c^{3} d \,e^{6}-2940 B \,x^{6} c^{3} d \,e^{6}-2940 B \ln \left (e x +d \right ) x \,c^{3} d^{6} e +315 B \,x^{3} a^{2} c \,e^{7}-30625 B \,x^{3} c^{3} d^{4} e^{3}+252 A \,x^{2} a^{2} c \,e^{7}+1260 A \,x^{2} c^{3} d^{4} e^{3}-20139 B \,x^{2} c^{3} d^{5} e^{2}+420 A x \,c^{3} d^{5} e^{2}-7203 B x \,c^{3} d^{6} e +630 B \,x^{5} a \,c^{2} e^{7}-13230 B \,x^{5} c^{3} d^{2} e^{5}+420 A \,x^{4} a \,c^{2} e^{7}+60 A \,c^{3} d^{6} e +10 B \,a^{3} d \,e^{6}+1050 B \,x^{3} a \,c^{2} d^{2} e^{5}+252 A \,x^{2} a \,c^{2} d^{2} e^{5}+189 B \,x^{2} a^{2} c d \,e^{6}+630 B \,x^{2} a \,c^{2} d^{3} e^{4}+84 A x \,a^{2} c d \,e^{6}+84 A x a \,c^{2} d^{3} e^{4}+63 B x \,a^{2} c \,d^{2} e^{5}+1050 B \,x^{4} a \,c^{2} d \,e^{6}+420 A \,x^{3} a \,c^{2} d \,e^{6}+60 A \,a^{3} e^{7}}{420 e^{8} \left (e x +d \right )^{7}}\) | \(630\) |
Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d)^8,x,method=_RETURNVERBOSE)
Output:
(-c^3*(A*e-7*B*d)/e^2*x^6-3/2*c^2*(2*A*c*d*e+B*a*e^2-21*B*c*d^2)/e^3*x^5-1 /6*c^2*(6*A*a*e^3+30*A*c*d^2*e+15*B*a*d*e^2-385*B*c*d^3)/e^4*x^4-1/12*c*(1 2*A*a*c*d*e^3+60*A*c^2*d^3*e+9*B*a^2*e^4+30*B*a*c*d^2*e^2-875*B*c^2*d^4)/e ^5*x^3-1/20*c*(12*A*a^2*e^5+12*A*a*c*d^2*e^3+60*A*c^2*d^4*e+9*B*a^2*d*e^4+ 30*B*a*c*d^3*e^2-959*B*c^2*d^5)/e^6*x^2-1/60*(12*A*a^2*c*d*e^5+12*A*a*c^2* d^3*e^3+60*A*c^3*d^5*e+10*B*a^3*e^6+9*B*a^2*c*d^2*e^4+30*B*a*c^2*d^4*e^2-1 029*B*c^3*d^6)/e^7*x-1/420*(60*A*a^3*e^7+12*A*a^2*c*d^2*e^5+12*A*a*c^2*d^4 *e^3+60*A*c^3*d^6*e+10*B*a^3*d*e^6+9*B*a^2*c*d^3*e^4+30*B*a*c^2*d^5*e^2-10 89*B*c^3*d^7)/e^8)/(e*x+d)^7+B*c^3*ln(e*x+d)/e^8
Time = 0.09 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.91 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {1089 \, B c^{3} d^{7} - 60 \, A c^{3} d^{6} e - 30 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 9 \, B a^{2} c d^{3} e^{4} - 12 \, A a^{2} c d^{2} e^{5} - 10 \, B a^{3} d e^{6} - 60 \, A a^{3} e^{7} + 420 \, {\left (7 \, B c^{3} d e^{6} - A c^{3} e^{7}\right )} x^{6} + 630 \, {\left (21 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} - B a c^{2} e^{7}\right )} x^{5} + 70 \, {\left (385 \, B c^{3} d^{3} e^{4} - 30 \, A c^{3} d^{2} e^{5} - 15 \, B a c^{2} d e^{6} - 6 \, A a c^{2} e^{7}\right )} x^{4} + 35 \, {\left (875 \, B c^{3} d^{4} e^{3} - 60 \, A c^{3} d^{3} e^{4} - 30 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 9 \, B a^{2} c e^{7}\right )} x^{3} + 21 \, {\left (959 \, B