Integrand size = 20, antiderivative size = 158 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^2} \, dx=-\frac {A b-a B}{2 (b d-a e)^2 (a+b x)^2}-\frac {b B d-2 A b e+a B e}{(b d-a e)^3 (a+b x)}-\frac {e (B d-A e)}{(b d-a e)^3 (d+e x)}-\frac {e (2 b B d-3 A b e+a B e) \log (a+b x)}{(b d-a e)^4}+\frac {e (2 b B d-3 A b e+a B e) \log (d+e x)}{(b d-a e)^4} \]
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Time = 0.12 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^2} \, dx=-\frac {A b-a B}{2 (a+b x)^2 (b d-a e)^2}-\frac {a B e-2 A b e+b B d}{(a+b x) (b d-a e)^3}-\frac {e (B d-A e)}{(d+e x) (b d-a e)^3}-\frac {e \log (a+b x) (a B e-3 A b e+2 b B d)}{(b d-a e)^4}+\frac {e \log (d+e x) (a B e-3 A b e+2 b B d)}{(b d-a e)^4} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b (A b-a B)}{(b d-a e)^2 (a+b x)^3}+\frac {b (b B d-2 A b e+a B e)}{(b d-a e)^3 (a+b x)^2}+\frac {b e (-2 b B d+3 A b e-a B e)}{(b d-a e)^4 (a+b x)}-\frac {e^2 (-B d+A e)}{(b d-a e)^3 (d+e x)^2}-\frac {e^2 (-2 b B d+3 A b e-a B e)}{(b d-a e)^4 (d+e x)}\right ) \, dx \\ & = -\frac {A b-a B}{2 (b d-a e)^2 (a+b x)^2}-\frac {b B d-2 A b e+a B e}{(b d-a e)^3 (a+b x)}-\frac {e (B d-A e)}{(b d-a e)^3 (d+e x)}-\frac {e (2 b B d-3 A b e+a B e) \log (a+b x)}{(b d-a e)^4}+\frac {e (2 b B d-3 A b e+a B e) \log (d+e x)}{(b d-a e)^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^2} \, dx=\frac {\frac {(-A b+a B) (b d-a e)^2}{(a+b x)^2}-\frac {2 (b d-a e) (b B d-2 A b e+a B e)}{a+b x}+\frac {2 e (b d-a e) (-B d+A e)}{d+e x}-2 e (2 b B d-3 A b e+a B e) \log (a+b x)+2 e (2 b B d-3 A b e+a B e) \log (d+e x)}{2 (b d-a e)^4} \]
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Time = 0.79 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.02
method | result | size |
default | \(-\frac {2 A b e -B a e -B b d}{\left (a e -b d \right )^{3} \left (b x +a \right )}-\frac {A b -B a}{2 \left (a e -b d \right )^{2} \left (b x +a \right )^{2}}+\frac {e \left (3 A b e -B a e -2 B b d \right ) \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}-\frac {\left (A e -B d \right ) e}{\left (a e -b d \right )^{3} \left (e x +d \right )}-\frac {e \left (3 A b e -B a e -2 B b d \right ) \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}\) | \(161\) |
norman | \(\frac {-\frac {2 A \,a^{2} b^{2} e^{3}+5 A a \,b^{3} d \,e^{2}-A \,b^{4} d^{2} e -5 B \,a^{2} b^{2} d \,e^{2}-B a \,b^{3} d^{2} e}{2 e \,b^{2} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {\left (3 A \,b^{3} e^{3}-B a \,b^{2} e^{3}-2 b^{3} B d \,e^{2}\right ) x^{2}}{e b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {\left (9 A a \,b^{3} e^{3}+3 A \,b^{4} d \,e^{2}-3 B \,a^{2} b^{2} e^{3}-7 B a \,b^{3} d \,e^{2}-2 B \,b^{4} d^{2} e \right ) x}{2 e \,b^{2} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}}{\left (b x +a \right )^{2} \left (e x +d \right )}+\frac {e \left (3 A b e -B a e -2 B b d \right ) \ln \left (b x +a \right )}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {e \left (3 A b e -B a e -2 B b d \right ) \ln \left (e x +d \right )}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}\) | \(453\) |
