Integrand size = 24, antiderivative size = 257 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{d+e x} \, dx=\frac {d^2 (B d-A e) (c d-b e)^3 x}{e^7}-\frac {d (B d-A e) (c d-b e)^3 x^2}{2 e^6}+\frac {(B d-A e) (c d-b e)^3 x^3}{3 e^5}-\frac {\left (B (c d-b e)^3-A c e \left (c^2 d^2-3 b c d e+3 b^2 e^2\right )\right ) x^4}{4 e^4}-\frac {c \left (A c e (c d-3 b e)-B \left (c^2 d^2-3 b c d e+3 b^2 e^2\right )\right ) x^5}{5 e^3}-\frac {c^2 (B c d-3 b B e-A c e) x^6}{6 e^2}+\frac {B c^3 x^7}{7 e}-\frac {d^3 (B d-A e) (c d-b e)^3 \log (d+e x)}{e^8} \] Output:
d^2*(-A*e+B*d)*(-b*e+c*d)^3*x/e^7-1/2*d*(-A*e+B*d)*(-b*e+c*d)^3*x^2/e^6+1/ 3*(-A*e+B*d)*(-b*e+c*d)^3*x^3/e^5-1/4*(B*(-b*e+c*d)^3-A*c*e*(3*b^2*e^2-3*b *c*d*e+c^2*d^2))*x^4/e^4-1/5*c*(A*c*e*(-3*b*e+c*d)-B*(3*b^2*e^2-3*b*c*d*e+ c^2*d^2))*x^5/e^3-1/6*c^2*(-A*c*e-3*B*b*e+B*c*d)*x^6/e^2+1/7*B*c^3*x^7/e-d ^3*(-A*e+B*d)*(-b*e+c*d)^3*ln(e*x+d)/e^8
Time = 0.13 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{d+e x} \, dx=\frac {420 d^2 e (B d-A e) (c d-b e)^3 x-210 d e^2 (B d-A e) (c d-b e)^3 x^2+140 e^3 (-B d+A e) (-c d+b e)^3 x^3+105 e^4 \left (-B (c d-b e)^3+A c e \left (c^2 d^2-3 b c d e+3 b^2 e^2\right )\right ) x^4+84 c e^5 \left (A c e (-c d+3 b e)+B \left (c^2 d^2-3 b c d e+3 b^2 e^2\right )\right ) x^5+70 c^2 e^6 (-B c d+3 b B e+A c e) x^6+60 B c^3 e^7 x^7-420 d^3 (B d-A e) (c d-b e)^3 \log (d+e x)}{420 e^8} \] Input:
Integrate[((A + B*x)*(b*x + c*x^2)^3)/(d + e*x),x]
Output:
(420*d^2*e*(B*d - A*e)*(c*d - b*e)^3*x - 210*d*e^2*(B*d - A*e)*(c*d - b*e) ^3*x^2 + 140*e^3*(-(B*d) + A*e)*(-(c*d) + b*e)^3*x^3 + 105*e^4*(-(B*(c*d - b*e)^3) + A*c*e*(c^2*d^2 - 3*b*c*d*e + 3*b^2*e^2))*x^4 + 84*c*e^5*(A*c*e* (-(c*d) + 3*b*e) + B*(c^2*d^2 - 3*b*c*d*e + 3*b^2*e^2))*x^5 + 70*c^2*e^6*( -(B*c*d) + 3*b*B*e + A*c*e)*x^6 + 60*B*c^3*e^7*x^7 - 420*d^3*(B*d - A*e)*( c*d - b*e)^3*Log[d + e*x])/(420*e^8)
Time = 1.09 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1195, 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.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^3}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {x^3 \left (A c e \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )-B (c d-b e)^3\right )}{e^4}+\frac {c x^4 \left (B \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )-A c e (c d-3 b e)\right )}{e^3}+\frac {c^2 x^5 (A c e+3 b B e-B c d)}{e^2}-\frac {d^3 (B d-A e) (c d-b e)^3}{e^7 (d+e x)}+\frac {d^2 (B d-A e) (c d-b e)^3}{e^7}-\frac {d x (B d-A e) (c d-b e)^3}{e^6}+\frac {x^2 (A e-B d) (b e-c d)^3}{e^5}+\frac {B c^3 x^6}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {x^4 \left (B (c d-b e)^3-A c e \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )\right )}{4 e^4}-\frac {c x^5 \left (A c e (c d-3 b e)-B \left (3 b^2 e^2-3 b c d e+c^2 d^2\right )\right )}{5 e^3}-\frac {c^2 x^6 (-A c e-3 b B e+B c d)}{6 e^2}-\frac {d^3 (B d-A e) (c d-b e)^3 \log (d+e x)}{e^8}+\frac {d^2 x (B d-A e) (c d-b e)^3}{e^7}-\frac {d x^2 (B d-A e) (c d-b e)^3}{2 e^6}+\frac {x^3 (B d-A e) (c d-b e)^3}{3 e^5}+\frac {B c^3 x^7}{7 e}\) |
Input:
Int[((A + B*x)*(b*x + c*x^2)^3)/(d + e*x),x]
Output:
(d^2*(B*d - A*e)*(c*d - b*e)^3*x)/e^7 - (d*(B*d - A*e)*(c*d - b*e)^3*x^2)/ (2*e^6) + ((B*d - A*e)*(c*d - b*e)^3*x^3)/(3*e^5) - ((B*(c*d - b*e)^3 - A* c*e*(c^2*d^2 - 3*b*c*d*e + 3*b^2*e^2))*x^4)/(4*e^4) - (c*(A*c*e*(c*d - 3*b *e) - B*(c^2*d^2 - 3*b*c*d*e + 3*b^2*e^2))*x^5)/(5*e^3) - (c^2*(B*c*d - 3* b*B*e - A*c*e)*x^6)/(6*e^2) + (B*c^3*x^7)/(7*e) - (d^3*(B*d - A*e)*(c*d - b*e)^3*Log[d + e*x])/e^8
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(546\) vs. \(2(245)=490\).
