Integrand size = 20, antiderivative size = 157 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^3} \, dx=-\frac {b (A b-a B)}{(b d-a e)^3 (a+b x)}+\frac {B d-A e}{2 (b d-a e)^2 (d+e x)^2}+\frac {b B d-2 A b e+a B e}{(b d-a e)^3 (d+e x)}+\frac {b (b B d-3 A b e+2 a B e) \log (a+b x)}{(b d-a e)^4}-\frac {b (b B d-3 A b e+2 a B e) \log (d+e x)}{(b d-a e)^4} \]
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Time = 0.10 (sec) , antiderivative size = 157, 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} \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^3} \, dx=-\frac {b (A b-a B)}{(a+b x) (b d-a e)^3}+\frac {a B e-2 A b e+b B d}{(d+e x) (b d-a e)^3}+\frac {B d-A e}{2 (d+e x)^2 (b d-a e)^2}+\frac {b \log (a+b x) (2 a B e-3 A b e+b B d)}{(b d-a e)^4}-\frac {b \log (d+e x) (2 a B e-3 A b e+b B d)}{(b d-a e)^4} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b^2 (A b-a B)}{(b d-a e)^3 (a+b x)^2}+\frac {b^2 (b B d-3 A b e+2 a B e)}{(b d-a e)^4 (a+b x)}+\frac {e (-B d+A e)}{(b d-a e)^2 (d+e x)^3}+\frac {e (-b B d+2 A b e-a B e)}{(b d-a e)^3 (d+e x)^2}+\frac {b e (-b B d+3 A b e-2 a B e)}{(b d-a e)^4 (d+e x)}\right ) \, dx \\ & = -\frac {b (A b-a B)}{(b d-a e)^3 (a+b x)}+\frac {B d-A e}{2 (b d-a e)^2 (d+e x)^2}+\frac {b B d-2 A b e+a B e}{(b d-a e)^3 (d+e x)}+\frac {b (b B d-3 A b e+2 a B e) \log (a+b x)}{(b d-a e)^4}-\frac {b (b B d-3 A b e+2 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 = 0.93 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^3} \, dx=\frac {-\frac {2 b (A b-a B) (b d-a e)}{a+b x}+\frac {(b d-a e)^2 (B d-A e)}{(d+e x)^2}+\frac {2 (b d-a e) (b B d-2 A b e+a B e)}{d+e x}+2 b (b B d-3 A b e+2 a B e) \log (a+b x)-2 b (b B d-3 A b e+2 a B e) \log (d+e x)}{2 (b d-a e)^4} \]
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Time = 0.80 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {b \left (3 A b e -2 B a e -B b d \right ) \ln \left (b x +a \right )}{\left (a e -b d \right )^{4}}+\frac {\left (A b -B a \right ) b}{\left (a e -b d \right )^{3} \left (b x +a \right )}-\frac {A e -B d}{2 \left (a e -b d \right )^{2} \left (e x +d \right )^{2}}+\frac {b \left (3 A b e -2 B a e -B b d \right ) \ln \left (e x +d \right )}{\left (a e -b d \right )^{4}}+\frac {2 A b e -B a e -B b d}{\left (a e -b d \right )^{3} \left (e x +d \right )}\) | \(159\) |
norman | \(\frac {\frac {\left (3 A \,b^{3} e^{3}-2 B a \,b^{2} e^{3}-b^{3} B d \,e^{2}\right ) x^{2}}{e b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}-\frac {A \,a^{2} b \,e^{4}-5 A a \,b^{2} d \,e^{3}-2 A \,b^{3} d^{2} e^{2}+B \,a^{2} b d \,e^{3}+5 B a \,b^{2} d^{2} e^{2}}{2 e^{2} b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}+\frac {\left (3 A a \,b^{2} e^{4}+9 A \,b^{3} d \,e^{3}-2 B \,a^{2} b \,e^{4}-7 B a \,b^{2} d \,e^{3}-3 b^{3} B \,d^{2} e^{2}\right ) x}{2 e^{2} b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}}{\left (b x +a \right ) \left (e x +d \right )^{2}}+\frac {b \left (3 A b e -2 B a e -B b d \right ) \ln \left (e x +d \right )}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {b \left (3 A b e -2 B a e -B b d \right ) \ln \left (b x +a \right )}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}\) | \(450\) |