risch | \(\frac {-\frac {b e \left (3 A b e -B a e -2 B b d \right ) x^{2}}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}-\frac {\left (3 a e +b d \right ) \left (3 A b e -B a e -2 B b d \right ) x}{2 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {2 a^{2} A \,e^{2}+5 A a b d e -A \,b^{2} d^{2}-5 B \,a^{2} d e -B a b \,d^{2}}{2 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}}{\left (b x +a \right )^{2} \left (e x +d \right )}-\frac {3 e^{2} \ln \left (e x +d \right ) A b}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {e^{2} \ln \left (e x +d \right ) B a}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {2 e \ln \left (e x +d \right ) B b d}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {3 e^{2} \ln \left (-b x -a \right ) A b}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {e^{2} \ln \left (-b x -a \right ) B a}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {2 e \ln \left (-b x -a \right ) B b d}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}\) | \(616\) |
parallelrisch | \(\frac {6 A \ln \left (b x +a \right ) x \,a^{2} b^{3} e^{4}+2 B \ln \left (e x +d \right ) x^{3} a \,b^{4} e^{4}+4 B \ln \left (e x +d \right ) x^{3} b^{5} d \,e^{3}+12 A \ln \left (b x +a \right ) x^{2} a \,b^{4} e^{4}+6 A \ln \left (b x +a \right ) x^{2} b^{5} d \,e^{3}-12 A \ln \left (e x +d \right ) x^{2} a \,b^{4} e^{4}-6 A \ln \left (e x +d \right ) x^{2} b^{5} d \,e^{3}-4 B \ln \left (b x +a \right ) x^{2} a^{2} b^{3} e^{4}-4 B \ln \left (b x +a \right ) x^{2} b^{5} d^{2} e^{2}+4 B x \,a^{2} b^{3} d \,e^{3}-5 B x a \,b^{4} d^{2} e^{2}-2 B \ln \left (b x +a \right ) x^{3} a \,b^{4} e^{4}-4 B \ln \left (b x +a \right ) x^{3} b^{5} d \,e^{3}+10 B \ln \left (e x +d \right ) x^{2} a \,b^{4} d \,e^{3}+12 A \ln \left (b x +a \right ) x a \,b^{4} d \,e^{3}-12 A \ln \left (e x +d \right ) x a \,b^{4} d \,e^{3}-6 A \ln \left (e x +d \right ) x \,a^{2} b^{3} e^{4}-2 B \ln \left (b x +a \right ) x \,a^{3} b^{2} e^{4}+2 B \ln \left (e x +d \right ) x \,a^{3} b^{2} e^{4}+6 A \ln \left (b x +a \right ) a^{2} b^{3} d \,e^{3}-6 A \ln \left (e x +d \right ) a^{2} b^{3} d \,e^{3}-2 B \ln \left (b x +a \right ) a^{3} b^{2} d \,e^{3}-4 B \ln \left (b x +a \right ) a^{2} b^{3} d^{2} e^{2}+2 B \ln \left (e x +d \right ) a^{3} b^{2} d \,e^{3}-8 B \ln \left (b x +a \right ) x \,a^{2} b^{3} d \,e^{3}-2 A \,a^{3} b^{2} e^{4}-A \,b^{5} d^{3} e +6 A x a \,b^{4} d \,e^{3}+4 B \ln \left (e x +d \right ) a^{2} b^{3} d^{2} e^{2}+2 B \,x^{2} a \,b^{4} d \,e^{3}+4 B \ln \left (e x +d \right ) x^{2} a^{2} b^{3} e^{4}+4 B \ln \left (e x +d \right ) x^{2} b^{5} d^{2} e^{2}-8 B \ln \left (b x +a \right ) x a \,b^{4} d^{2} e^{2}+8 B \ln \left (e x +d \right ) x \,a^{2} b^{3} d \,e^{3}+8 B \ln \left (e x +d \right ) x a \,b^{4} d^{2} e^{2}-10 B \ln \left (b x +a \right ) x^{2} a \,b^{4} d \,e^{3}+3 A x \,b^{5} d^{2} e^{2}+6 A \ln \left (b x +a \right ) x^{3} b^{5} e^{4}-6 A \ln \left (e x +d \right ) x^{3} b^{5} e^{4}+3 B x \,a^{3} b^{2} e^{4}-6 A \,x^{2} a \,b^{4} e^{4}+6 A \,x^{2} b^{5} d \,e^{3}+2 B \,x^{2} a^{2} b^{3} e^{4}-4 B \,x^{2} b^{5} d^{2} e^{2}-9 A x \,a^{2} b^{3} e^{4}-2 B x \,b^{5} d^{3} e -3 A \,a^{2} b^{3} d \,e^{3}+6 A a \,b^{4} d^{2} e^{2}+5 B \,a^{3} b^{2} d \,e^{3}-4 B \,a^{2} b^{3} d^{2} e^{2}-B a \,b^{4} d^{3} e}{2 \left (a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (e x +d \right ) \left (b x +a \right )^{2} b^{2} e}\) | \(944\) |
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Leaf count of result is larger than twice the leaf count of optimal. 803 vs. \(2 (156) = 312\).