Time = 0.78 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.13
| method | result | size |
| norman | \(\frac {d^{2} \left (A \,b^{3} e^{4}-3 A \,b^{2} c d \,e^{3}+3 A b \,c^{2} d^{2} e^{2}-A \,c^{3} d^{3} e -B \,b^{3} d \,e^{3}+3 B \,b^{2} c \,d^{2} e^{2}-3 B b \,c^{2} d^{3} e +B \,c^{3} d^{4}\right ) x}{e^{7}}+\frac {\left (A \,b^{3} e^{4}-3 A \,b^{2} c d \,e^{3}+3 A b \,c^{2} d^{2} e^{2}-A \,c^{3} d^{3} e -B \,b^{3} d \,e^{3}+3 B \,b^{2} c \,d^{2} e^{2}-3 B b \,c^{2} d^{3} e +B \,c^{3} d^{4}\right ) x^{3}}{3 e^{5}}+\frac {\left (3 A \,b^{2} c \,e^{3}-3 A b \,c^{2} d \,e^{2}+A \,c^{3} d^{2} e +B \,e^{3} b^{3}-3 B \,b^{2} c d \,e^{2}+3 B b \,c^{2} d^{2} e -B \,c^{3} d^{3}\right ) x^{4}}{4 e^{4}}+\frac {B \,c^{3} x^{7}}{7 e}+\frac {c \left (3 A b c \,e^{2}-A \,c^{2} d e +3 B \,e^{2} b^{2}-3 B b c d e +B \,c^{2} d^{2}\right ) x^{5}}{5 e^{3}}+\frac {c^{2} \left (A c e +3 B b e -B c d \right ) x^{6}}{6 e^{2}}-\frac {d \left (A \,b^{3} e^{4}-3 A \,b^{2} c d \,e^{3}+3 A b \,c^{2} d^{2} e^{2}-A \,c^{3} d^{3} e -B \,b^{3} d \,e^{3}+3 B \,b^{2} c \,d^{2} e^{2}-3 B b \,c^{2} d^{3} e +B \,c^{3} d^{4}\right ) x^{2}}{2 e^{6}}-\frac {d^{3} \left (A \,b^{3} e^{4}-3 A \,b^{2} c d \,e^{3}+3 A b \,c^{2} d^{2} e^{2}-A \,c^{3} d^{3} e -B \,b^{3} d \,e^{3}+3 B \,b^{2} c \,d^{2} e^{2}-3 B b \,c^{2} d^{3} e +B \,c^{3} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(547\) |
| default | \(\frac {-A \,b^{2} c d \,e^{5} x^{3}+3 A b \,c^{2} d^{4} e^{2} x +B \,b^{2} c \,d^{2} e^{4} x^{3}+A b \,c^{2} d^{2} e^{4} x^{3}+\frac {3}{5} B \,b^{2} c \,e^{6} x^{5}+\frac {1}{2} B b \,c^{2} e^{6} x^{6}+\frac {1}{2} B \,b^{3} d^{2} e^{4} x^{2}+A \,b^{3} d^{2} e^{4} x -B \,b^{3} d^{3} e^{3} x -3 A \,b^{2} c \,d^{3} e^{3} x -\frac {3}{4} A b \,c^{2} d \,e^{5} x^{4}-3 B b \,c^{2} d^{5} e x +3 B \,b^{2} c \,d^{4} e^{2} x +\frac {3}{4} A \,b^{2} c \,e^{6} x^{4}-\frac {3}{2} A b \,c^{2} d^{3} e^{3} x^{2}+\frac {3}{2} A \,b^{2} c \,d^{2} e^{4} x^{2}-\frac {3}{5} B b \,c^{2} d \,e^{5} x^{5}-\frac {3}{2} B \,b^{2} c \,d^{3} e^{3} x^{2}+\frac {1}{3} B \,c^{3} d^{4} e^{2} x^{3}-\frac {1}{6} B \,c^{3} d \,e^{5} x^{6}-\frac {1}{5} A \,c^{3} d \,e^{5} x^{5}-\frac {1}{4} B \,c^{3} d^{3} e^{3} x^{4}-\frac {1}{3} A \,c^{3} d^{3} e^{3} x^{3}-A \,c^{3} d^{5} e x +\frac {3}{5} A b \,c^{2} e^{6} x^{5}+\frac {1}{5} B \,c^{3} d^{2} e^{4} x^{5}+\frac {1}{4} A \,c^{3} d^{2} e^{4} x^{4}-\frac {1}{3} B \,b^{3} d \,e^{5} x^{3}-\frac {1}{2} A \,b^{3} d \,e^{5} x^{2}-\frac {3}{4} B \,b^{2} c d \,e^{5} x^{4}+\frac {3}{4} B b \,c^{2} d^{2} e^{4} x^{4}+\frac {1}{2} A \,c^{3} d^{4} e^{2} x^{2}-\frac {1}{2} B \,c^{3} d^{5} e \,x^{2}-B b \,c^{2} d^{3} e^{3} x^{3}+B \,c^{3} d^{6} x +\frac {1}{3} A \,b^{3} e^{6} x^{3}+\frac {3}{2} B b \,c^{2} d^{4} e^{2} x^{2}+\frac {1}{4} B \,b^{3} e^{6} x^{4}+\frac {1}{7} B \,c^{3} x^{7} e^{6}+\frac {1}{6} A \,c^{3} e^{6} x^{6}}{e^{7}}-\frac {d^{3} \left (A \,b^{3} e^{4}-3 A \,b^{2} c d \,e^{3}+3 A b \,c^{2} d^{2} e^{2}-A \,c^{3} d^{3} e -B \,b^{3} d \,e^{3}+3 B \,b^{2} c \,d^{2} e^{2}-3 B b \,c^{2} d^{3} e +B \,c^{3} d^{4}\right ) \ln \left (e x +d \right )}{e^{8}}\) | \(657\) |