risch | \(\frac {\frac {b e \left (3 A b e -2 B a e -B b d \right ) x^{2}}{a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}}+\frac {\left (a e +3 b d \right ) \left (3 A b e -2 B a e -B b d \right ) x}{2 a^{3} e^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}}-\frac {a^{2} A \,e^{2}-5 A a b d e -2 A \,b^{2} d^{2}+B \,a^{2} d e +5 B a b \,d^{2}}{2 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}}{\left (b x +a \right ) \left (e x +d \right )^{2}}+\frac {3 b^{2} \ln \left (-e x -d \right ) A e}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {2 b \ln \left (-e x -d \right ) B a e}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {b^{2} \ln \left (-e x -d \right ) B d}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}-\frac {3 b^{2} \ln \left (b x +a \right ) A e}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {2 b \ln \left (b x +a \right ) B a e}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}+\frac {b^{2} \ln \left (b x +a \right ) B d}{a^{4} e^{4}-4 a^{3} b d \,e^{3}+6 a^{2} b^{2} d^{2} e^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}\) | \(613\) |
parallelrisch | \(-\frac {-2 B \,x^{2} a \,b^{3} d \,e^{4}-6 A x a \,b^{3} d \,e^{4}+5 B x \,a^{2} b^{2} d \,e^{4}-4 B x a \,b^{3} d^{2} e^{3}-4 B \ln \left (b x +a \right ) x^{3} a \,b^{3} e^{5}-2 B \ln \left (b x +a \right ) x^{3} b^{4} d \,e^{4}+4 B \ln \left (e x +d \right ) x^{3} a \,b^{3} e^{5}+2 B \ln \left (e x +d \right ) x \,b^{4} d^{3} e^{2}+6 A \ln \left (b x +a \right ) a \,b^{3} d^{2} e^{3}-6 A \ln \left (e x +d \right ) a \,b^{3} d^{2} e^{3}-4 B \ln \left (b x +a \right ) a^{2} b^{2} d^{2} e^{3}+8 B \ln \left (e x +d \right ) x a \,b^{3} d^{2} e^{3}-10 B \ln \left (b x +a \right ) x^{2} a \,b^{3} d \,e^{4}+10 B \ln \left (e x +d \right ) x^{2} a \,b^{3} d \,e^{4}+12 A \ln \left (b x +a \right ) x a \,b^{3} d \,e^{4}+4 B \,x^{2} a^{2} b^{2} e^{5}-2 B \,x^{2} b^{4} d^{2} e^{3}-3 A x \,a^{2} b^{2} e^{5}+9 A x \,b^{4} d^{2} e^{3}+2 B x \,a^{3} b \,e^{5}-3 B x \,b^{4} d^{3} e^{2}+6 A \ln \left (b x +a \right ) x^{3} b^{4} e^{5}-6 A \ln \left (e x +d \right ) x^{3} b^{4} e^{5}-2 B \ln \left (b x +a \right ) a \,b^{3} d^{3} e^{2}+4 B \ln \left (e x +d \right ) a^{2} b^{2} d^{2} e^{3}+2 B \ln \left (e x +d \right ) a \,b^{3} d^{3} e^{2}+2 B \ln \left (e x +d \right ) x^{3} b^{4} d \,e^{4}+6 A \ln \left (b x +a \right ) x^{2} a \,b^{3} e^{5}+12 A \ln \left (b x +a \right ) x^{2} b^{4} d \,e^{4}-12 A \ln \left (e x +d \right ) x^{2} b^{4} d \,e^{4}-4 B \ln \left (b x +a \right ) x^{2} a^{2} b^{2} e^{5}-4 B \ln \left (b x +a \right ) x^{2} b^{4} d^{2} e^{3}+4 B \ln \left (e x +d \right ) x^{2} a^{2} b^{2} e^{5}+4 B \ln \left (e x +d \right ) x^{2} b^{4} d^{2} e^{3}+6 A \ln \left (b x +a \right ) x \,b^{4} d^{2} e^{3}-6 A \ln \left (e x +d \right ) x \,b^{4} d^{2} e^{3}-2 B \ln \left (b x +a \right ) x \,b^{4} d^{3} e^{2}-6 A \ln \left (e x +d \right ) x^{2} a \,b^{3} e^{5}-8 B \ln \left (b x +a \right ) x \,a^{2} b^{2} d \,e^{4}-8 B \ln \left (b x +a \right ) x a \,b^{3} d^{2} e^{3}+8 B \ln \left (e x +d \right ) x \,a^{2} b^{2} d \,e^{4}+A \,a^{3} b \,e^{5}+2 A \,b^{4} d^{3} e^{2}-12 A \ln \left (e x +d \right ) x a \,b^{3} d \,e^{4}-6 A \,a^{2} b^{2} d \,e^{4}+3 A a \,b^{3} d^{2} e^{3}+B \,a^{3} b d \,e^{4}+4 B \,a^{2} b^{2} d^{2} e^{3}-5 B a \,b^{3} d^{3} e^{2}-6 A \,x^{2} a \,b^{3} e^{5}+6 A \,x^{2} b^{4} d \,e^{4}}{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 )^{2} \left (b x +a \right ) b \,e^{2}}\) | \(942\) |
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Leaf count of result is larger than twice the leaf count of optimal. 801 vs. \(2 (154) = 308\).
Time = 0.24 (sec) , antiderivative size = 801, normalized size of antiderivative = 5.10 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^3} \, dx=-\frac {A a^{3} e^{3} - {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} d^{3} + {\left (4 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{2} e + {\left (B a^{3} - 6 \, A a^{2} b\right )} d e^{2} - 2 \, {\left (B b^{3} d^{2} e + {\left (B a b^{2} - 3 \, A b^{3}\right )} d e^{2} - {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{3}\right )} x^{2} - {\left (3 \, B b^{3} d^{3} + {\left (4 \, B a b^{2} - 9 \, A b^{3}\right )} d^{2} e - {\left (5 \, B a^{2} b - 6 \, A a b^{2}\right )} d e^{2} - {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} e^{3}\right )} x - 2 \, {\left (B a b^{2} d^{3} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e + {\left (B b^{3} d e^{2} + {\left (2 \, 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} - 6 \, A b^{3}\right )} d e^{2} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{3}\right )} x^{2} + {\left (B b^{3} d^{3} + {\left (4 \, 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}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (B a b^{2} d^{3} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} d^{2} e + {\left (B b^{3} d e^{2} + {\left (2 \, 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} - 6 \, A b^{3}\right )} d e^{2} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} e^{3}\right )} x^{2} + {\left (B b^{3} d^{3} + {\left (4 \, 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}\right )} x\right )} \log \left (e x + d\right )}{2 \, {\left (a b^{4} d^{6} - 4 \, a^{2} b^{3} d^{5} e + 6 \, a^{3} b^{2} d^{4} e^{2} - 4 \, a^{4} b d^{3} e^{3} + a^{5} d^{2} e^{4} + {\left (b^{5} d^{4} e^{2} - 4 \, a