Time = 0.25 (sec) , antiderivative size = 803, normalized size of antiderivative = 5.08 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^2} \, dx=-\frac {2 \, A a^{3} e^{3} + {\left (B a b^{2} + A b^{3}\right )} d^{3} + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e - {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} d e^{2} + 2 \, {\left (2 \, B b^{3} d^{2} e - {\left (B a b^{2} + 3 \, A b^{3}\right )} d e^{2} - {\left (B a^{2} b - 3 \, A a b^{2}\right )} e^{3}\right )} x^{2} + {\left (2 \, B b^{3} d^{3} + {\left (5 \, B a b^{2} - 3 \, A b^{3}\right )} d^{2} e - 2 \, {\left (2 \, B a^{2} b + 3 \, A a b^{2}\right )} d e^{2} - 3 \, {\left (B a^{3} - 3 \, A a^{2} b\right )} e^{3}\right )} x + 2 \, {\left (2 \, B a^{2} b d^{2} e + {\left (B a^{3} - 3 \, A a^{2} b\right )} d e^{2} + {\left (2 \, B b^{3} d e^{2} + {\left (B a b^{2} - 3 \, A b^{3}\right )} e^{3}\right )} x^{3} + {\left (2 \, B b^{3} d^{2} e + {\left (5 \, B a b^{2} - 3 \, A b^{3}\right )} d e^{2} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} e^{3}\right )} x^{2} + {\left (4 \, B a b^{2} d^{2} e + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} e^{3}\right )} x\right )} \log \left (b x + a\right ) - 2 \, {\left (2 \, B a^{2} b d^{2} e + {\left (B a^{3} - 3 \, A a^{2} b\right )} d e^{2} + {\left (2 \, B b^{3} d e^{2} + {\left (B a b^{2} - 3 \, A b^{3}\right )} e^{3}\right )} x^{3} + {\left (2 \, B b^{3} d^{2} e + {\left (5 \, B a b^{2} - 3 \, A b^{3}\right )} d e^{2} + 2 \, {\left (B a^{2} b - 3 \, A a b^{2}\right )} e^{3}\right )} x^{2} + {\left (4 \, B a b^{2} d^{2} e + 2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d e^{2} + {\left (B a^{3} - 3 \, A a^{2} b\right )} e^{3}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a^{2} b^{4} d^{5} - 4 \, a^{3} b^{3} d^{4} e + 6 \, a^{4} b^{2} d^{3} e^{2} - 4 \, a^{5} b d^{2} e^{3} + a^{6} d e^{4} + {\left (b^{6} d^{4} e - 4 \, a b^{5} d^{3} e^{2} + 6 \, a^{2} b^{4} d^{2} e^{3} - 4 \, a^{3} b^{3} d e^{4} + a^{4} b^{2} e^{5}\right )} x^{3} + {\left (b^{6} d^{5} - 2 \, a b^{5} d^{4} e - 2 \, a^{2} b^{4} d^{3} e^{2} + 8 \, a^{3} b^{3} d^{2} e^{3} - 7 \, a^{4} b^{2} d e^{4} + 2 \, a^{5} b e^{5}\right )} x^{2} + {\left (2 \, a b^{5} d^{5} - 7 \, a^{2} b^{4} d^{4} e + 8 \, a^{3} b^{3} d^{3} e^{2} - 2 \, a^{4} b^{2} d^{2} e^{3} - 2 \, a^{5} b d e^{4} + a^{6} e^{5}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1066 vs. \(2 (148) = 296\).
