Optimal. Leaf size=107 \[ -\frac{e^2}{(a+b x) (b d-a e)^3}-\frac{e^3 \log (a+b x)}{(b d-a e)^4}+\frac{e^3 \log (d+e x)}{(b d-a e)^4}+\frac{e}{2 (a+b x)^2 (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (b d-a e)} \]
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Rubi [A] time = 0.0680284, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 44} \[ -\frac{e^2}{(a+b x) (b d-a e)^3}-\frac{e^3 \log (a+b x)}{(b d-a e)^4}+\frac{e^3 \log (d+e x)}{(b d-a e)^4}+\frac{e}{2 (a+b x)^2 (b d-a e)^2}-\frac{1}{3 (a+b x)^3 (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^4 (d+e x)} \, dx\\ &=\int \left (\frac{b}{(b d-a e) (a+b x)^4}-\frac{b e}{(b d-a e)^2 (a+b x)^3}+\frac{b e^2}{(b d-a e)^3 (a+b x)^2}-\frac{b e^3}{(b d-a e)^4 (a+b x)}+\frac{e^4}{(b d-a e)^4 (d+e x)}\right ) \, dx\\ &=-\frac{1}{3 (b d-a e) (a+b x)^3}+\frac{e}{2 (b d-a e)^2 (a+b x)^2}-\frac{e^2}{(b d-a e)^3 (a+b x)}-\frac{e^3 \log (a+b x)}{(b d-a e)^4}+\frac{e^3 \log (d+e x)}{(b d-a e)^4}\\ \end{align*}
Mathematica [A] time = 0.0448062, size = 107, normalized size = 1. \[ -\frac{e^2}{(a+b x) (b d-a e)^3}-\frac{e^3 \log (a+b x)}{(b d-a e)^4}+\frac{e^3 \log (d+e x)}{(b d-a e)^4}+\frac{e}{2 (a+b x)^2 (b d-a e)^2}+\frac{1}{3 (a+b x)^3 (a e-b d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 103, normalized size = 1. \begin{align*}{\frac{{e}^{3}\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{4}}}+{\frac{1}{ \left ( 3\,ae-3\,bd \right ) \left ( bx+a \right ) ^{3}}}+{\frac{e}{2\, \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) ^{2}}}+{\frac{{e}^{2}}{ \left ( ae-bd \right ) ^{3} \left ( bx+a \right ) }}-{\frac{{e}^{3}\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.21259, size = 487, normalized size = 4.55 \begin{align*} -\frac{e^{3} \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{e^{3} \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{6 \, b^{2} e^{2} x^{2} + 2 \, b^{2} d^{2} - 7 \, a b d e + 11 \, a^{2} e^{2} - 3 \,{\left (b^{2} d e - 5 \, a b e^{2}\right )} x}{6 \,{\left (a^{3} b^{3} d^{3} - 3 \, a^{4} b^{2} d^{2} e + 3 \, a^{5} b d e^{2} - a^{6} e^{3} +{\left (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}\right )} x^{3} + 3 \,{\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76155, size = 856, normalized size = 8. \begin{align*} -\frac{2 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 5 \, a^{2} b e^{3}\right )} x + 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \log \left (e x + d\right )}{6 \,{\left (a^{3} b^{4} d^{4} - 4 \, a^{4} b^{3} d^{3} e + 6 \, a^{5} b^{2} d^{2} e^{2} - 4 \, a^{6} b d e^{3} + a^{7} e^{4} +{\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} x^{3} + 3 \,{\left (a b^{6} d^{4} - 4 \, a^{2} b^{5} d^{3} e + 6 \, a^{3} b^{4} d^{2} e^{2} - 4 \, a^{4} b^{3} d e^{3} + a^{5} b^{2} e^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{5} d^{4} - 4 \, a^{3} b^{4} d^{3} e + 6 \, a^{4} b^{3} d^{2} e^{2} - 4 \, a^{5} b^{2} d e^{3} + a^{6} b e^{4}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.20426, size = 570, normalized size = 5.33 \begin{align*} \frac{e^{3} \log{\left (x + \frac{- \frac{a^{5} e^{8}}{\left (a e - b d\right )^{4}} + \frac{5 a^{4} b d e^{7}}{\left (a e - b d\right )^{4}} - \frac{10 a^{3} b^{2} d^{2} e^{6}}{\left (a e - b d\right )^{4}} + \frac{10 a^{2} b^{3} d^{3} e^{5}}{\left (a e - b d\right )^{4}} - \frac{5 a b^{4} d^{4} e^{4}}{\left (a e - b d\right )^{4}} + a e^{4} + \frac{b^{5} d^{5} e^{3}}{\left (a e - b d\right )^{4}} + b d e^{3}}{2 b e^{4}} \right )}}{\left (a e - b d\right )^{4}} - \frac{e^{3} \log{\left (x + \frac{\frac{a^{5} e^{8}}{\left (a e - b d\right )^{4}} - \frac{5 a^{4} b d e^{7}}{\left (a e - b d\right )^{4}} + \frac{10 a^{3} b^{2} d^{2} e^{6}}{\left (a e - b d\right )^{4}} - \frac{10 a^{2} b^{3} d^{3} e^{5}}{\left (a e - b d\right )^{4}} + \frac{5 a b^{4} d^{4} e^{4}}{\left (a e - b d\right )^{4}} + a e^{4} - \frac{b^{5} d^{5} e^{3}}{\left (a e - b d\right )^{4}} + b d e^{3}}{2 b e^{4}} \right )}}{\left (a e - b d\right )^{4}} + \frac{11 a^{2} e^{2} - 7 a b d e + 2 b^{2} d^{2} + 6 b^{2} e^{2} x^{2} + x \left (15 a b e^{2} - 3 b^{2} d e\right )}{6 a^{6} e^{3} - 18 a^{5} b d e^{2} + 18 a^{4} b^{2} d^{2} e - 6 a^{3} b^{3} d^{3} + x^{3} \left (6 a^{3} b^{3} e^{3} - 18 a^{2} b^{4} d e^{2} + 18 a b^{5} d^{2} e - 6 b^{6} d^{3}\right ) + x^{2} \left (18 a^{4} b^{2} e^{3} - 54 a^{3} b^{3} d e^{2} + 54 a^{2} b^{4} d^{2} e - 18 a b^{5} d^{3}\right ) + x \left (18 a^{5} b e^{3} - 54 a^{4} b^{2} d e^{2} + 54 a^{3} b^{3} d^{2} e - 18 a^{2} b^{4} d^{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17333, size = 316, normalized size = 2.95 \begin{align*} -\frac{b e^{3} \log \left ({\left | b x + a \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{e^{4} \log \left ({\left | x e + 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{2 \, b^{3} d^{3} - 9 \, a b^{2} d^{2} e + 18 \, a^{2} b d e^{2} - 11 \, a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} x^{2} - 3 \,{\left (b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + 5 \, a^{2} b e^{3}\right )} x}{6 \,{\left (b d - a e\right )}^{4}{\left (b x + a\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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