b^{4} d^{3} e^{3} + 6 \, a^{2} b^{3} d^{2} e^{4} - 4 \, a^{3} b^{2} d e^{5} + a^{4} b e^{6}\right )} x^{3} + {\left (2 \, b^{5} d^{5} e - 7 \, a b^{4} d^{4} e^{2} + 8 \, a^{2} b^{3} d^{3} e^{3} - 2 \, a^{3} b^{2} d^{2} e^{4} - 2 \, a^{4} b d e^{5} + a^{5} e^{6}\right )} x^{2} + {\left (b^{5} d^{6} - 2 \, a b^{4} d^{5} e - 2 \, a^{2} b^{3} d^{4} e^{2} + 8 \, a^{3} b^{2} d^{3} e^{3} - 7 \, a^{4} b d^{2} e^{4} + 2 \, a^{5} d 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 = 2.11 (sec) , antiderivative size = 1066, normalized size of antiderivative = 6.79 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^3} \, dx=- \frac {b \left (- 3 A b e + 2 B a e + B b d\right ) \log {\left (x + \frac {- 3 A a b^{2} e^{2} - 3 A b^{3} d e + 2 B a^{2} b e^{2} + 3 B a b^{2} d e + B b^{3} d^{2} - \frac {a^{5} b e^{5} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {5 a^{4} b^{2} d e^{4} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{3} b^{3} d^{2} e^{3} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{2} b^{4} d^{3} e^{2} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {5 a b^{5} d^{4} e \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {b^{6} d^{5} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}}}{- 6 A b^{3} e^{2} + 4 B a b^{2} e^{2} + 2 B b^{3} d e} \right )}}{\left (a e - b d\right )^{4}} + \frac {b \left (- 3 A b e + 2 B a e + B b d\right ) \log {\left (x + \frac {- 3 A a b^{2} e^{2} - 3 A b^{3} d e + 2 B a^{2} b e^{2} + 3 B a b^{2} d e + B b^{3} d^{2} + \frac {a^{5} b e^{5} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {5 a^{4} b^{2} d e^{4} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {10 a^{3} b^{3} d^{2} e^{3} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {10 a^{2} b^{4} d^{3} e^{2} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} + \frac {5 a b^{5} d^{4} e \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}} - \frac {b^{6} d^{5} \left (- 3 A b e + 2 B a e + B b d\right )}{\left (a e - b d\right )^{4}}}{- 6 A b^{3} e^{2} + 4 B a b^{2} e^{2} + 2 B b^{3} d e} \right )}}{\left (a e - b d\right )^{4}} + \frac {- A a^{2} e^{2} + 5 A a b d e + 2 A b^{2} d^{2} - B a^{2} d e - 5 B a b d^{2} + x^{2} \cdot \left (6 A b^{2} e^{2} - 4 B a b e^{2} - 2 B b^{2} d e\right ) + x \left (3 A a b e^{2} + 9 A b^{2} d e - 2 B a^{2} e^{2} - 7 B a b d e - 3 B b^{2} d^{2}\right )}{2 a^{4} d^{2} e^{3} - 6 a^{3} b d^{3} e^{2} + 6 a^{2} b^{2} d^{4} e - 2 a b^{3} d^{5} + x^{3} \cdot \left (2 a^{3} b e^{5} - 6 a^{2} b^{2} d e^{4} + 6 a b^{3} d^{2} e^{3} - 2 b^{4} d^{3} e^{2}\right ) + x^{2} \cdot \left (2 a^{4} e^{5} - 2 a^{3} b d e^{4} - 6 a^{2} b^{2} d^{2} e^{3} + 10 a b^{3} d^{3} e^{2} - 4 b^{4} d^{4} e\right ) + x \left (4 a^{4} d e^{4} - 10 a^{3} b d^{2} e^{3} + 6 a^{2} b^{2} d^{3} e^{2} + 2 a b^{3} d^{4} e - 2 b^{4} d^{5}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (154) = 308\).