| risch | \(-\frac {A \,b^{3} d \,x^{2}}{2 e^{2}}+\frac {A \,c^{3} d^{4} x^{2}}{2 e^{5}}-\frac {B \,c^{3} d^{5} x^{2}}{2 e^{6}}+\frac {B \,c^{3} d^{6} x}{e^{7}}+\frac {3 d^{6} \ln \left (e x +d \right ) B b \,c^{2}}{e^{7}}+\frac {3 d^{4} \ln \left (e x +d \right ) A \,b^{2} c}{e^{5}}+\frac {A \,b^{3} d^{2} x}{e^{3}}-\frac {B \,b^{3} d^{3} x}{e^{4}}+\frac {3 A \,b^{2} c \,x^{4}}{4 e}-\frac {3 d^{5} \ln \left (e x +d \right ) A b \,c^{2}}{e^{6}}-\frac {A \,c^{3} d^{3} x^{3}}{3 e^{4}}-\frac {A \,c^{3} d^{5} x}{e^{6}}+\frac {3 A b \,c^{2} x^{5}}{5 e}+\frac {B \,c^{3} d^{2} x^{5}}{5 e^{3}}+\frac {B \,c^{3} d^{4} x^{3}}{3 e^{5}}-\frac {B \,c^{3} d \,x^{6}}{6 e^{2}}-\frac {A \,c^{3} d \,x^{5}}{5 e^{2}}-\frac {B \,c^{3} d^{3} x^{4}}{4 e^{4}}+\frac {A \,c^{3} d^{2} x^{4}}{4 e^{3}}-\frac {B \,b^{3} d \,x^{3}}{3 e^{2}}+\frac {3 B \,b^{2} c \,x^{5}}{5 e}+\frac {B b \,c^{2} x^{6}}{2 e}-\frac {3 B b \,c^{2} d \,x^{5}}{5 e^{2}}-\frac {3 B \,b^{2} c \,d^{3} x^{2}}{2 e^{4}}-\frac {3 B \,b^{2} c d \,x^{4}}{4 e^{2}}+\frac {3 B b \,c^{2} d^{2} x^{4}}{4 e^{3}}-\frac {B b \,c^{2} d^{3} x^{3}}{e^{4}}+\frac {3 B b \,c^{2} d^{4} x^{2}}{2 e^{5}}-\frac {d^{3} \ln \left (e x +d \right ) A \,b^{3}}{e^{4}}+\frac {d^{6} \ln \left (e x +d \right ) A \,c^{3}}{e^{7}}+\frac {d^{4} \ln \left (e x +d \right ) B \,b^{3}}{e^{5}}-\frac {d^{7} \ln \left (e x +d \right ) B \,c^{3}}{e^{8}}+\frac {B \,b^{3} d^{2} x^{2}}{2 e^{3}}-\frac {3 A \,b^{2} c \,d^{3} x}{e^{4}}-\frac {3 A b \,c^{2} d \,x^{4}}{4 e^{2}}+\frac {A \,b^{3} x^{3}}{3 e}+\frac {B \,b^{3} x^{4}}{4 e}+\frac {A \,c^{3} x^{6}}{6 e}-\frac {3 B b \,c^{2} d^{5} x}{e^{6}}+\frac {3 B \,b^{2} c \,d^{4} x}{e^{5}}-\frac {3 A b \,c^{2} d^{3} x^{2}}{2 e^{4}}+\frac {3 A \,b^{2} c \,d^{2} x^{2}}{2 e^{3}}+\frac {B \,b^{2} c \,d^{2} x^{3}}{e^{3}}+\frac {A b \,c^{2} d^{2} x^{3}}{e^{3}}-\frac {3 d^{5} \ln \left (e x +d \right ) B \,b^{2} c}{e^{6}}+\frac {B \,c^{3} x^{7}}{7 e}+\frac {3 A b \,c^{2} d^{4} x}{e^{5}}-\frac {A \,b^{2} c d \,x^{3}}{e^{2}}\) | \(708\) |
| parallelrisch | \(-\frac {-140 B \,x^{3} c^{3} d^{4} e^{3}+210 A \,x^{2} b^{3} d \,e^{6}-210 B \,x^{6} b \,c^{2} e^{7}+420 A \ln \left (e x +d \right ) b^{3} d^{3} e^{4}-420 A \ln \left (e x +d \right ) c^{3} d^{6} e -84 B \,x^{5} c^{3} d^{2} e^{5}-315 A \,x^{4} b^{2} c \,e^{7}-420 A x \,b^{3} d^{2} e^{5}-252 A \,x^{5} b \,c^{2} e^{7}+84 A \,x^{5} c^{3} d \,e^{6}+70 B \,x^{6} c^{3} d \,e^{6}+420 B \ln \left (e x +d \right ) c^{3} d^{7}+252 B \,x^{5} b \,c^{2} d \,e^{6}+140 A \,x^{3} c^{3} d^{3} e^{4}+140 B \,x^{3} b^{3} d \,e^{6}-420 B x \,c^{3} d^{6} e +210 B \,x^{2} c^{3} d^{5} e^{2}-252 B \,x^{5} b^{2} c \,e^{7}+420 A x \,c^{3} d^{5} e^{2}+420 B x \,b^{3} d^{3} e^{4}-210 A \,x^{2} c^{3} d^{4} e^{3}-420 B \ln \left (e x +d \right ) b^{3} d^{4} e^{3}-210 B \,x^{2} b^{3} d^{2} e^{5}-105 A \,x^{4} c^{3} d^{2} e^{5}+105 B \,x^{4} c^{3} d^{3} e^{4}-105 B \,x^{4} b^{3} e^{7}-140 A \,x^{3} b^{3} e^{7}-60 B \,x^{7} c^{3} e^{7}-70 A \,x^{6} c^{3} e^{7}-1260 A x b \,c^{2} d^{4} e^{3}-1260 B x \,b^{2} c \,d^{4} e^{3}-1260 A \ln \left (e x +d \right ) b^{2} c \,d^{4} e^{3}+1260 A \ln \left (e x +d \right ) b \,c^{2} d^{5} e^{2}+420 A \,x^{3} b^{2} c d \,e^{6}-420 A \,x^{3} b \,c^{2} d^{2} e^{5}-420 B \,x^{3} b^{2} c \,d^{2} e^{5}+420 B \,x^{3} b \,c^{2} d^{3} e^{4}-630 A \,x^{2} b^{2} c \,d^{2} e^{5}+630 A \,x^{2} b \,c^{2} d^{3} e^{4}+630 B \,x^{2} b^{2} c \,d^{3} e^{4}+315 A \,x^{4} b \,c^{2} d \,e^{6}+1260 B \ln \left (e x +d \right ) b^{2} c \,d^{5} e^{2}-1260 B \ln \left (e x +d \right ) b \,c^{2} d^{6} e +315 B \,x^{4} b^{2} c d \,e^{6}-315 B \,x^{4} b \,c^{2} d^{2} e^{5}+1260 B x b \,c^{2} d^{5} e^{2}-630 B \,x^{2} b \,c^{2} d^{4} e^{3}+1260 A x \,b^{2} c \,d^{3} e^{4}}{420 e^{8}}\) | \(710\) |
