Integrand size = 24, antiderivative size = 185 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^3} \, dx=-\frac {A d^3}{2 b^3 x^2}-\frac {d^2 (b B d-3 A c d+3 A b e)}{b^4 x}-\frac {(b B-A c) (c d-b e)^3}{2 b^3 c^2 (b+c x)^2}-\frac {(c d-b e)^2 \left (2 b B c d-3 A c^2 d+b^2 B e\right )}{b^4 c^2 (b+c x)}-\frac {3 d (c d-b e) (b B d-2 A c d+A b e) \log (x)}{b^5}+\frac {3 d (c d-b e) (b B d-2 A c d+A b e) \log (b+c x)}{b^5} \]
-1/2*A*d^3/b^3/x^2-d^2*(3*A*b*e-3*A*c*d+B*b*d)/b^4/x-1/2*(-A*c+B*b)*(-b*e+ c*d)^3/b^3/c^2/(c*x+b)^2-(-b*e+c*d)^2*(-3*A*c^2*d+B*b^2*e+2*B*b*c*d)/b^4/c ^2/(c*x+b)-3*d*(-b*e+c*d)*(A*b*e-2*A*c*d+B*b*d)*ln(x)/b^5+3*d*(-b*e+c*d)*( A*b*e-2*A*c*d+B*b*d)*ln(c*x+b)/b^5
Time = 0.08 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^3} \, dx=-\frac {\frac {A b^2 d^3}{x^2}+\frac {2 b d^2 (b B d-3 A c d+3 A b e)}{x}-\frac {b^2 (b B-A c) (-c d+b e)^3}{c^2 (b+c x)^2}+\frac {2 b (c d-b e)^2 \left (2 b B c d-3 A c^2 d+b^2 B e\right )}{c^2 (b+c x)}-6 d (-c d+b e) (b B d-2 A c d+A b e) \log (x)+6 d (-c d+b e) (b B d-2 A c d+A b e) \log (b+c x)}{2 b^5} \]
-1/2*((A*b^2*d^3)/x^2 + (2*b*d^2*(b*B*d - 3*A*c*d + 3*A*b*e))/x - (b^2*(b* B - A*c)*(-(c*d) + b*e)^3)/(c^2*(b + c*x)^2) + (2*b*(c*d - b*e)^2*(2*b*B*c *d - 3*A*c^2*d + b^2*B*e))/(c^2*(b + c*x)) - 6*d*(-(c*d) + b*e)*(b*B*d - 2 *A*c*d + A*b*e)*Log[x] + 6*d*(-(c*d) + b*e)*(b*B*d - 2*A*c*d + A*b*e)*Log[ b + c*x])/b^5
Time = 0.45 (sec) , antiderivative size = 185, 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 = {1206, 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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 1206 |
| \(\displaystyle \int \left (\frac {3 d (b e-c d) (A b e-2 A c d+b B d)}{b^5 x}-\frac {3 c d (b e-c d) (A b e-2 A c d+b B d)}{b^5 (b+c x)}+\frac {d^2 (3 A b e-3 A c d+b B d)}{b^4 x^2}-\frac {(b B-A c) (b e-c d)^3}{b^3 c (b+c x)^3}+\frac {A d^3}{b^3 x^3}+\frac {(b e-c d)^2 \left (-3 A c^2 d+b^2 B e+2 b B c d\right )}{b^4 c (b+c x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 d \log (x) (c d-b e) (A b e-2 A c d+b B d)}{b^5}+\frac {3 d (c d-b e) \log (b+c x) (A b e-2 A c d+b B d)}{b^5}-\frac {d^2 (3 A b e-3 A c d+b B d)}{b^4 x}-\frac {(b B-A c) (c d-b e)^3}{2 b^3 c^2 (b+c x)^2}-\frac {A d^3}{2 b^3 x^2}-\frac {(c d-b e)^2 \left (-3 A c^2 d+b^2 B e+2 b B c d\right )}{b^4 c^2 (b+c x)}\) |