c^{3} d^{5} e^{2} - 60 \, A c^{3} d^{4} e^{3} - 30 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 9 \, B a^{2} c d e^{6} - 12 \, A a^{2} c e^{7}\right )} x^{2} + 7 \, {\left (1029 \, B c^{3} d^{6} e - 60 \, A c^{3} d^{5} e^{2} - 30 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 9 \, B a^{2} c d^{2} e^{5} - 12 \, A a^{2} c d e^{6} - 10 \, B a^{3} e^{7}\right )} x + 420 \, {\left (B c^{3} e^{7} x^{7} + 7 \, B c^{3} d e^{6} x^{6} + 21 \, B c^{3} d^{2} e^{5} x^{5} + 35 \, B c^{3} d^{3} e^{4} x^{4} + 35 \, B c^{3} d^{4} e^{3} x^{3} + 21 \, B c^{3} d^{5} e^{2} x^{2} + 7 \, B c^{3} d^{6} e x + B c^{3} d^{7}\right )} \log \left (e x + d\right )}{420 \, {\left (e^{15} x^{7} + 7 \, d e^{14} x^{6} + 21 \, d^{2} e^{13} x^{5} + 35 \, d^{3} e^{12} x^{4} + 35 \, d^{4} e^{11} x^{3} + 21 \, d^{5} e^{10} x^{2} + 7 \, d^{6} e^{9} x + d^{7} e^{8}\right )}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^8,x, algorithm="fricas")
Output:
1/420*(1089*B*c^3*d^7 - 60*A*c^3*d^6*e - 30*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d ^4*e^3 - 9*B*a^2*c*d^3*e^4 - 12*A*a^2*c*d^2*e^5 - 10*B*a^3*d*e^6 - 60*A*a^ 3*e^7 + 420*(7*B*c^3*d*e^6 - A*c^3*e^7)*x^6 + 630*(21*B*c^3*d^2*e^5 - 2*A* c^3*d*e^6 - B*a*c^2*e^7)*x^5 + 70*(385*B*c^3*d^3*e^4 - 30*A*c^3*d^2*e^5 - 15*B*a*c^2*d*e^6 - 6*A*a*c^2*e^7)*x^4 + 35*(875*B*c^3*d^4*e^3 - 60*A*c^3*d ^3*e^4 - 30*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 - 9*B*a^2*c*e^7)*x^3 + 21*( 959*B*c^3*d^5*e^2 - 60*A*c^3*d^4*e^3 - 30*B*a*c^2*d^3*e^4 - 12*A*a*c^2*d^2 *e^5 - 9*B*a^2*c*d*e^6 - 12*A*a^2*c*e^7)*x^2 + 7*(1029*B*c^3*d^6*e - 60*A* c^3*d^5*e^2 - 30*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*e^4 - 9*B*a^2*c*d^2*e^5 - 12*A*a^2*c*d*e^6 - 10*B*a^3*e^7)*x + 420*(B*c^3*e^7*x^7 + 7*B*c^3*d*e^6* x^6 + 21*B*c^3*d^2*e^5*x^5 + 35*B*c^3*d^3*e^4*x^4 + 35*B*c^3*d^4*e^3*x^3 + 21*B*c^3*d^5*e^2*x^2 + 7*B*c^3*d^6*e*x + B*c^3*d^7)*log(e*x + d))/(e^15*x ^7 + 7*d*e^14*x^6 + 21*d^2*e^13*x^5 + 35*d^3*e^12*x^4 + 35*d^4*e^11*x^3 + 21*d^5*e^10*x^2 + 7*d^6*e^9*x + d^7*e^8)
Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^8} \, dx=\text {Timed out} \] Input:
integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**8,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.61 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {1089 \, B c^{3} d^{7} - 60 \, A c^{3} d^{6} e - 30 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 9 \, B a^{2} c d^{3} e^{4} - 12 \, A a^{2} c d^{2} e^{5} - 10 \, B a^{3} d e^{6} - 60 \, A a^{3} e^{7} + 420 \, {\left (7 \, B c^{3} d e^{6} - A c^{3} e^{7}\right )} x^{6} + 630 \, {\left (21 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} - B a c^{2} e^{7}\right )} x^{5} + 70 \, {\left (385 \, B c^{3} d^{3} e^{4} - 30 \, A c^{3} d^{2} e^{5} - 15 \, B a c^{2} d e^{6} - 6 \, A a c^{2} e^{7}\right )} x^{4} + 35 \, {\left (875 \, B c^{3} d^{4} e^{3} - 60 \, A c^{3} d^{3} e^{4} - 30 \, B a c^{2} d^{2} e^{5} - 12 \, A a c^{2} d e^{6} - 9 \, B a^{2} c e^{7}\right )} x^{3} + 21 \, {\left (959 \, B c^{3} d^{5} e^{2} - 60 \, A c^{3} d^{4} e^{3} - 30 \, B a c^{2} d^{3} e^{4} - 12 \, A a c^{2} d^{2} e^{5} - 9 \, B a^{2} c d e^{6} - 12 \, A a^{2} c e^{7}\right )} x^{2} + 7 \, {\left (1029 \, B c^{3} d^{6} e - 60 \, A c^{3} d^{5} e^{2} - 30 \, B a c^{2} d^{4} e^{3} - 12 \, A a c^{2} d^{3} e^{4} - 9 \, B a^{2} c d^{2} e^{5} - 12 \, A a^{2} c d e^{6} - 10 \, B a^{3} e^{7}\right )} x}{420 \, {\left (e^{15} x^{7} + 7 \, d e^{14} x^{6} + 21 \, d^{2} e^{13} x^{5} + 35 \, d^{3} e^{12} x^{4} + 35 \, d^{4} e^{11} x^{3} + 21 \, d^{5} e^{10} x^{2} + 7 \, d^{6} e^{9} x + d^{7} e^{8}\right )}} + \frac {B c^{3} \log \left (e x + d\right )}{e^{8}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^8,x, algorithm="maxima")
Output:
1/420*(1089*B*c^3*d^7 - 60*A*c^3*d^6*e - 30*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d ^4*e^3 - 9*B*a^2*c*d^3*e^4 - 12*A*a^2*c*d^2*e^5 - 10*B*a^3*d*e^6 - 60*A*a^ 3*e^7 + 420*(7*B*c^3*d*e^6 - A*c^3*e^7)*x^6 + 630*(21*B*c^3*d^2*e^5 - 2*A* c^3*d*e^6 - B*a*c^2*e^7)*x^5 + 70*(385*B*c^3*d^3*e^4 - 30*A*c^3*d^2*e^5 - 15*B*a*c^2*d*e^6 - 6*A*a*c^2*e^7)*x^4 + 35*(875*B*c^3*d^4*e^3 - 60*A*c^3*d ^3*e^4 - 30*B*a*c^2*d^2*e^5 - 12*A*a*c^2*d*e^6 - 9*B*a^2*c*e^7)*x^3 + 21*( 959*B*c^3*d^5*e^2 - 60*A*c^3*d^4*e^3 - 30*B*a*c^2*d^3*e^4 - 12*A*a*c^2*d^2 *e^5 - 9*B*a^2*c*d*e^6 - 12*A*a^2*c*e^7)*x^2 + 7*(1029*B*c^3*d^6*e - 60*A* c^3*d^5*e^2 - 30*B*a*c^2*d^4*e^3 - 12*A*a*c^2*d^3*e^4 - 9*B*a^2*c*d^2*e^5 - 12*A*a^2*c*d*e^6 - 10*B*a^3*e^7)*x)/(e^15*x^7 + 7*d*e^14*x^6 + 21*d^2*e^ 13*x^5 + 35*d^3*e^12*x^4 + 35*d^4*e^11*x^3 + 21*d^5*e^10*x^2 + 7*d^6*e^9*x + d^7*e^8) + B*c^3*log(e*x + d)/e^8