Input:
int((B*x+A)*(c*x^2+b*x)^3/(e*x+d),x,method=_RETURNVERBOSE)
Output:
d^2*(A*b^3*e^4-3*A*b^2*c*d*e^3+3*A*b*c^2*d^2*e^2-A*c^3*d^3*e-B*b^3*d*e^3+3 *B*b^2*c*d^2*e^2-3*B*b*c^2*d^3*e+B*c^3*d^4)/e^7*x+1/3/e^5*(A*b^3*e^4-3*A*b ^2*c*d*e^3+3*A*b*c^2*d^2*e^2-A*c^3*d^3*e-B*b^3*d*e^3+3*B*b^2*c*d^2*e^2-3*B *b*c^2*d^3*e+B*c^3*d^4)*x^3+1/4/e^4*(3*A*b^2*c*e^3-3*A*b*c^2*d*e^2+A*c^3*d ^2*e+B*b^3*e^3-3*B*b^2*c*d*e^2+3*B*b*c^2*d^2*e-B*c^3*d^3)*x^4+1/7*B*c^3*x^ 7/e+1/5*c/e^3*(3*A*b*c*e^2-A*c^2*d*e+3*B*b^2*e^2-3*B*b*c*d*e+B*c^2*d^2)*x^ 5+1/6*c^2/e^2*(A*c*e+3*B*b*e-B*c*d)*x^6-1/2*d/e^6*(A*b^3*e^4-3*A*b^2*c*d*e ^3+3*A*b*c^2*d^2*e^2-A*c^3*d^3*e-B*b^3*d*e^3+3*B*b^2*c*d^2*e^2-3*B*b*c^2*d ^3*e+B*c^3*d^4)*x^2-d^3*(A*b^3*e^4-3*A*b^2*c*d*e^3+3*A*b*c^2*d^2*e^2-A*c^3 *d^3*e-B*b^3*d*e^3+3*B*b^2*c*d^2*e^2-3*B*b*c^2*d^3*e+B*c^3*d^4)/e^8*ln(e*x +d)
Leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (245) = 490\).
Time = 0.08 (sec) , antiderivative size = 531, normalized size of antiderivative = 2.07 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{d+e x} \, dx=\frac {60 \, B c^{3} e^{7} x^{7} - 70 \, {\left (B c^{3} d e^{6} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 84 \, {\left (B c^{3} d^{2} e^{5} - {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e^{7}\right )} x^{5} - 105 \, {\left (B c^{3} d^{3} e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{6} - {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{7}\right )} x^{4} + 140 \, {\left (B c^{3} d^{4} e^{3} + A b^{3} e^{7} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{5} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{6}\right )} x^{3} - 210 \, {\left (B c^{3} d^{5} e^{2} + A b^{3} d e^{6} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{4} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{5}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} e + A b^{3} d^{2} e^{5} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x - 420 \, {\left (B c^{3} d^{7} + A b^{3} d^{3} e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3}\right )} \log \left (e x + d\right )}{420 \, e^{8}} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^3/(e*x+d),x, algorithm="fricas")
Output:
1/420*(60*B*c^3*e^7*x^7 - 70*(B*c^3*d*e^6 - (3*B*b*c^2 + A*c^3)*e^7)*x^6 + 84*(B*c^3*d^2*e^5 - (3*B*b*c^2 + A*c^3)*d*e^6 + 3*(B*b^2*c + A*b*c^2)*e^7 )*x^5 - 105*(B*c^3*d^3*e^4 - (3*B*b*c^2 + A*c^3)*d^2*e^5 + 3*(B*b^2*c + A* b*c^2)*d*e^6 - (B*b^3 + 3*A*b^2*c)*e^7)*x^4 + 140*(B*c^3*d^4*e^3 + A*b^3*e ^7 - (3*B*b*c^2 + A*c^3)*d^3*e^4 + 3*(B*b^2*c + A*b*c^2)*d^2*e^5 - (B*b^3 + 3*A*b^2*c)*d*e^6)*x^3 - 210*(B*c^3*d^5*e^2 + A*b^3*d*e^6 - (3*B*b*c^2 + A*c^3)*d^4*e^3 + 3*(B*b^2*c + A*b*c^2)*d^3*e^4 - (B*b^3 + 3*A*b^2*c)*d^2*e ^5)*x^2 + 420*(B*c^3*d^6*e + A*b^3*d^2*e^5 - (3*B*b*c^2 + A*c^3)*d^5*e^2 + 3*(B*b^2*c + A*b*c^2)*d^4*e^3 - (B*b^3 + 3*A*b^2*c)*d^3*e^4)*x - 420*(B*c ^3*d^7 + A*b^3*d^3*e^4 - (3*B*b*c^2 + A*c^3)*d^6*e + 3*(B*b^2*c + A*b*c^2) *d^5*e^2 - (B*b^3 + 3*A*b^2*c)*d^4*e^3)*log(e*x + d))/e^8
Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (241) = 482\).