-1/2*(A*d^3)/(b^3*x^2) - (d^2*(b*B*d - 3*A*c*d + 3*A*b*e))/(b^4*x) - ((b*B - A*c)*(c*d - b*e)^3)/(2*b^3*c^2*(b + c*x)^2) - ((c*d - b*e)^2*(2*b*B*c*d - 3*A*c^2*d + b^2*B*e))/(b^4*c^2*(b + c*x)) - (3*d*(c*d - b*e)*(b*B*d - 2 *A*c*d + A*b*e)*Log[x])/b^5 + (3*d*(c*d - b*e)*(b*B*d - 2*A*c*d + A*b*e)*L og[b + c*x])/b^5
3.12.56.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((b_.)*(x_) + (c_.) *(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[x^p*(d + e*x)^m*(f + g*x)^n *(b + c*x)^p, x], x] /; FreeQ[{b, c, d, e, f, g}, x] && ILtQ[p, -1] && Inte gersQ[m, n]
Time = 0.24 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.72
| method | result | size |
| norman | \(\frac {\frac {\left (3 A \,b^{2} c^{2} d \,e^{2}-9 A b \,c^{3} d^{2} e +6 A \,c^{4} d^{3}-b^{4} B \,e^{3}+3 b^{2} B \,c^{2} d^{2} e -3 B b \,c^{3} d^{3}\right ) x^{3}}{b^{4} c}-\frac {A \,d^{3}}{2 b}-\frac {d^{2} \left (3 A b e -2 A c d +B b d \right ) x}{b^{2}}-\frac {\left (A \,b^{3} c \,e^{3}-9 A \,b^{2} c^{2} d \,e^{2}+27 A b \,c^{3} d^{2} e -18 A \,c^{4} d^{3}+b^{4} B \,e^{3}+3 b^{3} B c d \,e^{2}-9 b^{2} B \,c^{2} d^{2} e +9 B b \,c^{3} d^{3}\right ) x^{2}}{2 c^{2} b^{3}}}{x^{2} \left (c x +b \right )^{2}}+\frac {3 d \left (A \,b^{2} e^{2}-3 A b c d e +2 A \,c^{2} d^{2}+B \,b^{2} d e -B b c \,d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {3 d \left (A \,b^{2} e^{2}-3 A b c d e +2 A \,c^{2} d^{2}+B \,b^{2} d e -B b c \,d^{2}\right ) \ln \left (c x +b \right )}{b^{5}}\) | \(319\) |
| default | \(-\frac {A \,d^{3}}{2 b^{3} x^{2}}-\frac {d^{2} \left (3 A b e -3 A c d +B b d \right )}{b^{4} x}+\frac {3 d \left (A \,b^{2} e^{2}-3 A b c d e +2 A \,c^{2} d^{2}+B \,b^{2} d e -B b c \,d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {-3 A \,b^{2} c^{2} d \,e^{2}+6 A b \,c^{3} d^{2} e -3 A \,c^{4} d^{3}+b^{4} B \,e^{3}-3 b^{2} B \,c^{2} d^{2} e +2 B b \,c^{3} d^{3}}{b^{4} c^{2} \left (c x +b \right )}-\frac {A \,b^{3} c \,e^{3}-3 A \,b^{2} c^{2} d \,e^{2}+3 A b \,c^{3} d^{2} e -A \,c^{4} d^{3}-b^{4} B \,e^{3}+3 b^{3} B c d \,e^{2}-3 b^{2} B \,c^{2} d^{2} e +B b \,c^{3} d^{3}}{2 c^{2} b^{3} \left (c x +b \right )^{2}}-\frac {3 d \left (A \,b^{2} e^{2}-3 A b c d e +2 A \,c^{2} d^{2}+B \,b^{2} d e -B b c \,d^{2}\right ) \ln \left (c x +b \right )}{b^{5}}\) | \(320\) |
| risch | \(\frac {\frac {\left (3 A \,b^{2} c^{2} d \,e^{2}-9 A b \,c^{3} d^{2} e +6 A \,c^{4} d^{3}-b^{4} B \,e^{3}+3 b^{2} B \,c^{2} d^{2} e -3 B b \,c^{3} d^{3}\right ) x^{3}}{b^{4} c}-\frac {A \,d^{3}}{2 b}-\frac {d^{2} \left (3 A b e -2 A c d +B b d \right ) x}{b^{2}}-\frac {\left (A \,b^{3} c \,e^{3}-9 A \,b^{2} c^{2} d \,e^{2}+27 A b \,c^{3} d^{2} e -18 A \,c^{4} d^{3}+b^{4} B \,e^{3}+3 b^{3} B c d \,e^{2}-9 b^{2} B \,c^{2} d^{2} e +9 B b \,c^{3} d^{3}\right ) x^{2}}{2 c^{2} b^{3}}}{x^{2} \left (c x +b \right )^{2}}-\frac {3 d \ln \left (c x +b \right ) A \,e^{2}}{b^{3}}+\frac {9 d^{2} \ln \left (c x +b \right ) A c e}{b^{4}}-\frac {6 d^{3} \ln \left (c x +b \right ) A \,c^{2}}{b^{5}}-\frac {3 d^{2} \ln \left (c x +b \right ) B e}{b^{3}}+\frac {3 d^{3} \ln \left (c x +b \right ) B c}{b^{4}}+\frac {3 d \ln \left (-x \right ) A \,e^{2}}{b^{3}}-\frac {9 d^{2} \ln \left (-x \right ) A c e}{b^{4}}+\frac {6 d^{3} \ln \left (-x \right ) A \,c^{2}}{b^{5}}+\frac {3 d^{2} \ln \left (-x \right ) B e}{b^{3}}-\frac {3 d^{3} \ln \left (-x \right ) B c}{b^{4}}\) | \(375\) |
| parallelrisch | \(\frac {-27 A \,x^{2} b^{3} c^{3} d^{2} e -3 B \,x^{2} b^{5} c d \,e^{2}+9 B \,x^{2} b^{4} c^{2} d^{2} e -6 A x \,b^{4} c^{2} d^{2} e -6 B \ln \left (x \right ) x^{4} b \,c^{5} d^{3}+6 B \ln \left (c x +b \right ) x^{4} b \,c^{5} d^{3}+24 A \ln \left (x \right ) x^{3} b \,c^{5} d^{3}-24 A \ln \left (c x +b \right ) x^{3} b \,c^{5} d^{3}-12 B \ln \left (x \right ) x^{3} b^{2} c^{4} d^{3}+12 B \ln \left (c x +b \right ) x^{3} b^{2} c^{4} d^{3}+12 A \ln \left (x \right ) x^{2} b^{2} c^{4} d^{3}-12 A \ln \left (c x +b \right ) x^{2} b^{2} c^{4} d^{3}-6 B \ln \left (x \right ) x^{2} b^{3} c^{3} d^{3}+6 B \ln \left (c x +b \right ) x^{2} b^{3} c^{3} d^{3}+6 