Time = 0.14 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.40 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {B c^{3} \log \left ({\left | e x + d \right |}\right )}{e^{8}} + \frac {420 \, {\left (7 \, B c^{3} d e^{5} - A c^{3} e^{6}\right )} x^{6} + 630 \, {\left (21 \, B c^{3} d^{2} e^{4} - 2 \, A c^{3} d e^{5} - B a c^{2} e^{6}\right )} x^{5} + 70 \, {\left (385 \, B c^{3} d^{3} e^{3} - 30 \, A c^{3} d^{2} e^{4} - 15 \, B a c^{2} d e^{5} - 6 \, A a c^{2} e^{6}\right )} x^{4} + 35 \, {\left (875 \, B c^{3} d^{4} e^{2} - 60 \, A c^{3} d^{3} e^{3} - 30 \, B a c^{2} d^{2} e^{4} - 12 \, A a c^{2} d e^{5} - 9 \, B a^{2} c e^{6}\right )} x^{3} + 21 \, {\left (959 \, B c^{3} d^{5} e - 60 \, A c^{3} d^{4} e^{2} - 30 \, B a c^{2} d^{3} e^{3} - 12 \, A a c^{2} d^{2} e^{4} - 9 \, B a^{2} c d e^{5} - 12 \, A a^{2} c e^{6}\right )} x^{2} + 7 \, {\left (1029 \, B c^{3} d^{6} - 60 \, A c^{3} d^{5} e - 30 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} - 9 \, B a^{2} c d^{2} e^{4} - 12 \, A a^{2} c d e^{5} - 10 \, B a^{3} e^{6}\right )} x + \frac {1089 \, B c^{3} d^{7} - 60 \, A c^{3} d^{6} e - 30 \, B a c^{2} d^{5} e^{2} - 12 \, A a c^{2} d^{4} e^{3} - 9 \, B a^{2} c d^{3} e^{4} - 12 \, A a^{2} c d^{2} e^{5} - 10 \, B a^{3} d e^{6} - 60 \, A a^{3} e^{7}}{e}}{420 \, {\left (e x + d\right )}^{7} e^{7}} \] Input:
integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^8,x, algorithm="giac")
Output:
B*c^3*log(abs(e*x + d))/e^8 + 1/420*(420*(7*B*c^3*d*e^5 - A*c^3*e^6)*x^6 + 630*(21*B*c^3*d^2*e^4 - 2*A*c^3*d*e^5 - B*a*c^2*e^6)*x^5 + 70*(385*B*c^3* d^3*e^3 - 30*A*c^3*d^2*e^4 - 15*B*a*c^2*d*e^5 - 6*A*a*c^2*e^6)*x^4 + 35*(8 75*B*c^3*d^4*e^2 - 60*A*c^3*d^3*e^3 - 30*B*a*c^2*d^2*e^4 - 12*A*a*c^2*d*e^ 5 - 9*B*a^2*c*e^6)*x^3 + 21*(959*B*c^3*d^5*e - 60*A*c^3*d^4*e^2 - 30*B*a*c ^2*d^3*e^3 - 12*A*a*c^2*d^2*e^4 - 9*B*a^2*c*d*e^5 - 12*A*a^2*c*e^6)*x^2 + 7*(1029*B*c^3*d^6 - 60*A*c^3*d^5*e - 30*B*a*c^2*d^4*e^2 - 12*A*a*c^2*d^3*e ^3 - 9*B*a^2*c*d^2*e^4 - 12*A*a^2*c*d*e^5 - 10*B*a^3*e^6)*x + (1089*B*c^3* d^7 - 60*A*c^3*d^6*e - 30*B*a*c^2*d^5*e^2 - 12*A*a*c^2*d^4*e^3 - 9*B*a^2*c *d^3*e^4 - 12*A*a^2*c*d^2*e^5 - 10*B*a^3*d*e^6 - 60*A*a^3*e^7)/e)/((e*x + d)^7*e^7)
Time = 6.22 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.37 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {B\,c^3\,\ln \left (d+e\,x\right )}{e^8}-\frac {x^3\,\left (\frac {3\,B\,a^2\,c\,e^7}{4}+\frac {5\,B\,a\,c^2\,d^2\,e^5}{2}+A\,a\,c^2\,d\,e^6-\frac {875\,B\,c^3\,d^4\,e^3}{12}+5\,A\,c^3\,d^3\,e^4\right )+x^6\,\left (A\,c^3\,e^7-7\,B\,c^3\,d\,e^6\right )+x^2\,\left (\frac {9\,B\,a^2\,c\,d\,e^6}{20}+\frac {3\,A\,a^2\,c\,e^7}{5}+\frac {3\,B\,a\,c^2\,d^3\,e^4}{2}+\frac {3\,A\,a\,c^2\,d^2\,e^5}{5}-\frac {959\,B\,c^3\,d^5\,e^2}{20}+3\,A\,c^3\,d^4\,e^3\right )+x^5\,\left (-\frac {63\,B\,c^3\,d^2\,e^5}{2}+3\,A\,c^3\,d\,e^6+\frac {3\,B\,a\,c^2\,e^7}{2}\right )+x\,\left (\frac {B\,a^3\,e^7}{6}+\frac {3\,B\,a^2\,c\,d^2\,e^5}{20}+\frac {A\,a^2\,c\,d\,e^6}{5}+\frac {B\,a\,c^2\,d^4\,e^3}{2}+\frac {A\,a\,c^2\,d^3\,e^4}{5}-\frac {343\,B\,c^3\,d^6\,e}{20}+A\,c^3\,d^5\,e^2\right )+x^4\,\left (-\frac {385\,B\,c^3\,d^3\,e^4}{6}+5\,A\,c^3\,d^2\,e^5+\frac {5\,B\,a\,c^2\,d\,e^6}{2}+A\,a\,c^2\,e^7\right )+\frac {A\,a^3\,e^7}{7}-\frac {363\,B\,c^3\,d^7}{140}+\frac {B\,a^3\,d\,e^6}{42}+\frac {A\,c^3\,d^6\,e}{7}+\frac {A\,a\,c^2\,d^4\,e^3}{35}+\frac {A\,a^2\,c\,d^2\,e^5}{35}+\frac {B\,a\,c^2\,d^5\,e^2}{14}+\frac {3\,B\,a^2\,c\,d^3\,e^4}{140}}{e^8\,{\left (d+e\,x\right )}^7} \] Input:
int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^8,x)
Output:
(B*c^3*log(d + e*x))/e^8 - (x^3*((3*B*a^2*c*e^7)/4 + 5*A*c^3*d^3*e^4 - (87 5*B*c^3*d^4*e^3)/12 + (5*B*a*c^2*d^2*e^5)/2 + A*a*c^2*d*e^6) + x^6*(A*c^3* e^7 - 7*B*c^3*d*e^6) + x^2*((3*A*a^2*c*e^7)/5 + 3*A*c^3*d^4*e^3 - (959*B*c ^3*d^5*e^2)/20 + (3*A*a*c^2*d^2*e^5)/5 + (3*B*a*c^2*d^3*e^4)/2 + (9*B*a^2* c*d*e^6)/20) + x^5*((3*B*a*c^2*e^7)/2 + 3*A*c^3*d*e^6 - (63*B*c^3*d^2*e^5) /2) + x*((B*a^3*e^7)/6 - (343*B*c^3*d^6*e)/20 + A*c^3*d^5*e^2 + (A*a*c^2*d ^3*e^4)/5 + (B*a*c^2*d^4*e^3)/2 + (3*B*a^2*c*d^2*e^5)/20 + (A*a^2*c*d*e^6) /5) + x^4*(A*a*c^2*e^7 + 5*A*c^3*d^2*e^5 - (385*B*c^3*d^3*e^4)/6 + (5*B*a* c^2*d*e^6)/2) + (A*a^3*e^7)/7 - (363*B*c^3*d^7)/140 + (B*a^3*d*e^6)/42 + ( A*c^3*d^6*e)/7 + (A*a*c^2*d^4*e^3)/35 + (A*a^2*c*d^2*e^5)/35 + (B*a*c^2*d^ 5*e^2)/14 + (3*B*a^2*c*d^3*e^4)/140)/(e^8*(d + e*x)^7)
Time = 0.23 (sec) , antiderivative size = 637, normalized size of antiderivative = 1.95 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^8} \, dx=\frac {60 a \,c^{3} e^{8} x^{7}-12 a^{3} c \,d^{3} e^{5}-12 a^{2} c^{2} d^{5} e^{3}+420 \,\mathrm {log}\left (e x +d \right ) b \,c^{3} d \,e^{7} x^{7}-10 a^{3} b \,d^{2} e^{6}+2940 \,\mathrm {log}\left (e x +d \right ) b \,c^{3} d^{7} e x -63 a^{2} b c \,d^{3} e^{5} x -189 a^{2} b c \,d^{2} e^{6} x^{2}-315 a^{2} b c d \,e^{7} x^{3}-210 a b \,c^{2} d^{5} e^{3} x -630 a b \,c^{2} d^{4} e^{4} x^{2}-1050 a b \,c^{2} d^{3} e^{5} x^{3}-1050 a b \,c^{2} d^{2} e^{6} x^{4}-630 