Optimal. Leaf size=82 \[ \frac{b^2 \log (a+b x)}{(b d-a e)^3}-\frac{b^2 \log (d+e x)}{(b d-a e)^3}+\frac{b}{(d+e x) (b d-a e)^2}+\frac{1}{2 (d+e x)^2 (b d-a e)} \]
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Rubi [A] time = 0.0481985, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 44} \[ \frac{b^2 \log (a+b x)}{(b d-a e)^3}-\frac{b^2 \log (d+e x)}{(b d-a e)^3}+\frac{b}{(d+e x) (b d-a e)^2}+\frac{1}{2 (d+e x)^2 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b x}{(d+e x)^3 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{1}{(a+b x) (d+e x)^3} \, dx\\ &=\int \left (\frac{b^3}{(b d-a e)^3 (a+b x)}-\frac{e}{(b d-a e) (d+e x)^3}-\frac{b e}{(b d-a e)^2 (d+e x)^2}-\frac{b^2 e}{(b d-a e)^3 (d+e x)}\right ) \, dx\\ &=\frac{1}{2 (b d-a e) (d+e x)^2}+\frac{b}{(b d-a e)^2 (d+e x)}+\frac{b^2 \log (a+b x)}{(b d-a e)^3}-\frac{b^2 \log (d+e x)}{(b d-a e)^3}\\ \end{align*}
Mathematica [A] time = 0.0486683, size = 67, normalized size = 0.82 \[ \frac{2 b^2 \log (a+b x)+\frac{(b d-a e) (-a e+3 b d+2 b e x)}{(d+e x)^2}-2 b^2 \log (d+e x)}{2 (b d-a e)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 81, normalized size = 1. \begin{align*} -{\frac{1}{ \left ( 2\,ae-2\,bd \right ) \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{2}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{3}}}+{\frac{b}{ \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) }}-{\frac{{b}^{2}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.986212, size = 273, normalized size = 3.33 \begin{align*} \frac{b^{2} \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{b^{2} \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{2 \, b e x + 3 \, b d - a e}{2 \,{\left (b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} +{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \,{\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.62136, size = 490, normalized size = 5.98 \begin{align*} \frac{3 \, b^{2} d^{2} - 4 \, a b d e + a^{2} e^{2} + 2 \,{\left (b^{2} d e - a b e^{2}\right )} x + 2 \,{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \log \left (b x + a\right ) - 2 \,{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \log \left (e x + d\right )}{2 \,{\left (b^{3} d^{5} - 3 \, a b^{2} d^{4} e + 3 \, a^{2} b d^{3} e^{2} - a^{3} d^{2} e^{3} +{\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} x^{2} + 2 \,{\left (b^{3} d^{4} e - 3 \, a b^{2} d^{3} e^{2} + 3 \, a^{2} b d^{2} e^{3} - a^{3} d e^{4}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.27972, size = 381, normalized size = 4.65 \begin{align*} \frac{b^{2} \log{\left (x + \frac{- \frac{a^{4} b^{2} e^{4}}{\left (a e - b d\right )^{3}} + \frac{4 a^{3} b^{3} d e^{3}}{\left (a e - b d\right )^{3}} - \frac{6 a^{2} b^{4} d^{2} e^{2}}{\left (a e - b d\right )^{3}} + \frac{4 a b^{5} d^{3} e}{\left (a e - b d\right )^{3}} + a b^{2} e - \frac{b^{6} d^{4}}{\left (a e - b d\right )^{3}} + b^{3} d}{2 b^{3} e} \right )}}{\left (a e - b d\right )^{3}} - \frac{b^{2} \log{\left (x + \frac{\frac{a^{4} b^{2} e^{4}}{\left (a e - b d\right )^{3}} - \frac{4 a^{3} b^{3} d e^{3}}{\left (a e - b d\right )^{3}} + \frac{6 a^{2} b^{4} d^{2} e^{2}}{\left (a e - b d\right )^{3}} - \frac{4 a b^{5} d^{3} e}{\left (a e - b d\right )^{3}} + a b^{2} e + \frac{b^{6} d^{4}}{\left (a e - b d\right )^{3}} + b^{3} d}{2 b^{3} e} \right )}}{\left (a e - b d\right )^{3}} + \frac{- a e + 3 b d + 2 b e x}{2 a^{2} d^{2} e^{2} - 4 a b d^{3} e + 2 b^{2} d^{4} + x^{2} \left (2 a^{2} e^{4} - 4 a b d e^{3} + 2 b^{2} d^{2} e^{2}\right ) + x \left (4 a^{2} d e^{3} - 8 a b d^{2} e^{2} + 4 b^{2} d^{3} e\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.08115, size = 224, normalized size = 2.73 \begin{align*} \frac{b^{3} \log \left ({\left | b x + a \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{b^{2} e \log \left ({\left | x e + d \right |}\right )}{b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}} + \frac{3 \, b^{2} d^{2} - 4 \, a b d e + a^{2} e^{2} + 2 \,{\left (b^{2} d e - a b e^{2}\right )} x}{2 \,{\left (b d - a e\right )}^{3}{\left (x e + d\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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