Time = 0.70 (sec) , antiderivative size = 578, normalized size of antiderivative = 2.25 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{d+e x} \, dx=\frac {B c^{3} x^{7}}{7 e} + \frac {d^{3} \left (- A e + B d\right ) \left (b e - c d\right )^{3} \log {\left (d + e x \right )}}{e^{8}} + x^{6} \left (\frac {A c^{3}}{6 e} + \frac {B b c^{2}}{2 e} - \frac {B c^{3} d}{6 e^{2}}\right ) + x^{5} \cdot \left (\frac {3 A b c^{2}}{5 e} - \frac {A c^{3} d}{5 e^{2}} + \frac {3 B b^{2} c}{5 e} - \frac {3 B b c^{2} d}{5 e^{2}} + \frac {B c^{3} d^{2}}{5 e^{3}}\right ) + x^{4} \cdot \left (\frac {3 A b^{2} c}{4 e} - \frac {3 A b c^{2} d}{4 e^{2}} + \frac {A c^{3} d^{2}}{4 e^{3}} + \frac {B b^{3}}{4 e} - \frac {3 B b^{2} c d}{4 e^{2}} + \frac {3 B b c^{2} d^{2}}{4 e^{3}} - \frac {B c^{3} d^{3}}{4 e^{4}}\right ) + x^{3} \left (\frac {A b^{3}}{3 e} - \frac {A b^{2} c d}{e^{2}} + \frac {A b c^{2} d^{2}}{e^{3}} - \frac {A c^{3} d^{3}}{3 e^{4}} - \frac {B b^{3} d}{3 e^{2}} + \frac {B b^{2} c d^{2}}{e^{3}} - \frac {B b c^{2} d^{3}}{e^{4}} + \frac {B c^{3} d^{4}}{3 e^{5}}\right ) + x^{2} \left (- \frac {A b^{3} d}{2 e^{2}} + \frac {3 A b^{2} c d^{2}}{2 e^{3}} - \frac {3 A b c^{2} d^{3}}{2 e^{4}} + \frac {A c^{3} d^{4}}{2 e^{5}} + \frac {B b^{3} d^{2}}{2 e^{3}} - \frac {3 B b^{2} c d^{3}}{2 e^{4}} + \frac {3 B b c^{2} d^{4}}{2 e^{5}} - \frac {B c^{3} d^{5}}{2 e^{6}}\right ) + x \left (\frac {A b^{3} d^{2}}{e^{3}} - \frac {3 A b^{2} c d^{3}}{e^{4}} + \frac {3 A b c^{2} d^{4}}{e^{5}} - \frac {A c^{3} d^{5}}{e^{6}} - \frac {B b^{3} d^{3}}{e^{4}} + \frac {3 B b^{2} c d^{4}}{e^{5}} - \frac {3 B b c^{2} d^{5}}{e^{6}} + \frac {B c^{3} d^{6}}{e^{7}}\right ) \] Input:
integrate((B*x+A)*(c*x**2+b*x)**3/(e*x+d),x)
Output:
B*c**3*x**7/(7*e) + d**3*(-A*e + B*d)*(b*e - c*d)**3*log(d + e*x)/e**8 + x **6*(A*c**3/(6*e) + B*b*c**2/(2*e) - B*c**3*d/(6*e**2)) + x**5*(3*A*b*c**2 /(5*e) - A*c**3*d/(5*e**2) + 3*B*b**2*c/(5*e) - 3*B*b*c**2*d/(5*e**2) + B* c**3*d**2/(5*e**3)) + x**4*(3*A*b**2*c/(4*e) - 3*A*b*c**2*d/(4*e**2) + A*c **3*d**2/(4*e**3) + B*b**3/(4*e) - 3*B*b**2*c*d/(4*e**2) + 3*B*b*c**2*d**2 /(4*e**3) - B*c**3*d**3/(4*e**4)) + x**3*(A*b**3/(3*e) - A*b**2*c*d/e**2 + A*b*c**2*d**2/e**3 - A*c**3*d**3/(3*e**4) - B*b**3*d/(3*e**2) + B*b**2*c* d**2/e**3 - B*b*c**2*d**3/e**4 + B*c**3*d**4/(3*e**5)) + x**2*(-A*b**3*d/( 2*e**2) + 3*A*b**2*c*d**2/(2*e**3) - 3*A*b*c**2*d**3/(2*e**4) + A*c**3*d** 4/(2*e**5) + B*b**3*d**2/(2*e**3) - 3*B*b**2*c*d**3/(2*e**4) + 3*B*b*c**2* d**4/(2*e**5) - B*c**3*d**5/(2*e**6)) + x*(A*b**3*d**2/e**3 - 3*A*b**2*c*d **3/e**4 + 3*A*b*c**2*d**4/e**5 - A*c**3*d**5/e**6 - B*b**3*d**3/e**4 + 3* B*b**2*c*d**4/e**5 - 3*B*b*c**2*d**5/e**6 + B*c**3*d**6/e**7)
Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (245) = 490\).