A \,x^{3} b^{3} c^{3} d \,e^{2}-18 A \,x^{3} b^{2} c^{4} d^{2} e +6 B \,x^{3} b^{3} c^{3} d^{2} e +9 A \,x^{2} b^{4} c^{2} d \,e^{2}-A \,b^{4} c^{2} d^{3}-B \,x^{2} b^{6} e^{3}-2 B x \,b^{4} c^{2} d^{3}+12 A \ln \left (x \right ) x^{4} c^{6} d^{3}-12 A \ln \left (c x +b \right ) x^{4} c^{6} d^{3}+6 A \ln \left (x \right ) x^{4} b^{2} c^{4} d \,e^{2}-18 A \ln \left (x \right ) x^{4} b \,c^{5} d^{2} e -6 A \ln \left (c x +b \right ) x^{4} b^{2} c^{4} d \,e^{2}+18 A \ln \left (c x +b \right ) x^{4} b \,c^{5} d^{2} e +6 B \ln \left (x \right ) x^{4} b^{2} c^{4} d^{2} e -6 B \ln \left (c x +b \right ) x^{4} b^{2} c^{4} d^{2} e +12 A \ln \left (x \right ) x^{3} b^{3} c^{3} d \,e^{2}-36 A \ln \left (x \right ) x^{3} b^{2} c^{4} d^{2} e -12 A \ln \left (c x +b \right ) x^{3} b^{3} c^{3} d \,e^{2}+36 A \ln \left (c x +b \right ) x^{3} b^{2} c^{4} d^{2} e +6 A \ln \left (x \right ) x^{2} b^{4} c^{2} d \,e^{2}-18 A \ln \left (x \right ) x^{2} b^{3} c^{3} d^{2} e -6 A \ln \left (c x +b \right ) x^{2} b^{4} c^{2} d \,e^{2}+18 A \ln \left (c x +b \right ) x^{2} b^{3} c^{3} d^{2} e +6 B \ln \left (x \right ) x^{2} b^{4} c^{2} d^{2} e -6 B \ln \left (c x +b \right ) x^{2} b^{4} c^{2} d^{2} e +12 B \ln \left (x \right ) x^{3} b^{3} c^{3} d^{2} e -12 B \ln \left (c x +b \right ) x^{3} b^{3} c^{3} d^{2} e +12 A \,x^{3} b \,c^{5} d^{3}-2 B \,x^{3} b^{5} c \,e^{3}-6 B \,x^{3} b^{2} c^{4} d^{3}-A \,x^{2} b^{5} c \,e^{3}+18 A \,x^{2} b^{2} c^{4} d^{3}-9 B \,x^{2} b^{3} c^{3} d^{3}+4 A x \,b^{3} c^{3} d^{3}}{2 c^{2} b^{5} x^{2} \left (c x +b \right )^{2}}\) | \(848\) |
((3*A*b^2*c^2*d*e^2-9*A*b*c^3*d^2*e+6*A*c^4*d^3-B*b^4*e^3+3*B*b^2*c^2*d^2* e-3*B*b*c^3*d^3)/b^4/c*x^3-1/2*A*d^3/b-d^2*(3*A*b*e-2*A*c*d+B*b*d)/b^2*x-1 /2*(A*b^3*c*e^3-9*A*b^2*c^2*d*e^2+27*A*b*c^3*d^2*e-18*A*c^4*d^3+B*b^4*e^3+ 3*B*b^3*c*d*e^2-9*B*b^2*c^2*d^2*e+9*B*b*c^3*d^3)/c^2/b^3*x^2)/x^2/(c*x+b)^ 2+3*d*(A*b^2*e^2-3*A*b*c*d*e+2*A*c^2*d^2+B*b^2*d*e-B*b*c*d^2)/b^5*ln(x)-3* d*(A*b^2*e^2-3*A*b*c*d*e+2*A*c^2*d^2+B*b^2*d*e-B*b*c*d^2)/b^5*ln(c*x+b)
Leaf count of result is larger than twice the leaf count of optimal. 627 vs. \(2 (181) = 362\).