Time = 2.08 (sec) , antiderivative size = 1066, normalized size of antiderivative = 6.75 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^2} \, dx=\frac {e \left (- 3 A b e + B a e + 2 B b d\right ) \log {\left (x + \frac {- 3 A a b e^{3} - 3 A b^{2} d e^{2} + B a^{2} e^{3} + 3 B a b d e^{2} + 2 B b^{2} d^{2} e - \frac {a^{5} e^{6} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} + \frac {5 a^{4} b d e^{5} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{3} b^{2} d^{2} e^{4} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{2} b^{3} d^{3} e^{3} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} - \frac {5 a b^{4} d^{4} e^{2} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} + \frac {b^{5} d^{5} e \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}}}{- 6 A b^{2} e^{3} + 2 B a b e^{3} + 4 B b^{2} d e^{2}} \right )}}{\left (a e - b d\right )^{4}} - \frac {e \left (- 3 A b e + B a e + 2 B b d\right ) \log {\left (x + \frac {- 3 A a b e^{3} - 3 A b^{2} d e^{2} + B a^{2} e^{3} + 3 B a b d e^{2} + 2 B b^{2} d^{2} e + \frac {a^{5} e^{6} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} - \frac {5 a^{4} b d e^{5} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{3} b^{2} d^{2} e^{4} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{2} b^{3} d^{3} e^{3} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} + \frac {5 a b^{4} d^{4} e^{2} \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}} - \frac {b^{5} d^{5} e \left (- 3 A b e + B a e + 2 B b d\right )}{\left (a e - b d\right )^{4}}}{- 6 A b^{2} e^{3} + 2 B a b e^{3} + 4 B b^{2} d e^{2}} \right )}}{\left (a e - b d\right )^{4}} + \frac {- 2 A a^{2} e^{2} - 5 A a b d e + A b^{2} d^{2} + 5 B a^{2} d e + B a b d^{2} + x^{2} \left (- 6 A b^{2} e^{2} + 2 B a b e^{2} + 4 B b^{2} d e\right ) + x \left (- 9 A a b e^{2} - 3 A b^{2} d e + 3 B a^{2} e^{2} + 7 B a b d e + 2 B b^{2} d^{2}\right )}{2 a^{5} d e^{3} - 6 a^{4} b d^{2} e^{2} + 6 a^{3} b^{2} d^{3} e - 2 a^{2} b^{3} d^{4} + x^{3} \cdot \left (2 a^{3} b^{2} e^{4} - 6 a^{2} b^{3} d e^{3} + 6 a b^{4} d^{2} e^{2} - 2 b^{5} d^{3} e\right ) + x^{2} \cdot \left (4 a^{4} b e^{4} - 10 a^{3} b^{2} d e^{3} + 6 a^{2} b^{3} d^{2} e^{2} + 2 a b^{4} d^{3} e - 2 b^{5} d^{4}\right ) + x \left (2 a^{5} e^{4} - 2 a^{4} b d e^{3} - 6 a^{3} b^{2} d^{2} e^{2} + 10 a^{2} b^{3} d^{3} e - 4 a b^{4} d^{4}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (156) = 312\).