Time = 0.04 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.06 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{d+e x} \, dx=\frac {60 \, B c^{3} e^{6} x^{7} - 70 \, {\left (B c^{3} d e^{5} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{6}\right )} x^{6} + 84 \, {\left (B c^{3} d^{2} e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{5} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} e^{6}\right )} x^{5} - 105 \, {\left (B c^{3} d^{3} e^{3} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{4} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{5} - {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{6}\right )} x^{4} + 140 \, {\left (B c^{3} d^{4} e^{2} + A b^{3} e^{6} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{3} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{4} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{5}\right )} x^{3} - 210 \, {\left (B c^{3} d^{5} e + A b^{3} d e^{5} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{2} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{3} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{4}\right )} x^{2} + 420 \, {\left (B c^{3} d^{6} + A b^{3} d^{2} e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{2} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{3}\right )} x}{420 \, e^{7}} - \frac {{\left (B c^{3} d^{7} + A b^{3} d^{3} e^{4} - {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^3/(e*x+d),x, algorithm="maxima")
Output:
1/420*(60*B*c^3*e^6*x^7 - 70*(B*c^3*d*e^5 - (3*B*b*c^2 + A*c^3)*e^6)*x^6 + 84*(B*c^3*d^2*e^4 - (3*B*b*c^2 + A*c^3)*d*e^5 + 3*(B*b^2*c + A*b*c^2)*e^6 )*x^5 - 105*(B*c^3*d^3*e^3 - (3*B*b*c^2 + A*c^3)*d^2*e^4 + 3*(B*b^2*c + A* b*c^2)*d*e^5 - (B*b^3 + 3*A*b^2*c)*e^6)*x^4 + 140*(B*c^3*d^4*e^2 + A*b^3*e ^6 - (3*B*b*c^2 + A*c^3)*d^3*e^3 + 3*(B*b^2*c + A*b*c^2)*d^2*e^4 - (B*b^3 + 3*A*b^2*c)*d*e^5)*x^3 - 210*(B*c^3*d^5*e + A*b^3*d*e^5 - (3*B*b*c^2 + A* c^3)*d^4*e^2 + 3*(B*b^2*c + A*b*c^2)*d^3*e^3 - (B*b^3 + 3*A*b^2*c)*d^2*e^4 )*x^2 + 420*(B*c^3*d^6 + A*b^3*d^2*e^4 - (3*B*b*c^2 + A*c^3)*d^5*e + 3*(B* b^2*c + A*b*c^2)*d^4*e^2 - (B*b^3 + 3*A*b^2*c)*d^3*e^3)*x)/e^7 - (B*c^3*d^ 7 + A*b^3*d^3*e^4 - (3*B*b*c^2 + A*c^3)*d^6*e + 3*(B*b^2*c + A*b*c^2)*d^5* e^2 - (B*b^3 + 3*A*b^2*c)*d^4*e^3)*log(e*x + d)/e^8
Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (245) = 490\).
Time = 0.26 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.59 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{d+e x} \, dx=\frac {60 \, B c^{3} e^{6} x^{7} - 70 \, B c^{3} d e^{5} x^{6} + 210 \, B b c^{2} e^{6} x^{6} + 70 \, A c^{3} e^{6} x^{6} + 84 \, B c^{3} d^{2} e^{4} x^{5} - 252 \, B b c^{2} d e^{5} x^{5} - 84 \, A c^{3} d e^{5} x^{5} + 252 \, B b^{2} c e^{6} x^{5} + 252 \, A b c^{2} e^{6} x^{5} - 105 \, B c^{3} d^{3} e^{3} x^{4} + 315 \, B b c^{2} d^{2} e^{4} x^{4} + 105 \, A c^{3} d^{2} e^{4} x^{4} - 315 \, B b^{2} c d e^{5} x^{4} - 315 \, A b c^{2} d e^{5} x^{4} + 105 \, B b^{3} e^{6} x^{4} + 315 \, A b^{2} c e^{6} x^{4} + 140 \, B c^{3} d^{4} e^{2} x^{3} - 420 \, B b c^{2} d^{3} e^{3} x^{3} - 140 \, A c^{3} d^{3} e^{3} x^{3} + 420 \, B b^{2} c d^{2} e^{4} x^{3} + 420 \, A b c^{2} d^{2} e^{4} x^{3} - 140 \, B b^{3} d e^{5} x^{3} - 420 \, A b^{2} c d e^{5} x^{3} + 140 \, A b^{3} e^{6} x^{3} - 210 \, B c^{3} d^{5} e x^{2} + 630 \, B b c^{2} d^{4} e^{2} x^{2} + 210 \, A c^{3} d^{4} e^{2} x^{2} - 630 \, B b^{2} c d^{3} e^{3} x^{2} - 630 \, A b c^{2} d^{3} e^{3} x^{2} + 210 \, B b^{3} d^{2} e^{4} x^{2} + 630 \, A b^{2} c d^{2} e^{4} x^{2} - 210 \, A b^{3} d e^{5} x^{2} + 420 \, B c^{3} d^{6} x - 1260 \, B b c^{2} d^{5} e x - 420 \, A c^{3} d^{5} e x + 1260 \, B b^{2} c d^{4} e^{2} x + 1260 \, A b c^{2} d^{4} e^{2} x - 420 \, B b^{3} d^{3} e^{3} x - 1260 \, A b^{2} c d^{3} e^{3} x + 420 \, A b^{3} d^{2} e^{4} x}{420 \, e^{7}} - \frac {{\left (B c^{3} d^{7} - 3 \, B b c^{2} d^{6} e - A c^{3} d^{6} e + 3 \, B b^{2} c d^{5} e^{2} + 3 \, A b c^{2} d^{5} e^{2} - B b^{3} d^{4} e^{3} - 3 \, A b^{2} c d^{4} e^{3} + A b^{3} d^{3} e^{4}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{8}} \] Input:
integrate((B*x+A)*(c*x^2+b*x)^3/(e*x+d),x, algorithm="giac")
Output:
1/420*(60*B*c^3*e^6*x^7 - 70*B*c^3*d*e^5*x^6 + 210*B*b*c^2*e^6*x^6 + 70*A* c^3*e^6*x^6 + 84*B*c^3*d^2*e^4*x^5 - 252*B*b*c^2*d*e^5*x^5 - 84*A*c^3*d*e^ 5*x^5 + 252*B*b^2*c*e^6*x^5 + 252*A*b*c^2*e^6*x^5 - 105*B*c^3*d^3*e^3*x^4 + 315*B*b*c^2*d^2*e^4*x^4 + 105*A*c^3*d^2*e^4*x^4 - 315*B*b^2*c*d*e^5*x^4 - 315*A*b*c^2*d*e^5*x^4 + 105*B*b^3*e^6*x^4 + 315*A*b^2*c*e^6*x^4 + 140*B* c^3*d^4*e^2*x^3 - 420*B*b*c^2*d^3*e^3*x^3 - 140*A*c^3*d^3*e^3*x^3 + 420*B* b^2*c*d^2*e^4*x^3 + 420*A*b*c^2*d^2*e^4*x^3 - 140*B*b^3*d*e^5*x^3 - 420*A* b^2*c*d*e^5*x^3 + 140*A*b^3*e^6*x^3 - 210*B*c^3*d^5*e*x^2 + 630*B*b*c^2*d^ 4*e^2*x^2 + 210*A*c^3*d^4*e^2*x^2 - 630*B*b^2*c*d^3*e^3*x^2 - 630*A*b*c^2* d^3*e^3*x^2 + 210*B*b^3*d^2*e^4*x^2 + 630*A*b^2*c*d^2*e^4*x^2 - 210*A*b^3* d*e^5*x^2 + 420*B*c^3*d^6*x - 1260*B*b*c^2*d^5*e*x - 420*A*c^3*d^5*e*x + 1 260*B*b^2*c*d^4*e^2*x + 1260*A*b*c^2*d^4*e^2*x - 420*B*b^3*d^3*e^3*x - 126 0*A*b^2*c*d^3*e^3*x + 420*A*b^3*d^2*e^4*x)/e^7 - (B*c^3*d^7 - 3*B*b*c^2*d^ 6*e - A*c^3*d^6*e + 3*B*b^2*c*d^5*e^2 + 3*A*b*c^2*d^5*e^2 - B*b^3*d^4*e^3 - 3*A*b^2*c*d^4*e^3 + A*b^3*d^3*e^4)*log(abs(e*x + d))/e^8
Time = 0.10 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.18 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{d+e x} \, dx=x^4\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{4\,e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{e}\right )}{4\,e}\right )-x^5\,\left (\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{5\,e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{5\,e}\right )+x^3\,\left (\frac {A\,b^3}{3\,e}-\frac {d\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{e}\right )}{e}\right )}{3\,e}\right )+x^6\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{6\,e}-\frac {B\,c^3\,d}{6\,e^2}\right )-\frac {\ln \left (d+e\,x\right )\,\left (-B\,b^3\,d^4\,e^3+A\,b^3\,d^3\,e^4+3\,B\,b^2\,c\,d^5\,e^2-3\,A\,b^2\,c\,d^4\,e^3-3\,B\,b\,c^2\,d^6\,e+3\,A\,b\,c^2\,d^5\,e^2+B\,c^3\,d^7-A\,c^3\,d^6\,e\right )}{e^8}-\frac {d\,x^2\,\left (\frac {A\,b^3}{e}-\frac {d\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{e}\right )}{e}\right )}{e}\right )}{2\,e}+\frac {d^2\,x\,\left (\frac {A\,b^3}{e}-\frac {d\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e}-\frac {B\,c^3\,d}{e^2}\right )}{e}-\frac {3\,b\,c\,\left (A\,c+B\,b\right )}{e}\right )}{e}\right )}{e}\right )}{e^2}+\frac {B\,c^3\,x^7}{7\,e} \] Input:
int(((b*x + c*x^2)^3*(A + B*x))/(d + e*x),x)
Output:
x^4*((B*b^3 + 3*A*b^2*c)/(4*e) + (d*((d*((A*c^3 + 3*B*b*c^2)/e - (B*c^3*d) /e^2))/e - (3*b*c*(A*c + B*b))/e))/(4*e)) - x^5*((d*((A*c^3 + 3*B*b*c^2)/e - (B*c^3*d)/e^2))/(5*e) - (3*b*c*(A*c + B*b))/(5*e)) + x^3*((A*b^3)/(3*e) - (d*((B*b^3 + 3*A*b^2*c)/e + (d*((d*((A*c^3 + 3*B*b*c^2)/e - (B*c^3*d)/e ^2))/e - (3*b*c*(A*c + B*b))/e))/e))/(3*e)) + x^6*((A*c^3 + 3*B*b*c^2)/(6* e) - (B*c^3*d)/(6*e^2)) - (log(d + e*x)*(B*c^3*d^7 - A*c^3*d^6*e + A*b^3*d ^3*e^4 - B*b^3*d^4*e^3 + 3*A*b*c^2*d^5*e^2 - 3*A*b^2*c*d^4*e^3 + 3*B*b^2*c *d^5*e^2 - 3*B*b*c^2*d^6*e))/e^8 - (d*x^2*((A*b^3)/e - (d*((B*b^3 + 3*A*b^ 2*c)/e + (d*((d*((A*c^3 + 3*B*b*c^2)/e - (B*c^3*d)/e^2))/e - (3*b*c*(A*c + B*b))/e))/e))/e))/(2*e) + (d^2*x*((A*b^3)/e - (d*((B*b^3 + 3*A*b^2*c)/e + (d*((d*((A*c^3 + 3*B*b*c^2)/e - (B*c^3*d)/e^2))/e - (3*b*c*(A*c + B*b))/e ))/e))/e))/e^2 + (B*c^3*x^7)/(7*e)
