Integrand size = 22, antiderivative size = 320 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {B c^3 x}{e^7}+\frac {(B d-A e) \left (c d^2+a e^2\right )^3}{6 e^8 (d+e x)^6}-\frac {\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{5 e^8 (d+e x)^5}+\frac {3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{4 e^8 (d+e x)^4}+\frac {c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right )}{3 e^8 (d+e x)^3}+\frac {c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right )}{2 e^8 (d+e x)^2}-\frac {3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right )}{e^8 (d+e x)}-\frac {c^3 (7 B d-A e) \log (d+e x)}{e^8} \]
B*c^3*x/e^7+1/6*(-A*e+B*d)*(a*e^2+c*d^2)^3/e^8/(e*x+d)^6-1/5*(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)^5+3/4*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)^4+1/3*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)^3+1/2*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)^2-3*c^2*( -2*A*c*d*e+B*a*e^2+7*B*c*d^2)/e^8/(e*x+d)-c^3*(-A*e+7*B*d)*ln(e*x+d)/e^8
Time = 0.13 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.18 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^7} \, dx=-\frac {A e \left (10 a^3 e^6+3 a^2 c e^4 \left (d^2+6 d e x+15 e^2 x^2\right )+6 a c^2 e^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )-c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )\right )+B \left (2 a^3 e^6 (d+6 e x)+3 a^2 c e^4 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+30 a c^2 e^2 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )+c^3 \left (669 d^7+3594 d^6 e x+7725 d^5 e^2 x^2+8200 d^4 e^3 x^3+4050 d^3 e^4 x^4+360 d^2 e^5 x^5-360 d e^6 x^6-60 e^7 x^7\right )\right )+60 c^3 (7 B d-A e) (d+e x)^6 \log (d+e x)}{60 e^8 (d+e x)^6} \]
-1/60*(A*e*(10*a^3*e^6 + 3*a^2*c*e^4*(d^2 + 6*d*e*x + 15*e^2*x^2) + 6*a*c^ 2*e^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) - c^3 *d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e ^4*x^4 + 360*e^5*x^5)) + B*(2*a^3*e^6*(d + 6*e*x) + 3*a^2*c*e^4*(d^3 + 6*d ^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 30*a*c^2*e^2*(d^5 + 6*d^4*e*x + 15*d ^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5) + c^3*(669*d^7 + 3 594*d^6*e*x + 7725*d^5*e^2*x^2 + 8200*d^4*e^3*x^3 + 4050*d^3*e^4*x^4 + 360 *d^2*e^5*x^5 - 360*d*e^6*x^6 - 60*e^7*x^7)) + 60*c^3*(7*B*d - A*e)*(d + e* x)^6*Log[d + e*x])/(e^8*(d + e*x)^6)