Time = 0.36 (sec) , antiderivative size = 627, normalized size of antiderivative = 3.39 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^3} \, dx=-\frac {A b^{4} c^{2} d^{3} - 2 \, {\left (3 \, A b^{3} c^{3} d e^{2} - B b^{5} c e^{3} - 3 \, {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + 3 \, {\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{3} + {\left (9 \, {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} - 9 \, {\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e + 3 \, {\left (B b^{5} c - 3 \, A b^{4} c^{2}\right )} d e^{2} + {\left (B b^{6} + A b^{5} c\right )} e^{3}\right )} x^{2} + 2 \, {\left (3 \, A b^{4} c^{2} d^{2} e + {\left (B b^{4} c^{2} - 2 \, A b^{3} c^{3}\right )} d^{3}\right )} x + 6 \, {\left ({\left (A b^{2} c^{4} d e^{2} - {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} + {\left (B b^{2} c^{4} - 3 \, A b c^{5}\right )} d^{2} e\right )} x^{4} + 2 \, {\left (A b^{3} c^{3} d e^{2} - {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + {\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{3} + {\left (A b^{4} c^{2} d e^{2} - {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} + {\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e\right )} x^{2}\right )} \log \left (c x + b\right ) - 6 \, {\left ({\left (A b^{2} c^{4} d e^{2} - {\left (B b c^{5} - 2 \, A c^{6}\right )} d^{3} + {\left (B b^{2} c^{4} - 3 \, A b c^{5}\right )} d^{2} e\right )} x^{4} + 2 \, {\left (A b^{3} c^{3} d e^{2} - {\left (B b^{2} c^{4} - 2 \, A b c^{5}\right )} d^{3} + {\left (B b^{3} c^{3} - 3 \, A b^{2} c^{4}\right )} d^{2} e\right )} x^{3} + {\left (A b^{4} c^{2} d e^{2} - {\left (B b^{3} c^{3} - 2 \, A b^{2} c^{4}\right )} d^{3} + {\left (B b^{4} c^{2} - 3 \, A b^{3} c^{3}\right )} d^{2} e\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{4} x^{4} + 2 \, b^{6} c^{3} x^{3} + b^{7} c^{2} x^{2}\right )}} \]
-1/2*(A*b^4*c^2*d^3 - 2*(3*A*b^3*c^3*d*e^2 - B*b^5*c*e^3 - 3*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + 3*(B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (9*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 9*(B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e + 3*(B*b^5*c - 3*A*b^ 4*c^2)*d*e^2 + (B*b^6 + A*b^5*c)*e^3)*x^2 + 2*(3*A*b^4*c^2*d^2*e + (B*b^4* c^2 - 2*A*b^3*c^3)*d^3)*x + 6*((A*b^2*c^4*d*e^2 - (B*b*c^5 - 2*A*c^6)*d^3 + (B*b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(A*b^3*c^3*d*e^2 - (B*b^2*c^4 - 2 *A*b*c^5)*d^3 + (B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (A*b^4*c^2*d*e^2 - (B*b^3*c^3 - 2*A*b^2*c^4)*d^3 + (B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e)*x^2)*log( c*x + b) - 6*((A*b^2*c^4*d*e^2 - (B*b*c^5 - 2*A*c^6)*d^3 + (B*b^2*c^4 - 3* A*b*c^5)*d^2*e)*x^4 + 2*(A*b^3*c^3*d*e^2 - (B*b^2*c^4 - 2*A*b*c^5)*d^3 + ( B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (A*b^4*c^2*d*e^2 - (B*b^3*c^3 - 2*A* b^2*c^4)*d^3 + (B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e)*x^2)*log(x))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (187) = 374\).
