Optimal. Leaf size=117 \[ -\frac{A b-a B}{(a+b x) (b d-a e)^2}+\frac{B d-A e}{(d+e x) (b d-a e)^2}+\frac{\log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^3}-\frac{\log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^3} \]
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Rubi [A] time = 0.102761, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{A b-a B}{(a+b x) (b d-a e)^2}+\frac{B d-A e}{(d+e x) (b d-a e)^2}+\frac{\log (a+b x) (a B e-2 A b e+b B d)}{(b d-a e)^3}-\frac{\log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x}{(a+b x)^2 (d+e x)^2} \, dx &=\int \left (\frac{b (A b-a B)}{(b d-a e)^2 (a+b x)^2}+\frac{b (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)^2 (d+e x)^2}+\frac{e (-b B d+2 A b e-a B e)}{(b d-a e)^3 (d+e x)}\right ) \, dx\\ &=-\frac{A b-a B}{(b d-a e)^2 (a+b x)}+\frac{B d-A e}{(b d-a e)^2 (d+e x)}+\frac{(b B d-2 A b e+a B e) \log (a+b x)}{(b d-a e)^3}-\frac{(b B d-2 A b e+a B e) \log (d+e x)}{(b d-a e)^3}\\ \end{align*}
Mathematica [A] time = 0.108737, size = 103, normalized size = 0.88 \[ \frac{\frac{(a B-A b) (b d-a e)}{a+b x}+\frac{(b d-a e) (B d-A e)}{d+e x}+\log (a+b x) (a B e-2 A b e+b B d)-\log (d+e x) (a B e-2 A b e+b B d)}{(b d-a e)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 208, normalized size = 1.8 \begin{align*} -{\frac{Ae}{ \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) }}+{\frac{Bd}{ \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) }}-2\,{\frac{\ln \left ( ex+d \right ) Abe}{ \left ( ae-bd \right ) ^{3}}}+{\frac{\ln \left ( ex+d \right ) Bae}{ \left ( ae-bd \right ) ^{3}}}+{\frac{\ln \left ( ex+d \right ) Bbd}{ \left ( ae-bd \right ) ^{3}}}+2\,{\frac{\ln \left ( bx+a \right ) Abe}{ \left ( ae-bd \right ) ^{3}}}-{\frac{\ln \left ( bx+a \right ) Bae}{ \left ( ae-bd \right ) ^{3}}}-{\frac{\ln \left ( bx+a \right ) Bbd}{ \left ( ae-bd \right ) ^{3}}}-{\frac{Ab}{ \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) }}+{\frac{Ba}{ \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.31904, size = 346, normalized size = 2.96 \begin{align*} \frac{{\left (B b d +{\left (B a - 2 \, A b\right )} e\right )} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac{{\left (B b d +{\left (B a - 2 \, A b\right )} e\right )} \log \left (e x + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac{A a e -{\left (2 \, B a - A b\right )} d -{\left (B b d +{\left (B a - 2 \, A b\right )} e\right )} x}{a b^{2} d^{3} - 2 \, a^{2} b d^{2} e + a^{3} d e^{2} +{\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} +{\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.63504, size = 813, normalized size = 6.95 \begin{align*} -\frac{2 \, B a^{2} d e - A a^{2} e^{2} -{\left (2 \, B a b - A b^{2}\right )} d^{2} -{\left (B b^{2} d^{2} - 2 \, A b^{2} d e -{\left (B a^{2} - 2 \, A a b\right )} e^{2}\right )} x -{\left (B a b d^{2} +{\left (B a^{2} - 2 \, A a b\right )} d e +{\left (B b^{2} d e +{\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} x^{2} +{\left (B b^{2} d^{2} + 2 \,{\left (B a b - A b^{2}\right )} d e +{\left (B a^{2} - 2 \, A a b\right )} e^{2}\right )} x\right )} \log \left (b x + a\right ) +{\left (B a b d^{2} +{\left (B a^{2} - 2 \, A a b\right )} d e +{\left (B b^{2} d e +{\left (B a b - 2 \, A b^{2}\right )} e^{2}\right )} x^{2} +{\left (B b^{2} d^{2} + 2 \,{\left (B a b - A b^{2}\right )} d e +{\left (B a^{2} - 2 \, A a b\right )} e^{2}\right )} x\right )} \log \left (e x + d\right )}{a b^{3} d^{4} - 3 \, a^{2} b^{2} d^{3} e + 3 \, a^{3} b d^{2} e^{2} - a^{4} d e^{3} +{\left (b^{4} d^{3} e - 3 \, a b^{3} d^{2} e^{2} + 3 \, a^{2} b^{2} d e^{3} - a^{3} b e^{4}\right )} x^{2} +{\left (b^{4} d^{4} - 2 \, a b^{3} d^{3} e + 2 \, a^{3} b d e^{3} - a^{4} e^{4}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.38195, size = 706, normalized size = 6.03 \begin{align*} \frac{- A a e - A b d + 2 B a d + x \left (- 2 A b e + B a e + B b d\right )}{a^{3} d e^{2} - 2 a^{2} b d^{2} e + a b^{2} d^{3} + x^{2} \left (a^{2} b e^{3} - 2 a b^{2} d e^{2} + b^{3} d^{2} e\right ) + x \left (a^{3} e^{3} - a^{2} b d e^{2} - a b^{2} d^{2} e + b^{3} d^{3}\right )} + \frac{\left (- 2 A b e + B a e + B b d\right ) \log{\left (x + \frac{- 2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} + 2 B a b d e + B b^{2} d^{2} - \frac{a^{4} e^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac{4 a^{3} b d e^{3} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac{6 a^{2} b^{2} d^{2} e^{2} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac{4 a b^{3} d^{3} e \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac{b^{4} d^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}}}{- 4 A b^{2} e^{2} + 2 B a b e^{2} + 2 B b^{2} d e} \right )}}{\left (a e - b d\right )^{3}} - \frac{\left (- 2 A b e + B a e + B b d\right ) \log{\left (x + \frac{- 2 A a b e^{2} - 2 A b^{2} d e + B a^{2} e^{2} + 2 B a b d e + B b^{2} d^{2} + \frac{a^{4} e^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac{4 a^{3} b d e^{3} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac{6 a^{2} b^{2} d^{2} e^{2} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} - \frac{4 a b^{3} d^{3} e \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}} + \frac{b^{4} d^{4} \left (- 2 A b e + B a e + B b d\right )}{\left (a e - b d\right )^{3}}}{- 4 A b^{2} e^{2} + 2 B a b e^{2} + 2 B b^{2} d e} \right )}}{\left (a e - b d\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.50503, size = 273, normalized size = 2.33 \begin{align*} -\frac{{\left (B b^{2} d + B a b e - 2 \, A b^{2} e\right )} \log \left ({\left | -\frac{b d}{b x + a} + \frac{a e}{b x + a} - e \right |}\right )}{b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}} + \frac{\frac{B a b^{2}}{b x + a} - \frac{A b^{3}}{b x + a}}{b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}} - \frac{B b d e - A b e^{2}}{{\left (b d - a e\right )}^{3}{\left (\frac{b d}{b x + a} - \frac{a e}{b x + a} + e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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