Integrand size = 20, antiderivative size = 82 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)} \, dx=-\frac {A b-a B}{b (b d-a e) (a+b x)}+\frac {(B d-A e) \log (a+b x)}{(b d-a e)^2}-\frac {(B d-A e) \log (d+e x)}{(b d-a e)^2} \]
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Time = 0.04 (sec) , antiderivative size = 82, 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} \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)} \, dx=-\frac {A b-a B}{b (a+b x) (b d-a e)}+\frac {\log (a+b x) (B d-A e)}{(b d-a e)^2}-\frac {(B d-A e) \log (d+e x)}{(b d-a e)^2} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {A b-a B}{(b d-a e) (a+b x)^2}+\frac {b (B d-A e)}{(b d-a e)^2 (a+b x)}+\frac {e (-B d+A e)}{(b d-a e)^2 (d+e x)}\right ) \, dx \\ & = -\frac {A b-a B}{b (b d-a e) (a+b x)}+\frac {(B d-A e) \log (a+b x)}{(b d-a e)^2}-\frac {(B d-A e) \log (d+e x)}{(b d-a e)^2} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)} \, dx=\frac {\frac {(-A b+a B) (b d-a e)}{b (a+b x)}+(B d-A e) \log (a+b x)+(-B d+A e) \log (d+e x)}{(b d-a e)^2} \]
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Time = 2.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.01
method | result | size |
default | \(-\frac {-A b +B a}{\left (a e -b d \right ) b \left (b x +a \right )}-\frac {\left (A e -B d \right ) \ln \left (b x +a \right )}{\left (a e -b d \right )^{2}}+\frac {\left (A e -B d \right ) \ln \left (e x +d \right )}{\left (a e -b d \right )^{2}}\) | \(83\) |
norman | \(-\frac {\left (A b -B a \right ) x}{a \left (a e -b d \right ) \left (b x +a \right )}+\frac {\left (A e -B d \right ) \ln \left (e x +d \right )}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}-\frac {\left (A e -B d \right ) \ln \left (b x +a \right )}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}\) | \(110\) |
parallelrisch | \(-\frac {A \ln \left (b x +a \right ) x \,b^{2} e -A \ln \left (e x +d \right ) x \,b^{2} e -B \ln \left (b x +a \right ) x \,b^{2} d +B \ln \left (e x +d \right ) x \,b^{2} d +A \ln \left (b x +a \right ) a b e -A \ln \left (e x +d \right ) a b e -B \ln \left (b x +a \right ) a b d +B \ln \left (e x +d \right ) a b d -A a b e +A \,b^{2} d +B \,a^{2} e -B a b d}{\left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (b x +a \right ) b}\) | \(161\) |
risch | \(\frac {A}{\left (a e -b d \right ) \left (b x +a \right )}-\frac {B a}{\left (a e -b d \right ) b \left (b x +a \right )}+\frac {\ln \left (-e x -d \right ) A e}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}-\frac {\ln \left (-e x -d \right ) B d}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}-\frac {\ln \left (b x +a \right ) A e}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}+\frac {\ln \left (b x +a \right ) B d}{a^{2} e^{2}-2 a b d e +b^{2} d^{2}}\) | \(181\) |
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Time = 0.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.91 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)} \, dx=\frac {{\left (B a b - A b^{2}\right )} d - {\left (B a^{2} - A a b\right )} e + {\left (B a b d - A a b e + {\left (B b^{2} d - A b^{2} e\right )} x\right )} \log \left (b x + a\right ) - {\left (B a b d - A a b e + {\left (B b^{2} d - A b^{2} e\right )} x\right )} \log \left (e x + d\right )}{a b^{3} d^{2} - 2 \, a^{2} b^{2} d e + a^{3} b e^{2} + {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 355 vs. \(2 (63) = 126\).
Time = 0.70 (sec) , antiderivative size = 355, normalized size of antiderivative = 4.33 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)} \, dx=\frac {A b - B a}{a^{2} b e - a b^{2} d + x \left (a b^{2} e - b^{3} d\right )} - \frac {\left (- A e + B d\right ) \log {\left (x + \frac {- A a e^{2} - A b d e + B a d e + B b d^{2} - \frac {a^{3} e^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{2}} + \frac {3 a^{2} b d e^{2} \left (- A e + B d\right )}{\left (a e - b d\right )^{2}} - \frac {3 a b^{2} d^{2} e \left (- A e + B d\right )}{\left (a e - b d\right )^{2}} + \frac {b^{3} d^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{2}}}{- 2 A b e^{2} + 2 B b d e} \right )}}{\left (a e - b d\right )^{2}} + \frac {\left (- A e + B d\right ) \log {\left (x + \frac {- A a e^{2} - A b d e + B a d e + B b d^{2} + \frac {a^{3} e^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{2}} - \frac {3 a^{2} b d e^{2} \left (- A e + B d\right )}{\left (a e - b d\right )^{2}} + \frac {3 a b^{2} d^{2} e \left (- A e + B d\right )}{\left (a e - b d\right )^{2}} - \frac {b^{3} d^{3} \left (- A e + B d\right )}{\left (a e - b d\right )^{2}}}{- 2 A b e^{2} + 2 B b d e} \right )}}{\left (a e - b d\right )^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.44 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)} \, dx=\frac {{\left (B d - A e\right )} \log \left (b x + a\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} - \frac {{\left (B d - A e\right )} \log \left (e x + d\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} + \frac {B a - A b}{a b^{2} d - a^{2} b e + {\left (b^{3} d - a b^{2} e\right )} x} \]
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Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.22 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)} \, dx=-\frac {{\left (B b d - A b e\right )} \log \left ({\left | \frac {b d}{b x + a} - \frac {a e}{b x + a} + e \right |}\right )}{b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}} + \frac {\frac {B a}{b x + a} - \frac {A b}{b x + a}}{b^{2} d - a b e} \]
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Time = 1.32 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x}{(a+b x)^2 (d+e x)} \, dx=\frac {A\,b-B\,a}{b\,\left (a\,e-b\,d\right )\,\left (a+b\,x\right )}-\frac {2\,\mathrm {atanh}\left (\frac {a^2\,e^2-b^2\,d^2}{{\left (a\,e-b\,d\right )}^2}+\frac {2\,b\,e\,x}{a\,e-b\,d}\right )\,\left (A\,e-B\,d\right )}{{\left (a\,e-b\,d\right )}^2} \]
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