Time = 133.62 (sec) , antiderivative size = 653, normalized size of antiderivative = 3.53 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^3} \, dx=\frac {- A b^{3} c^{2} d^{3} + x^{3} \cdot \left (6 A b^{2} c^{3} d e^{2} - 18 A b c^{4} d^{2} e + 12 A c^{5} d^{3} - 2 B b^{4} c e^{3} + 6 B b^{2} c^{3} d^{2} e - 6 B b c^{4} d^{3}\right ) + x^{2} \left (- A b^{4} c e^{3} + 9 A b^{3} c^{2} d e^{2} - 27 A b^{2} c^{3} d^{2} e + 18 A b c^{4} d^{3} - B b^{5} e^{3} - 3 B b^{4} c d e^{2} + 9 B b^{3} c^{2} d^{2} e - 9 B b^{2} c^{3} d^{3}\right ) + x \left (- 6 A b^{3} c^{2} d^{2} e + 4 A b^{2} c^{3} d^{3} - 2 B b^{3} c^{2} d^{3}\right )}{2 b^{6} c^{2} x^{2} + 4 b^{5} c^{3} x^{3} + 2 b^{4} c^{4} x^{4}} + \frac {3 d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right ) \log {\left (x + \frac {3 A b^{3} d e^{2} - 9 A b^{2} c d^{2} e + 6 A b c^{2} d^{3} + 3 B b^{3} d^{2} e - 3 B b^{2} c d^{3} - 3 b d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right )}{6 A b^{2} c d e^{2} - 18 A b c^{2} d^{2} e + 12 A c^{3} d^{3} + 6 B b^{2} c d^{2} e - 6 B b c^{2} d^{3}} \right )}}{b^{5}} - \frac {3 d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right ) \log {\left (x + \frac {3 A b^{3} d e^{2} - 9 A b^{2} c d^{2} e + 6 A b c^{2} d^{3} + 3 B b^{3} d^{2} e - 3 B b^{2} c d^{3} + 3 b d \left (b e - c d\right ) \left (A b e - 2 A c d + B b d\right )}{6 A b^{2} c d e^{2} - 18 A b c^{2} d^{2} e + 12 A c^{3} d^{3} + 6 B b^{2} c d^{2} e - 6 B b c^{2} d^{3}} \right )}}{b^{5}} \]
(-A*b**3*c**2*d**3 + x**3*(6*A*b**2*c**3*d*e**2 - 18*A*b*c**4*d**2*e + 12* A*c**5*d**3 - 2*B*b**4*c*e**3 + 6*B*b**2*c**3*d**2*e - 6*B*b*c**4*d**3) + x**2*(-A*b**4*c*e**3 + 9*A*b**3*c**2*d*e**2 - 27*A*b**2*c**3*d**2*e + 18*A *b*c**4*d**3 - B*b**5*e**3 - 3*B*b**4*c*d*e**2 + 9*B*b**3*c**2*d**2*e - 9* B*b**2*c**3*d**3) + x*(-6*A*b**3*c**2*d**2*e + 4*A*b**2*c**3*d**3 - 2*B*b* *3*c**2*d**3))/(2*b**6*c**2*x**2 + 4*b**5*c**3*x**3 + 2*b**4*c**4*x**4) + 3*d*(b*e - c*d)*(A*b*e - 2*A*c*d + B*b*d)*log(x + (3*A*b**3*d*e**2 - 9*A*b **2*c*d**2*e + 6*A*b*c**2*d**3 + 3*B*b**3*d**2*e - 3*B*b**2*c*d**3 - 3*b*d *(b*e - c*d)*(A*b*e - 2*A*c*d + B*b*d))/(6*A*b**2*c*d*e**2 - 18*A*b*c**2*d **2*e + 12*A*c**3*d**3 + 6*B*b**2*c*d**2*e - 6*B*b*c**2*d**3))/b**5 - 3*d* (b*e - c*d)*(A*b*e - 2*A*c*d + B*b*d)*log(x + (3*A*b**3*d*e**2 - 9*A*b**2* c*d**2*e + 6*A*b*c**2*d**3 + 3*B*b**3*d**2*e - 3*B*b**2*c*d**3 + 3*b*d*(b* e - c*d)*(A*b*e - 2*A*c*d + B*b*d))/(6*A*b**2*c*d*e**2 - 18*A*b*c**2*d**2* e + 12*A*c**3*d**3 + 6*B*b**2*c*d**2*e - 6*B*b*c**2*d**3))/b**5
