Optimal. Leaf size=81 \[ -\frac{b}{(a+b x) (b d-a e)^2}-\frac{e}{(d+e x) (b d-a e)^2}-\frac{2 b e \log (a+b x)}{(b d-a e)^3}+\frac{2 b e \log (d+e x)}{(b d-a e)^3} \]
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Rubi [A] time = 0.0533884, antiderivative size = 81, 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} \[ -\frac{b}{(a+b x) (b d-a e)^2}-\frac{e}{(d+e x) (b d-a e)^2}-\frac{2 b e \log (a+b x)}{(b d-a e)^3}+\frac{2 b e \log (d+e x)}{(b d-a e)^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )} \, dx &=\int \frac{1}{(a+b x)^2 (d+e x)^2} \, dx\\ &=\int \left (\frac{b^2}{(b d-a e)^2 (a+b x)^2}-\frac{2 b^2 e}{(b d-a e)^3 (a+b x)}+\frac{e^2}{(b d-a e)^2 (d+e x)^2}+\frac{2 b e^2}{(b d-a e)^3 (d+e x)}\right ) \, dx\\ &=-\frac{b}{(b d-a e)^2 (a+b x)}-\frac{e}{(b d-a e)^2 (d+e x)}-\frac{2 b e \log (a+b x)}{(b d-a e)^3}+\frac{2 b e \log (d+e x)}{(b d-a e)^3}\\ \end{align*}
Mathematica [A] time = 0.0681853, size = 66, normalized size = 0.81 \[ \frac{\frac{b (a e-b d)}{a+b x}+\frac{e (a e-b d)}{d+e x}-2 b e \log (a+b x)+2 b e \log (d+e x)}{(b d-a e)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 82, normalized size = 1. \begin{align*} -{\frac{e}{ \left ( ae-bd \right ) ^{2} \left ( ex+d \right ) }}-2\,{\frac{be\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{3}}}-{\frac{b}{ \left ( ae-bd \right ) ^{2} \left ( bx+a \right ) }}+2\,{\frac{be\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 = 1.02128, size = 281, normalized size = 3.47 \begin{align*} -\frac{2 \, b e \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{2 \, b e \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 + b d + a e}{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 = 1.8012, size = 486, normalized size = 6. \begin{align*} -\frac{b^{2} d^{2} - a^{2} e^{2} + 2 \,{\left (b^{2} d e - a b e^{2}\right )} x + 2 \,{\left (b^{2} e^{2} x^{2} + a b d e +{\left (b^{2} d e + a b e^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \,{\left (b^{2} e^{2} x^{2} + a b d e +{\left (b^{2} d e + a b 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 = 1.32834, size = 405, normalized size = 5. \begin{align*} - \frac{2 b e \log{\left (x + \frac{- \frac{2 a^{4} b e^{5}}{\left (a e - b d\right )^{3}} + \frac{8 a^{3} b^{2} d e^{4}}{\left (a e - b d\right )^{3}} - \frac{12 a^{2} b^{3} d^{2} e^{3}}{\left (a e - b d\right )^{3}} + \frac{8 a b^{4} d^{3} e^{2}}{\left (a e - b d\right )^{3}} + 2 a b e^{2} - \frac{2 b^{5} d^{4} e}{\left (a e - b d\right )^{3}} + 2 b^{2} d e}{4 b^{2} e^{2}} \right )}}{\left (a e - b d\right )^{3}} + \frac{2 b e \log{\left (x + \frac{\frac{2 a^{4} b e^{5}}{\left (a e - b d\right )^{3}} - \frac{8 a^{3} b^{2} d e^{4}}{\left (a e - b d\right )^{3}} + \frac{12 a^{2} b^{3} d^{2} e^{3}}{\left (a e - b d\right )^{3}} - \frac{8 a b^{4} d^{3} e^{2}}{\left (a e - b d\right )^{3}} + 2 a b e^{2} + \frac{2 b^{5} d^{4} e}{\left (a e - b d\right )^{3}} + 2 b^{2} d e}{4 b^{2} e^{2}} \right )}}{\left (a e - b d\right )^{3}} - \frac{a e + b d + 2 b e x}{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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21836, size = 211, normalized size = 2.6 \begin{align*} -\frac{2 \, b e^{2} \log \left ({\left | b - \frac{b d}{x e + d} + \frac{a e}{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{e^{3}}{{\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}{\left (x e + d\right )}} - \frac{b^{2} e}{{\left (b d - a e\right )}^{3}{\left (b - \frac{b d}{x e + d} + \frac{a e}{x e + d}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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