Time = 0.25 (sec) , antiderivative size = 705, normalized size of antiderivative = 2.74 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^3}{d+e x} \, dx=\frac {-420 \,\mathrm {log}\left (e x +d \right ) a \,b^{3} d^{3} e^{4}+420 \,\mathrm {log}\left (e x +d \right ) a \,c^{3} d^{6} e -1260 \,\mathrm {log}\left (e x +d \right ) b^{3} c \,d^{5} e^{2}+1260 \,\mathrm {log}\left (e x +d \right ) b^{2} c^{2} d^{6} e +420 a \,b^{3} d^{2} e^{5} x -210 a \,b^{3} d \,e^{6} x^{2}+315 a \,b^{2} c \,e^{7} x^{4}+252 a b \,c^{2} e^{7} x^{5}-420 a \,c^{3} d^{5} e^{2} x +210 a \,c^{3} d^{4} e^{3} x^{2}-140 a \,c^{3} d^{3} e^{4} x^{3}+105 a \,c^{3} d^{2} e^{5} x^{4}-84 a \,c^{3} d \,e^{6} x^{5}+1260 b^{3} c \,d^{4} e^{3} x -630 b^{3} c \,d^{3} e^{4} x^{2}+420 b^{3} c \,d^{2} e^{5} x^{3}-315 b^{3} c d \,e^{6} x^{4}-1260 b^{2} c^{2} d^{5} e^{2} x +630 b^{2} c^{2} d^{4} e^{3} x^{2}-420 b^{2} c^{2} d^{3} e^{4} x^{3}+315 b^{2} c^{2} d^{2} e^{5} x^{4}-252 b^{2} c^{2} d \,e^{6} x^{5}+420 b \,c^{3} d^{6} e x -210 b \,c^{3} d^{5} e^{2} x^{2}+140 b \,c^{3} d^{4} e^{3} x^{3}-105 b \,c^{3} d^{3} e^{4} x^{4}+84 b \,c^{3} d^{2} e^{5} x^{5}-70 b \,c^{3} d \,e^{6} x^{6}+1260 \,\mathrm {log}\left (e x +d \right ) a \,b^{2} c \,d^{4} e^{3}-1260 \,\mathrm {log}\left (e x +d \right ) a b \,c^{2} d^{5} e^{2}-1260 a \,b^{2} c \,d^{3} e^{4} x +630 a \,b^{2} c \,d^{2} e^{5} x^{2}-420 a \,b^{2} c d \,e^{6} x^{3}+1260 a b \,c^{2} d^{4} e^{3} x -630 a b \,c^{2} d^{3} e^{4} x^{2}+420 a b \,c^{2} d^{2} e^{5} x^{3}-315 a b \,c^{2} d \,e^{6} x^{4}+105 b^{4} e^{7} x^{4}+420 \,\mathrm {log}\left (e x +d \right ) b^{4} d^{4} e^{3}-420 \,\mathrm {log}\left (e x +d \right ) b \,c^{3} d^{7}+140 a \,b^{3} e^{7} x^{3}+70 a \,c^{3} e^{7} x^{6}-420 b^{4} d^{3} e^{4} x +210 b^{4} d^{2} e^{5} x^{2}-140 b^{4} d \,e^{6} x^{3}+252 b^{3} c \,e^{7} x^{5}+210 b^{2} c^{2} e^{7} x^{6}+60 b \,c^{3} e^{7} x^{7}}{420 e^{8}} \] Input:
int((B*x+A)*(c*x^2+b*x)^3/(e*x+d),x)
Output:
( - 420*log(d + e*x)*a*b**3*d**3*e**4 + 1260*log(d + e*x)*a*b**2*c*d**4*e* *3 - 1260*log(d + e*x)*a*b*c**2*d**5*e**2 + 420*log(d + e*x)*a*c**3*d**6*e + 420*log(d + e*x)*b**4*d**4*e**3 - 1260*log(d + e*x)*b**3*c*d**5*e**2 + 1260*log(d + e*x)*b**2*c**2*d**6*e - 420*log(d + e*x)*b*c**3*d**7 + 420*a* b**3*d**2*e**5*x - 210*a*b**3*d*e**6*x**2 + 140*a*b**3*e**7*x**3 - 1260*a* b**2*c*d**3*e**4*x + 630*a*b**2*c*d**2*e**5*x**2 - 420*a*b**2*c*d*e**6*x** 3 + 315*a*b**2*c*e**7*x**4 + 1260*a*b*c**2*d**4*e**3*x - 630*a*b*c**2*d**3 *e**4*x**2 + 420*a*b*c**2*d**2*e**5*x**3 - 315*a*b*c**2*d*e**6*x**4 + 252* a*b*c**2*e**7*x**5 - 420*a*c**3*d**5*e**2*x + 210*a*c**3*d**4*e**3*x**2 - 140*a*c**3*d**3*e**4*x**3 + 105*a*c**3*d**2*e**5*x**4 - 84*a*c**3*d*e**6*x **5 + 70*a*c**3*e**7*x**6 - 420*b**4*d**3*e**4*x + 210*b**4*d**2*e**5*x**2 - 140*b**4*d*e**6*x**3 + 105*b**4*e**7*x**4 + 1260*b**3*c*d**4*e**3*x - 6 30*b**3*c*d**3*e**4*x**2 + 420*b**3*c*d**2*e**5*x**3 - 315*b**3*c*d*e**6*x **4 + 252*b**3*c*e**7*x**5 - 1260*b**2*c**2*d**5*e**2*x + 630*b**2*c**2*d* *4*e**3*x**2 - 420*b**2*c**2*d**3*e**4*x**3 + 315*b**2*c**2*d**2*e**5*x**4 - 252*b**2*c**2*d*e**6*x**5 + 210*b**2*c**2*e**7*x**6 + 420*b*c**3*d**6*e *x - 210*b*c**3*d**5*e**2*x**2 + 140*b*c**3*d**4*e**3*x**3 - 105*b*c**3*d* *3*e**4*x**4 + 84*b*c**3*d**2*e**5*x**5 - 70*b*c**3*d*e**6*x**6 + 60*b*c** 3*e**7*x**7)/(420*e**8)