a b \,c^{2} d \,e^{7} x^{5}+2940 \,\mathrm {log}\left (e x +d \right ) b \,c^{3} d^{2} e^{6} x^{6}+14700 \,\mathrm {log}\left (e x +d \right ) b \,c^{3} d^{4} e^{4} x^{4}+420 \,\mathrm {log}\left (e x +d \right ) b \,c^{3} d^{8}-60 a^{4} d \,e^{7}+669 b \,c^{3} d^{8}+8820 \,\mathrm {log}\left (e x +d \right ) b \,c^{3} d^{3} e^{5} x^{5}+8820 \,\mathrm {log}\left (e x +d \right ) b \,c^{3} d^{6} e^{2} x^{2}-9 a^{2} b c \,d^{4} e^{4}-30 a b \,c^{2} d^{6} e^{2}+14700 \,\mathrm {log}\left (e x +d \right ) b \,c^{3} d^{5} e^{3} x^{3}-70 a^{3} b d \,e^{7} x -84 a^{3} c \,d^{2} e^{6} x -252 a^{3} c d \,e^{7} x^{2}-84 a^{2} c^{2} d^{4} e^{4} x -252 a^{2} c^{2} d^{3} e^{5} x^{2}-420 a^{2} c^{2} d^{2} e^{6} x^{3}-420 a^{2} c^{2} d \,e^{7} x^{4}+4263 b \,c^{3} d^{7} e x +11319 b \,c^{3} d^{6} e^{2} x^{2}+15925 b \,c^{3} d^{5} e^{3} x^{3}+12250 b \,c^{3} d^{4} e^{4} x^{4}+4410 b \,c^{3} d^{3} e^{5} x^{5}-420 b \,c^{3} d \,e^{7} x^{7}}{420 d \,e^{8} \left (e^{7} x^{7}+7 d \,e^{6} x^{6}+21 d^{2} e^{5} x^{5}+35 d^{3} e^{4} x^{4}+35 d^{4} e^{3} x^{3}+21 d^{5} e^{2} x^{2}+7 d^{6} e x +d^{7}\right )} \] Input:
int((B*x+A)*(c*x^2+a)^3/(e*x+d)^8,x)
Output:
(420*log(d + e*x)*b*c**3*d**8 + 2940*log(d + e*x)*b*c**3*d**7*e*x + 8820*l og(d + e*x)*b*c**3*d**6*e**2*x**2 + 14700*log(d + e*x)*b*c**3*d**5*e**3*x* *3 + 14700*log(d + e*x)*b*c**3*d**4*e**4*x**4 + 8820*log(d + e*x)*b*c**3*d **3*e**5*x**5 + 2940*log(d + e*x)*b*c**3*d**2*e**6*x**6 + 420*log(d + e*x) *b*c**3*d*e**7*x**7 - 60*a**4*d*e**7 - 10*a**3*b*d**2*e**6 - 70*a**3*b*d*e **7*x - 12*a**3*c*d**3*e**5 - 84*a**3*c*d**2*e**6*x - 252*a**3*c*d*e**7*x* *2 - 9*a**2*b*c*d**4*e**4 - 63*a**2*b*c*d**3*e**5*x - 189*a**2*b*c*d**2*e* *6*x**2 - 315*a**2*b*c*d*e**7*x**3 - 12*a**2*c**2*d**5*e**3 - 84*a**2*c**2 *d**4*e**4*x - 252*a**2*c**2*d**3*e**5*x**2 - 420*a**2*c**2*d**2*e**6*x**3 - 420*a**2*c**2*d*e**7*x**4 - 30*a*b*c**2*d**6*e**2 - 210*a*b*c**2*d**5*e **3*x - 630*a*b*c**2*d**4*e**4*x**2 - 1050*a*b*c**2*d**3*e**5*x**3 - 1050* a*b*c**2*d**2*e**6*x**4 - 630*a*b*c**2*d*e**7*x**5 + 60*a*c**3*e**8*x**7 + 669*b*c**3*d**8 + 4263*b*c**3*d**7*e*x + 11319*b*c**3*d**6*e**2*x**2 + 15 925*b*c**3*d**5*e**3*x**3 + 12250*b*c**3*d**4*e**4*x**4 + 4410*b*c**3*d**3 *e**5*x**5 - 420*b*c**3*d*e**7*x**7)/(420*d*e**8*(d**7 + 7*d**6*e*x + 21*d **5*e**2*x**2 + 35*d**4*e**3*x**3 + 35*d**3*e**4*x**4 + 21*d**2*e**5*x**5 + 7*d*e**6*x**6 + e**7*x**7))