Time = 0.59 (sec) , antiderivative size = 320, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {c \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7 (d+e x)^4}-\frac {3 c^2 \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7 (d+e x)^2}+\frac {c^2 \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7 (d+e x)^3}+\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7 (d+e x)^6}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)^7}+\frac {3 c \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7 (d+e x)^5}+\frac {c^3 (A e-7 B d)}{e^7 (d+e x)}+\frac {B c^3}{e^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{3 e^8 (d+e x)^3}-\frac {3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (d+e x)}+\frac {c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{2 e^8 (d+e x)^2}-\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^5}+\frac {\left (a e^2+c d^2\right )^3 (B d-A e)}{6 e^8 (d+e x)^6}+\frac {3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{4 e^8 (d+e x)^4}-\frac {c^3 (7 B d-A e) \log (d+e x)}{e^8}+\frac {B c^3 x}{e^7}\) |
(B*c^3*x)/e^7 + ((B*d - A*e)*(c*d^2 + a*e^2)^3)/(6*e^8*(d + e*x)^6) - ((c* d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(5*e^8*(d + e*x)^5) + (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))/(4*e ^8*(d + e*x)^4) + (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)))/(3*e^8*(d + e*x)^3) + (c^2*(35*B*c*d^3 - 15*A*c* d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3))/(2*e^8*(d + e*x)^2) - (3*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2))/(e^8*(d + e*x)) - (c^3*(7*B*d - A*e)*Log[d + e*x] )/e^8
3.14.25.3.1 Defintions of rubi rules used
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 = 0.27 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.36
| method | result | size |
| risch | \(\frac {B \,c^{3} x}{e^{7}}+\frac {\left (6 A \,c^{3} d \,e^{5}-3 B a \,c^{2} e^{6}-21 B \,c^{3} d^{2} e^{4}\right ) x^{5}-\frac {c^{2} e^{3} \left (3 A a \,e^{3}-45 A c \,d^{2} e +15 B a d \,e^{2}+175 B c \,d^{3}\right ) x^{4}}{2}-\frac {c \,e^{2} \left (6 A a c d \,e^{3}-110 A \,c^{2} d^{3} e +3 B \,e^{4} a^{2}+30 B a c \,d^{2} e^{2}+455 B \,c^{2} d^{4}\right ) x^{3}}{3}-\frac {c e \left (3 A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}-125 A \,c^{2} d^{4} e +3 B \,a^{2} d \,e^{4}+30 B a c \,d^{3} e^{2}+539 B \,c^{2} d^{5}\right ) x^{2}}{4}+\left (-\frac {3}{10} A \,a^{2} c d \,e^{5}-\frac {3}{5} A a \,c^{2} d^{3} e^{3}+\frac {137}{10} A \,c^{3} d^{5} e -\frac {1}{5} B \,a^{3} e^{6}-\frac {3}{10} B \,a^{2} c \,d^{2} e^{4}-3 B a \,c^{2} d^{4} e^{2}-\frac {609}{10} B \,c^{3} d^{6}\right ) x -\frac {10 A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+6 A a \,c^{2} d^{4} e^{3}-147 A \,c^{3} d^{6} e +2 B \,a^{3} d \,e^{6}+3 B \,a^{2} c \,d^{3} e^{4}+30 B a \,c^{2} d^{5} e^{2}+669 B \,c^{3} d^{7}}{60 e}}{e^{7} \left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right ) A}{e^{7}}-\frac {7 c^{3} \ln \left (e x +d \right ) B d}{e^{8}}\) | \(435\) |
| default | \(\frac {B \,c^{3} x}{e^{7}}-\frac {A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}}{6 e^{8} \left (e x +d \right )^{6}}-\frac {-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}}{5 e^{8} \left (e x +d \right )^{5}}+\frac {c \left (12 A a c d \,e^{3}+20 A \,c^{2} d^{3} e -3 B \,e^{4} a^{2}-30 B a c \,d^{2} e^{2}-35 B \,c^{2} d^{4}\right )}{3 e^{8} \left (e x +d \right )^{3}}+\frac {3 c^{2} \left (2 A c d e -B a \,e^{2}-7 B c \,d^{2}\right )}{e^{8} \left (e x +d \right )}+\frac {c^{3} \left (A e -7 B d \right ) \ln \left (e x +d \right )}{e^{8}}-\frac {3 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{4 e^{8} \left (e x +d \right )^{4}}-\frac {c^{2} \left (3 A a \,e^{3}+15 A c \,d^{2} e -15 B a d \,e^{2}-35 B c \,d^{3}\right )}{2 e^{8} \left (e x +d \right )^{2}}\) | \(440\) |
| norman | \(\frac {\frac {B \,c^{3} x^{7}}{e}-\frac {10 A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+6 A a \,c^{2} d^{4} e^{3}-147 A \,c^{3} d^{6} e +2 B \,a^{3} d \,e^{6}+3 B \,a^{2} c \,d^{3} e^{4}+30 B a \,c^{2} d^{5} e^{2}+1029 B \,c^{3} d^{7}}{60 e^{8}}+\frac {3 \left (2 A \,c^{3} d e -B \,e^{2} a \,c^{2}-14 B \,c^{3} d^{2}\right ) x^{5}}{e^{3}}-\frac {3 \left (A a \,c^{2} e^{3}-15 A \,c^{3} d^{2} e +5 B a \,c^{2} d \,e^{2}+105 B \,c^{3} d^{3}\right ) x^{4}}{2 e^{4}}-\frac {\left (6 A a \,c^{2} d \,e^{3}-110 A \,c^{3} d^{3} e +3 B \,e^{4} c \,a^{2}+30 B a \,c^{2} d^{2} e^{2}+770 B \,c^{3} d^{4}\right ) x^{3}}{3 e^{5}}-\frac {\left (3 A \,a^{2} c \,e^{5}+6 A a \,c^{2} d^{2} e^{3}-125 A \,c^{3} d^{4} e +3 B \,a^{2} c d \,e^{4}+30 B a \,c^{2} d^{3} e^{2}+875 B \,c^{3} d^{5}\right ) x^{2}}{4 e^{6}}-\frac {\left (3 A \,a^{2} c d \,e^{5}+6 A a \,c^{2} d^{3} e^{3}-137 A \,c^{3} d^{5} e +2 B \,a^{3} e^{6}+3 B \,a^{2} c \,d^{2} e^{4}+30 B a \,c^{2} d^{4} e^{2}+959 B \,c^{3} d^{6}\right ) x}{10 e^{7}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \left (A e -7 B d \right ) \ln \left (e x +d \right )}{e^{8}}\) | \(445\) |