Time = 0.21 (sec) , antiderivative size = 479, normalized size of antiderivative = 3.05 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^3} \, dx=\frac {{\left (B b^{2} d + {\left (2 \, B a b - 3 \, A b^{2}\right )} e\right )} \log \left (b x + a\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} - \frac {{\left (B b^{2} d + {\left (2 \, B a b - 3 \, A b^{2}\right )} e\right )} \log \left (e x + d\right )}{b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}} + \frac {A a^{2} e^{2} + {\left (5 \, B a b - 2 \, A b^{2}\right )} d^{2} + {\left (B a^{2} - 5 \, A a b\right )} d e + 2 \, {\left (B b^{2} d e + {\left (2 \, B a b - 3 \, A b^{2}\right )} e^{2}\right )} x^{2} + {\left (3 \, B b^{2} d^{2} + {\left (7 \, B a b - 9 \, A b^{2}\right )} d e + {\left (2 \, B a^{2} - 3 \, A a b\right )} e^{2}\right )} x}{2 \, {\left (a b^{3} d^{5} - 3 \, a^{2} b^{2} d^{4} e + 3 \, a^{3} b d^{3} e^{2} - a^{4} d^{2} e^{3} + {\left (b^{4} d^{3} e^{2} - 3 \, a b^{3} d^{2} e^{3} + 3 \, a^{2} b^{2} d e^{4} - a^{3} b e^{5}\right )} x^{3} + {\left (2 \, b^{4} d^{4} e - 5 \, a b^{3} d^{3} e^{2} + 3 \, a^{2} b^{2} d^{2} e^{3} + a^{3} b d e^{4} - a^{4} e^{5}\right )} x^{2} + {\left (b^{4} d^{5} - a b^{3} d^{4} e - 3 \, a^{2} b^{2} d^{3} e^{2} + 5 \, a^{3} b d^{2} e^{3} - 2 \, a^{4} d e^{4}\right )} x\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.95 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^3} \, dx=-\frac {{\left (B b^{3} d + 2 \, B a b^{2} e - 3 \, A b^{3} e\right )} \log \left ({\left | \frac {b d}{b x + a} - \frac {a e}{b x + a} + e \right |}\right )}{b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}} + \frac {\frac {B a b^{4}}{b x + a} - \frac {A b^{5}}{b x + a}}{b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}} - \frac {3 \, B b^{2} d e^{2} + 2 \, B a b e^{3} - 5 \, A b^{2} e^{3} + \frac {2 \, {\left (2 \, B b^{4} d^{2} e - B a b^{3} d e^{2} - 3 \, A b^{4} d e^{2} - B a^{2} b^{2} e^{3} + 3 \, A a b^{3} e^{3}\right )}}{{\left (b x + a\right )} b}}{2 \, {\left (b d - a e\right )}^{4} {\left (\frac {b d}{b x + a} - \frac {a e}{b x + a} + e\right )}^{2}} \]
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Time = 1.55 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.89 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)^3} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {\left (b^2\,\left (3\,A\,e-B\,d\right )-2\,B\,a\,b\,e\right )\,\left (\frac {a^4\,e^4-2\,a^3\,b\,d\,e^3+2\,a\,b^3\,d^3\,e-b^4\,d^4}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}+2\,b\,e\,x\right )\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{{\left (a\,e-b\,d\right )}^4\,\left (B\,b^2\,d-3\,A\,b^2\,e+2\,B\,a\,b\,e\right )}\right )\,\left (b^2\,\left (3\,A\,e-B\,d\right )-2\,B\,a\,b\,e\right )}{{\left (a\,e-b\,d\right )}^4}-\frac {\frac {B\,a^2\,d\,e+A\,a^2\,e^2+5\,B\,a\,b\,d^2-5\,A\,a\,b\,d\,e-2\,A\,b^2\,d^2}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {x\,\left (a\,e+3\,b\,d\right )\,\left (2\,B\,a\,e-3\,A\,b\,e+B\,b\,d\right )}{2\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}+\frac {b\,e\,x^2\,\left (2\,B\,a\,e-3\,A\,b\,e+B\,b\,d\right )}{a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3}}{x\,\left (b\,d^2+2\,a\,e\,d\right )+a\,d^2+x^2\,\left (a\,e^2+2\,b\,d\,e\right )+b\,e^2\,x^3} \]
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