Time = 0.21 (sec) , antiderivative size = 477, normalized size of antiderivative = 3.02 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^2} \, dx=-\frac {{\left (2 \, B b d e + {\left (B a - 3 \, A b\right )} e^{2}\right )} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac {{\left (2 \, B b d e + {\left (B a - 3 \, A b\right )} e^{2}\right )} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac {2 \, A a^{2} e^{2} - {\left (B a b + A b^{2}\right )} d^{2} - 5 \, {\left (B a^{2} - A a b\right )} d e - 2 \, {\left (2 \, B b^{2} d e + {\left (B a b - 3 \, A b^{2}\right )} e^{2}\right )} x^{2} - {\left (2 \, B b^{2} d^{2} + {\left (7 \, B a b - 3 \, A b^{2}\right )} d e + 3 \, {\left (B a^{2} - 3 \, A a b\right )} e^{2}\right )} x}{2 \, {\left (a^{2} b^{3} d^{4} - 3 \, a^{3} b^{2} d^{3} e + 3 \, a^{4} b d^{2} e^{2} - a^{5} d e^{3} + {\left (b^{5} d^{3} e - 3 \, a b^{4} d^{2} e^{2} + 3 \, a^{2} b^{3} d e^{3} - a^{3} b^{2} e^{4}\right )} x^{3} + {\left (b^{5} d^{4} - a b^{4} d^{3} e - 3 \, a^{2} b^{3} d^{2} e^{2} + 5 \, a^{3} b^{2} d e^{3} - 2 \, a^{4} b e^{4}\right )} x^{2} + {\left (2 \, a b^{4} d^{4} - 5 \, a^{2} b^{3} d^{3} e + 3 \, a^{3} b^{2} d^{2} e^{2} + a^{4} b d e^{3} - a^{5} e^{4}\right )} x\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.93 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^2} \, dx=-\frac {{\left (2 \, B b d e^{2} + B a e^{3} - 3 \, A b e^{3}\right )} \log \left ({\left | b - \frac {b d}{e x + d} + \frac {a e}{e x + d} \right |}\right )}{b^{4} d^{4} e - 4 \, a b^{3} d^{3} e^{2} + 6 \, a^{2} b^{2} d^{2} e^{3} - 4 \, a^{3} b d e^{4} + a^{4} e^{5}} - \frac {\frac {B d e^{4}}{e x + d} - \frac {A e^{5}}{e x + d}}{b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} + 3 \, a^{2} b d e^{5} - a^{3} e^{6}} - \frac {2 \, B b^{3} d e + 3 \, B a b^{2} e^{2} - 5 \, A b^{3} e^{2} - \frac {2 \, {\left (B b^{3} d^{2} e^{2} + B a b^{2} d e^{3} - 3 \, A b^{3} d e^{3} - 2 \, B a^{2} b e^{4} + 3 \, A a b^{2} e^{4}\right )}}{{\left (e x + d\right )} e}}{2 \, {\left (b d - a e\right )}^{4} {\left (b - \frac {b d}{e x + d} + \frac {a e}{e x + d}\right )}^{2}} \]
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Time = 1.77 (sec) , antiderivative size = 453, normalized size of antiderivative = 2.87 \[ \int \frac {A+B x}{(a+b x)^3 (d+e x)^2} \, dx=\frac {\frac {5\,B\,a^2\,d\,e-2\,A\,a^2\,e^2+B\,a\,b\,d^2-5\,A\,a\,b\,d\,e+A\,b^2\,d^2}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {x\,\left (3\,a\,e+b\,d\right )\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {b\,e\,x^2\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}}{x\,\left (e\,a^2+2\,b\,d\,a\right )+a^2\,d+x^2\,\left (d\,b^2+2\,a\,e\,b\right )+b^2\,e\,x^3}-\frac {2\,\mathrm {atanh}\left (\frac {\left (e^2\,\left (3\,A\,b-B\,a\right )-2\,B\,b\,d\,e\right )\,\left (\frac {a^4\,e^4-2\,a^3\,b\,d\,e^3+2\,a\,b^3\,d^3\,e-b^4\,d^4}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}+2\,b\,e\,x\right )\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^4\,\left (B\,a\,e^2-3\,A\,b\,e^2+2\,B\,b\,d\,e\right )}\right )\,\left (e^2\,\left (3\,A\,b-B\,a\right )-2\,B\,b\,d\,e\right )}{{\left (a\,e-b\,d\right )}^4} \]
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