| parallelrisch | \(\frac {-90 A \,x^{2} a \,c^{2} d^{2} e^{5}-45 B \,x^{2} a^{2} c d \,e^{6}-450 B \,x^{2} a \,c^{2} d^{3} e^{4}-18 A x \,a^{2} c d \,e^{6}-36 A x a \,c^{2} d^{3} e^{4}-450 B \,x^{4} a \,c^{2} d \,e^{6}-120 A \,x^{3} a \,c^{2} d \,e^{6}-600 B \,x^{3} a \,c^{2} d^{2} e^{5}-18 B x \,a^{2} c \,d^{2} e^{5}-180 B x a \,c^{2} d^{4} e^{3}-10 A \,a^{3} e^{7}+1200 A \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{4}-8400 B \ln \left (e x +d \right ) x^{3} c^{3} d^{4} e^{3}+900 A \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{3}-6300 B \ln \left (e x +d \right ) x^{2} c^{3} d^{5} e^{2}+360 A \ln \left (e x +d \right ) x \,c^{3} d^{5} e^{2}-2520 B \ln \left (e x +d \right ) x \,c^{3} d^{6} e -12 B x \,a^{3} e^{7}+60 B \,x^{7} c^{3} e^{7}-420 B \ln \left (e x +d \right ) c^{3} d^{7}-2 B \,a^{3} d \,e^{6}+147 A \,c^{3} d^{6} e -1029 B \,c^{3} d^{7}+360 A \ln \left (e x +d \right ) x^{5} c^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}+900 A \ln \left (e x +d \right ) x^{4} c^{3} d^{2} e^{5}-6300 B \ln \left (e x +d \right ) x^{4} c^{3} d^{3} e^{4}-420 B \ln \left (e x +d \right ) x^{6} c^{3} d \,e^{6}+822 A x \,c^{3} d^{5} e^{2}-5754 B x \,c^{3} d^{6} e -30 B a \,c^{2} d^{5} e^{2}-3 A \,a^{2} c \,d^{2} e^{5}-6 A a \,c^{2} d^{4} e^{3}+60 A \ln \left (e x +d \right ) x^{6} c^{3} e^{7}+360 A \,x^{5} c^{3} d \,e^{6}+60 A \ln \left (e x +d \right ) c^{3} d^{6} e -2520 B \,x^{5} c^{3} d^{2} e^{5}+1350 A \,x^{4} c^{3} d^{2} e^{5}-9450 B \,x^{4} c^{3} d^{3} e^{4}+2200 A \,x^{3} c^{3} d^{3} e^{4}-15400 B \,x^{3} c^{3} d^{4} e^{3}+1875 A \,x^{2} c^{3} d^{4} e^{3}-13125 B \,x^{2} c^{3} d^{5} e^{2}-60 B \,x^{3} a^{2} c \,e^{7}-45 A \,x^{2} a^{2} c \,e^{7}-180 B \,x^{5} a \,c^{2} e^{7}-90 A \,x^{4} a \,c^{2} e^{7}-2520 B \ln \left (e x +d \right ) x^{5} c^{3} d^{2} e^{5}}{60 e^{8} \left (e x +d \right )^{6}}\) | \(734\) |
B*c^3*x/e^7+((6*A*c^3*d*e^5-3*B*a*c^2*e^6-21*B*c^3*d^2*e^4)*x^5-1/2*c^2*e^ 3*(3*A*a*e^3-45*A*c*d^2*e+15*B*a*d*e^2+175*B*c*d^3)*x^4-1/3*c*e^2*(6*A*a*c *d*e^3-110*A*c^2*d^3*e+3*B*a^2*e^4+30*B*a*c*d^2*e^2+455*B*c^2*d^4)*x^3-1/4 *c*e*(3*A*a^2*e^5+6*A*a*c*d^2*e^3-125*A*c^2*d^4*e+3*B*a^2*d*e^4+30*B*a*c*d ^3*e^2+539*B*c^2*d^5)*x^2+(-3/10*A*a^2*c*d*e^5-3/5*A*a*c^2*d^3*e^3+137/10* A*c^3*d^5*e-1/5*B*a^3*e^6-3/10*B*a^2*c*d^2*e^4-3*B*a*c^2*d^4*e^2-609/10*B* c^3*d^6)*x-1/60/e*(10*A*a^3*e^7+3*A*a^2*c*d^2*e^5+6*A*a*c^2*d^4*e^3-147*A* c^3*d^6*e+2*B*a^3*d*e^6+3*B*a^2*c*d^3*e^4+30*B*a*c^2*d^5*e^2+669*B*c^3*d^7 ))/e^7/(e*x+d)^6+c^3/e^7*ln(e*x+d)*A-7*c^3/e^8*ln(e*x+d)*B*d
Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (310) = 620\).
Time = 0.26 (sec) , antiderivative size = 695, normalized size of antiderivative = 2.17 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {60 \, B c^{3} e^{7} x^{7} + 360 \, B c^{3} d e^{6} x^{6} - 669 \, B c^{3} d^{7} + 147 \, A c^{3} d^{6} e - 30 \, B a c^{2} d^{5} e^{2} - 6 \, A a c^{2} d^{4} e^{3} - 3 \, B a^{2} c d^{3} e^{4} - 3 \, A a^{2} c d^{2} e^{5} - 2 \, B a^{3} d e^{6} - 10 \, A a^{3} e^{7} - 180 \, {\left (2 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} - 90 \, {\left (45 \, B c^{3} d^{3} e^{4} - 15 \, A c^{3} d^{2} e^{5} + 5 \, B a c^{2} d e^{6} + A a c^{2} e^{7}\right )} x^{4} - 20 \, {\left (410 \, B c^{3} d^{4} e^{3} - 110 \, A c^{3} d^{3} e^{4} + 30 \, B a c^{2} d^{2} e^{5} + 6 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} - 15 \, {\left (515 \, B c^{3} d^{5} e^{2} - 125 \, A c^{3} d^{4} e^{3} + 30 \, B a c^{2} d^{3} e^{4} + 6 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 3 \, A a^{2} c e^{7}\right )} x^{2} - 6 \, {\left (599 \, B c^{3} d^{6} e - 137 \, A c^{3} d^{5} e^{2} + 30 \, B a c^{2} d^{4} e^{3} + 6 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 3 \, A a^{2} c d e^{6} + 2 \, B a^{3} e^{7}\right )} x - 60 \, {\left (7 \, B c^{3} d^{7} - A c^{3} d^{6} e + {\left (7 \, B c^{3} d e^{6} - A c^{3} e^{7}\right )} x^{6} + 6 \, {\left (7 \, B c^{3} d^{2} e^{5} - A c^{3} d e^{6}\right )} x^{5} + 15 \, {\left (7 \, B c^{3} d^{3} e^{4} - A c^{3} d^{2} e^{5}\right )} x^{4} + 20 \, {\left (7 \, B c^{3} d^{4} e^{3} - A c^{3} d^{3} e^{4}\right )} x^{3} + 15 \, {\left (7 \, B c^{3} d^{5} e^{2} - A c^{3} d^{4} e^{3}\right )} x^{2} + 6 \, {\left (7 \, B c^{3} d^{6} e - A c^{3} d^{5} e^{2}\right )} x\right )} \log \left (e x + d\right )}{60 \, {\left (e^{14} x^{6} + 6 \, d e^{13} x^{5} + 15 \, d^{2} e^{12} x^{4} + 20 \, d^{3} e^{11} x^{3} + 15 \, d^{4} e^{10} x^{2} + 6 \, d^{5} e^{9} x + d^{6} e^{8}\right )}} \]