Time = 0.21 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.88 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^3} \, dx=-\frac {A b^{3} c^{2} d^{3} - 2 \, {\left (3 \, A b^{2} c^{3} d e^{2} - B b^{4} c e^{3} - 3 \, {\left (B b c^{4} - 2 \, A c^{5}\right )} d^{3} + 3 \, {\left (B b^{2} c^{3} - 3 \, A b c^{4}\right )} d^{2} e\right )} x^{3} + {\left (9 \, {\left (B b^{2} c^{3} - 2 \, A b c^{4}\right )} d^{3} - 9 \, {\left (B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} d^{2} e + 3 \, {\left (B b^{4} c - 3 \, A b^{3} c^{2}\right )} d e^{2} + {\left (B b^{5} + A b^{4} c\right )} e^{3}\right )} x^{2} + 2 \, {\left (3 \, A b^{3} c^{2} d^{2} e + {\left (B b^{3} c^{2} - 2 \, A b^{2} c^{3}\right )} d^{3}\right )} x}{2 \, {\left (b^{4} c^{4} x^{4} + 2 \, b^{5} c^{3} x^{3} + b^{6} c^{2} x^{2}\right )}} - \frac {3 \, {\left (A b^{2} d e^{2} - {\left (B b c - 2 \, A c^{2}\right )} d^{3} + {\left (B b^{2} - 3 \, A b c\right )} d^{2} e\right )} \log \left (c x + b\right )}{b^{5}} + \frac {3 \, {\left (A b^{2} d e^{2} - {\left (B b c - 2 \, A c^{2}\right )} d^{3} + {\left (B b^{2} - 3 \, A b c\right )} d^{2} e\right )} \log \left (x\right )}{b^{5}} \]
-1/2*(A*b^3*c^2*d^3 - 2*(3*A*b^2*c^3*d*e^2 - B*b^4*c*e^3 - 3*(B*b*c^4 - 2* A*c^5)*d^3 + 3*(B*b^2*c^3 - 3*A*b*c^4)*d^2*e)*x^3 + (9*(B*b^2*c^3 - 2*A*b* c^4)*d^3 - 9*(B*b^3*c^2 - 3*A*b^2*c^3)*d^2*e + 3*(B*b^4*c - 3*A*b^3*c^2)*d *e^2 + (B*b^5 + A*b^4*c)*e^3)*x^2 + 2*(3*A*b^3*c^2*d^2*e + (B*b^3*c^2 - 2* A*b^2*c^3)*d^3)*x)/(b^4*c^4*x^4 + 2*b^5*c^3*x^3 + b^6*c^2*x^2) - 3*(A*b^2* d*e^2 - (B*b*c - 2*A*c^2)*d^3 + (B*b^2 - 3*A*b*c)*d^2*e)*log(c*x + b)/b^5 + 3*(A*b^2*d*e^2 - (B*b*c - 2*A*c^2)*d^3 + (B*b^2 - 3*A*b*c)*d^2*e)*log(x) /b^5
Leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (181) = 362\).
Time = 0.26 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.11 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^3} \, dx=-\frac {3 \, {\left (B b c d^{3} - 2 \, A c^{2} d^{3} - B b^{2} d^{2} e + 3 \, A b c d^{2} e - A b^{2} d e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac {3 \, {\left (B b c^{2} d^{3} - 2 \, A c^{3} d^{3} - B b^{2} c d^{2} e + 3 \, A b c^{2} d^{2} e - A b^{2} c d e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} - \frac {6 \, B b c^{4} d^{3} x^{3} - 12 \, A c^{5} d^{3} x^{3} - 6 \, B b^{2} c^{3} d^{2} e x^{3} + 18 \, A b c^{4} d^{2} e x^{3} - 6 \, A