1/60*(60*B*c^3*e^7*x^7 + 360*B*c^3*d*e^6*x^6 - 669*B*c^3*d^7 + 147*A*c^3*d ^6*e - 30*B*a*c^2*d^5*e^2 - 6*A*a*c^2*d^4*e^3 - 3*B*a^2*c*d^3*e^4 - 3*A*a^ 2*c*d^2*e^5 - 2*B*a^3*d*e^6 - 10*A*a^3*e^7 - 180*(2*B*c^3*d^2*e^5 - 2*A*c^ 3*d*e^6 + B*a*c^2*e^7)*x^5 - 90*(45*B*c^3*d^3*e^4 - 15*A*c^3*d^2*e^5 + 5*B *a*c^2*d*e^6 + A*a*c^2*e^7)*x^4 - 20*(410*B*c^3*d^4*e^3 - 110*A*c^3*d^3*e^ 4 + 30*B*a*c^2*d^2*e^5 + 6*A*a*c^2*d*e^6 + 3*B*a^2*c*e^7)*x^3 - 15*(515*B* c^3*d^5*e^2 - 125*A*c^3*d^4*e^3 + 30*B*a*c^2*d^3*e^4 + 6*A*a*c^2*d^2*e^5 + 3*B*a^2*c*d*e^6 + 3*A*a^2*c*e^7)*x^2 - 6*(599*B*c^3*d^6*e - 137*A*c^3*d^5 *e^2 + 30*B*a*c^2*d^4*e^3 + 6*A*a*c^2*d^3*e^4 + 3*B*a^2*c*d^2*e^5 + 3*A*a^ 2*c*d*e^6 + 2*B*a^3*e^7)*x - 60*(7*B*c^3*d^7 - A*c^3*d^6*e + (7*B*c^3*d*e^ 6 - A*c^3*e^7)*x^6 + 6*(7*B*c^3*d^2*e^5 - A*c^3*d*e^6)*x^5 + 15*(7*B*c^3*d ^3*e^4 - A*c^3*d^2*e^5)*x^4 + 20*(7*B*c^3*d^4*e^3 - A*c^3*d^3*e^4)*x^3 + 1 5*(7*B*c^3*d^5*e^2 - A*c^3*d^4*e^3)*x^2 + 6*(7*B*c^3*d^6*e - A*c^3*d^5*e^2 )*x)*log(e*x + d))/(e^14*x^6 + 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^1 1*x^3 + 15*d^4*e^10*x^2 + 6*d^5*e^9*x + d^6*e^8)
Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.60 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^7} \, dx=-\frac {669 \, B c^{3} d^{7} - 147 \, A c^{3} d^{6} e + 30 \, B a c^{2} d^{5} e^{2} + 6 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + 3 \, A a^{2} c d^{2} e^{5} + 2 \, B a^{3} d e^{6} + 10 \, A a^{3} e^{7} + 180 \, {\left (7 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 30 \, {\left (175 \, B c^{3} d^{3} e^{4} - 45 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 20 \, {\left (455 \, B c^{3} d^{4} e^{3} - 110 \, A c^{3} d^{3} e^{4} + 30 \, B a c^{2} d^{2} e^{5} + 6 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 15 \, {\left (539 \, B c^{3} d^{5} e^{2} - 125 \, A c^{3} d^{4} e^{3} + 30 \, B a c^{2} d^{3} e^{4} + 6 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 3 \, A a^{2} c e^{7}\right )} x^{2} + 6 \, {\left (609 \, B c^{3} d^{6} e - 137 \, A c^{3} d^{5} e^{2} + 30 \, B a c^{2} d^{4} e^{3} + 6 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 3 \, A a^{2} c d e^{6} + 2 \, B a^{3} e^{7}\right )} x}{60 \, {\left (e^{14} x^{6} + 6 \, d e^{13} x^{5} + 15 \, d^{2} e^{12} x^{4} + 20 \, d^{3} e^{11} x^{3} + 15 \, d^{4} e^{10} x^{2} + 6 \, d^{5} e^{9} x + d^{6} e^{8}\right )}} + \frac {B c^{3} x}{e^{7}} - \frac {{\left (7 \, B c^{3} d - A c^{3} e\right )} \log \left (e x + d\right )}{e^{8}} \]
-1/60*(669*B*c^3*d^7 - 147*A*c^3*d^6*e + 30*B*a*c^2*d^5*e^2 + 6*A*a*c^2*d^ 4*e^3 + 3*B*a^2*c*d^3*e^4 + 3*A*a^2*c*d^2*e^5 + 2*B*a^3*d*e^6 + 10*A*a^3*e ^7 + 180*(7*B*c^3*d^2*e^5 - 2*A*c^3*d*e^6 + B*a*c^2*e^7)*x^5 + 30*(175*B*c ^3*d^3*e^4 - 45*A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^6 + 3*A*a*c^2*e^7)*x^4 + 20 *(455*B*c^3*d^4*e^3 - 110*A*c^3*d^3*e^4 + 30*B*a*c^2*d^2*e^5 + 6*A*a*c^2*d *e^6 + 3*B*a^2*c*e^7)*x^3 + 15*(539*B*c^3*d^5*e^2 - 125*A*c^3*d^4*e^3 + 30 *B*a*c^2*d^3*e^4 + 6*A*a*c^2*d^2*e^5 + 3*B*a^2*c*d*e^6 + 3*A*a^2*c*e^7)*x^ 2 + 6*(609*B*c^3*d^6*e - 137*A*c^3*d^5*e^2 + 30*B*a*c^2*d^4*e^3 + 6*A*a*c^ 2*d^3*e^4 + 3*B*a^2*c*d^2*e^5 + 3*A*a^2*c*d*e^6 + 2*B*a^3*e^7)*x)/(e^14*x^ 6 + 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^11*x^3 + 15*d^4*e^10*x^2 + 6 *d^5*e^9*x + d^6*e^8) + B*c^3*x/e^7 - (7*B*c^3*d - A*c^3*e)*log(e*x + d)/e ^8
Time = 0.26 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {B c^{3} x}{e^{7}} - \frac {{\left (7 \, B c^{3} d - A c^{3} e\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} - \frac {669 \, B c^{3} d^{7} - 147 \, A c^{3} d^{6} e + 30 \, B a c^{2} d^{5} e^{2} + 6 \, A a c^{2} d^{4} e^{3} + 3 \, B a^{2} c d^{3} e^{4} + 3 \, A a^{2} c d^{2} e^{5} + 2 \, B a^{3} d e^{6} + 10 \, A a^{3} e^{7} + 180 \, {\left (7 \, B c^{3} d^{2} e^{5} - 2 \, A c^{3} d e^{6} + B a c^{2} e^{7}\right )} x^{5} + 30 \, {\left (175 \, B c^{3} d^{3} e^{4} - 45 \, A c^{3} d^{2} e^{5} + 15 \, B a c^{2} d e^{6} + 3 \, A a c^{2} e^{7}\right )} x^{4} + 20 \, {\left (455 \, B c^{3} d^{4} e^{3} - 110 \, A c^{3} d^{3} e^{4} + 30 \, B a c^{2} d^{2} e^{5} + 6 \, A a c^{2} d e^{6} + 3 \, B a^{2} c e^{7}\right )} x^{3} + 15 \, {\left (539 \, B c^{3} d^{5} e^{2} - 125 \, A c^{3} d^{4} e^{3} + 30 \, B a c^{2} d^{3} e^{4} + 6 \, A a c^{2} d^{2} e^{5} + 3 \, B a^{2} c d e^{6} + 3 \, A a^{2} c e^{7}\right )} x^{2} + 6 \, {\left (609 \, B c^{3} d^{6} e - 137 \, A c^{3} d^{5} e^{2} + 30 \, B a c^{2} d^{4} e^{3} + 6 \, A a c^{2} d^{3} e^{4} + 3 \, B a^{2} c d^{2} e^{5} + 3 \, A a^{2} c d e^{6} + 2 \, B a^{3} e^{7}\right )} x}{60 \, {\left (e x + d\right )}^{6} e^{8}} \]
B*c^3*x/e^7 - (7*B*c^3*d - A*c^3*e)*log(abs(e*x + d))/e^8 - 1/60*(669*B*c^ 3*d^7 - 147*A*c^3*d^6*e + 30*B*a*c^2*d^5*e^2 + 6*A*a*c^2*d^4*e^3 + 3*B*a^2 *c*d^3*e^4 + 3*A*a^2*c*d^2*e^5 + 2*B*a^3*d*e^6 + 10*A*a^3*e^7 + 180*(7*B*c ^3*d^2*e^5 - 2*A*c^3*d*e^6 + B*a*c^2*e^7)*x^5 + 30*(175*B*c^3*d^3*e^4 - 45 *A*c^3*d^2*e^5 + 15*B*a*c^2*d*e^6 + 3*A*a*c^2*e^7)*x^4 + 20*(455*B*c^3*d^4 *e^3 - 110*A*c^3*d^3*e^4 + 30*B*a*c^2*d^2*e^5 + 6*A*a*c^2*d*e^6 + 3*B*a^2* c*e^7)*x^3 + 15*(539*B*c^3*d^5*e^2 - 125*A*c^3*d^4*e^3 + 30*B*a*c^2*d^3*e^ 4 + 6*A*a*c^2*d^2*e^5 + 3*B*a^2*c*d*e^6 + 3*A*a^2*c*e^7)*x^2 + 6*(609*B*c^ 3*d^6*e - 137*A*c^3*d^5*e^2 + 30*B*a*c^2*d^4*e^3 + 6*A*a*c^2*d^3*e^4 + 3*B *a^2*c*d^2*e^5 + 3*A*a^2*c*d*e^6 + 2*B*a^3*e^7)*x)/((e*x + d)^6*e^8)