b^{2} c^{3} d e^{2} x^{3} + 2 \, B b^{4} c e^{3} x^{3} + 9 \, B b^{2} c^{3} d^{3} x^{2} - 18 \, A b c^{4} d^{3} x^{2} - 9 \, B b^{3} c^{2} d^{2} e x^{2} + 27 \, A b^{2} c^{3} d^{2} e x^{2} + 3 \, B b^{4} c d e^{2} x^{2} - 9 \, A b^{3} c^{2} d e^{2} x^{2} + B b^{5} e^{3} x^{2} + A b^{4} c e^{3} x^{2} + 2 \, B b^{3} c^{2} d^{3} x - 4 \, A b^{2} c^{3} d^{3} x + 6 \, A b^{3} c^{2} d^{2} e x + A b^{3} c^{2} d^{3}}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4} c^{2}} \]
-3*(B*b*c*d^3 - 2*A*c^2*d^3 - B*b^2*d^2*e + 3*A*b*c*d^2*e - A*b^2*d*e^2)*l og(abs(x))/b^5 + 3*(B*b*c^2*d^3 - 2*A*c^3*d^3 - B*b^2*c*d^2*e + 3*A*b*c^2* d^2*e - A*b^2*c*d*e^2)*log(abs(c*x + b))/(b^5*c) - 1/2*(6*B*b*c^4*d^3*x^3 - 12*A*c^5*d^3*x^3 - 6*B*b^2*c^3*d^2*e*x^3 + 18*A*b*c^4*d^2*e*x^3 - 6*A*b^ 2*c^3*d*e^2*x^3 + 2*B*b^4*c*e^3*x^3 + 9*B*b^2*c^3*d^3*x^2 - 18*A*b*c^4*d^3 *x^2 - 9*B*b^3*c^2*d^2*e*x^2 + 27*A*b^2*c^3*d^2*e*x^2 + 3*B*b^4*c*d*e^2*x^ 2 - 9*A*b^3*c^2*d*e^2*x^2 + B*b^5*e^3*x^2 + A*b^4*c*e^3*x^2 + 2*B*b^3*c^2* d^3*x - 4*A*b^2*c^3*d^3*x + 6*A*b^3*c^2*d^2*e*x + A*b^3*c^2*d^3)/((c*x^2 + b*x)^2*b^4*c^2)
Time = 10.50 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.86 \[ \int \frac {(A+B x) (d+e x)^3}{\left (b x+c x^2\right )^3} \, dx=-\frac {\frac {A\,d^3}{2\,b}-\frac {x^3\,\left (-B\,b^4\,e^3+3\,B\,b^2\,c^2\,d^2\,e+3\,A\,b^2\,c^2\,d\,e^2-3\,B\,b\,c^3\,d^3-9\,A\,b\,c^3\,d^2\,e+6\,A\,c^4\,d^3\right )}{b^4\,c}+\frac {x^2\,\left (B\,b^4\,e^3+3\,B\,b^3\,c\,d\,e^2+A\,b^3\,c\,e^3-9\,B\,b^2\,c^2\,d^2\,e-9\,A\,b^2\,c^2\,d\,e^2+9\,B\,b\,c^3\,d^3+27\,A\,b\,c^3\,d^2\,e-18\,A\,c^4\,d^3\right )}{2\,b^3\,c^2}+\frac {d^2\,x\,\left (3\,A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}{b^2}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {6\,d\,\mathrm {atanh}\left (\frac {3\,d\,\left (b\,e-c\,d\right )\,\left (b+2\,c\,x\right )\,\left (A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}{b\,\left (3\,B\,b^2\,d^2\,e+3\,A\,b^2\,d\,e^2-3\,B\,b\,c\,d^3-9\,A\,b\,c\,d^2\,e+6\,A\,c^2\,d^3\right )}\right )\,\left (b\,e-c\,d\right )\,\left (A\,b\,e-2\,A\,c\,d+B\,b\,d\right )}{b^5} \]
- ((A*d^3)/(2*b) - (x^3*(6*A*c^4*d^3 - B*b^4*e^3 - 3*B*b*c^3*d^3 + 3*A*b^2 *c^2*d*e^2 + 3*B*b^2*c^2*d^2*e - 9*A*b*c^3*d^2*e))/(b^4*c) + (x^2*(B*b^4*e ^3 - 18*A*c^4*d^3 + A*b^3*c*e^3 + 9*B*b*c^3*d^3 - 9*A*b^2*c^2*d*e^2 - 9*B* b^2*c^2*d^2*e + 27*A*b*c^3*d^2*e + 3*B*b^3*c*d*e^2))/(2*b^3*c^2) + (d^2*x* (3*A*b*e - 2*A*c*d + B*b*d))/b^2)/(b^2*x^2 + c^2*x^4 + 2*b*c*x^3) - (6*d*a tanh((3*d*(b*e - c*d)*(b + 2*c*x)*(A*b*e - 2*A*c*d + B*b*d))/(b*(6*A*c^2*d ^3 - 3*B*b*c*d^3 + 3*A*b^2*d*e^2 + 3*B*b^2*d^2*e - 9*A*b*c*d^2*e)))*(b*e - c*d)*(A*b*e - 2*A*c*d + B*b*d))/b^5