Time = 10.77 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.58 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {\ln \left (d+e\,x\right )\,\left (A\,c^3\,e-7\,B\,c^3\,d\right )}{e^8}-\frac {\frac {2\,B\,a^3\,d\,e^6+10\,A\,a^3\,e^7+3\,B\,a^2\,c\,d^3\,e^4+3\,A\,a^2\,c\,d^2\,e^5+30\,B\,a\,c^2\,d^5\,e^2+6\,A\,a\,c^2\,d^4\,e^3+669\,B\,c^3\,d^7-147\,A\,c^3\,d^6\,e}{60\,e}+x^2\,\left (\frac {3\,B\,a^2\,c\,d\,e^5}{4}+\frac {3\,A\,a^2\,c\,e^6}{4}+\frac {15\,B\,a\,c^2\,d^3\,e^3}{2}+\frac {3\,A\,a\,c^2\,d^2\,e^4}{2}+\frac {539\,B\,c^3\,d^5\,e}{4}-\frac {125\,A\,c^3\,d^4\,e^2}{4}\right )+x^3\,\left (B\,a^2\,c\,e^6+10\,B\,a\,c^2\,d^2\,e^4+2\,A\,a\,c^2\,d\,e^5+\frac {455\,B\,c^3\,d^4\,e^2}{3}-\frac {110\,A\,c^3\,d^3\,e^3}{3}\right )+x^5\,\left (21\,B\,c^3\,d^2\,e^4-6\,A\,c^3\,d\,e^5+3\,B\,a\,c^2\,e^6\right )+x\,\left (\frac {B\,a^3\,e^6}{5}+\frac {3\,B\,a^2\,c\,d^2\,e^4}{10}+\frac {3\,A\,a^2\,c\,d\,e^5}{10}+3\,B\,a\,c^2\,d^4\,e^2+\frac {3\,A\,a\,c^2\,d^3\,e^3}{5}+\frac {609\,B\,c^3\,d^6}{10}-\frac {137\,A\,c^3\,d^5\,e}{10}\right )+x^4\,\left (\frac {175\,B\,c^3\,d^3\,e^3}{2}-\frac {45\,A\,c^3\,d^2\,e^4}{2}+\frac {15\,B\,a\,c^2\,d\,e^5}{2}+\frac {3\,A\,a\,c^2\,e^6}{2}\right )}{d^6\,e^7+6\,d^5\,e^8\,x+15\,d^4\,e^9\,x^2+20\,d^3\,e^{10}\,x^3+15\,d^2\,e^{11}\,x^4+6\,d\,e^{12}\,x^5+e^{13}\,x^6}+\frac {B\,c^3\,x}{e^7} \]
(log(d + e*x)*(A*c^3*e - 7*B*c^3*d))/e^8 - ((10*A*a^3*e^7 + 669*B*c^3*d^7 + 2*B*a^3*d*e^6 - 147*A*c^3*d^6*e + 6*A*a*c^2*d^4*e^3 + 3*A*a^2*c*d^2*e^5 + 30*B*a*c^2*d^5*e^2 + 3*B*a^2*c*d^3*e^4)/(60*e) + x^2*((3*A*a^2*c*e^6)/4 + (539*B*c^3*d^5*e)/4 - (125*A*c^3*d^4*e^2)/4 + (3*A*a*c^2*d^2*e^4)/2 + (1 5*B*a*c^2*d^3*e^3)/2 + (3*B*a^2*c*d*e^5)/4) + x^3*(B*a^2*c*e^6 - (110*A*c^ 3*d^3*e^3)/3 + (455*B*c^3*d^4*e^2)/3 + 10*B*a*c^2*d^2*e^4 + 2*A*a*c^2*d*e^ 5) + x^5*(3*B*a*c^2*e^6 - 6*A*c^3*d*e^5 + 21*B*c^3*d^2*e^4) + x*((B*a^3*e^ 6)/5 + (609*B*c^3*d^6)/10 - (137*A*c^3*d^5*e)/10 + (3*A*a*c^2*d^3*e^3)/5 + 3*B*a*c^2*d^4*e^2 + (3*B*a^2*c*d^2*e^4)/10 + (3*A*a^2*c*d*e^5)/10) + x^4* ((3*A*a*c^2*e^6)/2 - (45*A*c^3*d^2*e^4)/2 + (175*B*c^3*d^3*e^3)/2 + (15*B* a*c^2*d*e^5)/2))/(d^6*e^7 + e^13*x^6 + 6*d^5*e^8*x + 6*d*e^12*x^5 + 15*d^4 *e^9*x^2 + 20*d^3*e^10*x^3 + 15*d^2*e^11*x^4) + (B*c^3*x)/e^7