Optimal. Leaf size=57 \[ -\frac{1}{(a+b x) (b d-a e)}-\frac{e \log (a+b x)}{(b d-a e)^2}+\frac{e \log (d+e x)}{(b d-a e)^2} \]
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Rubi [A] time = 0.0351234, antiderivative size = 57, 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} \[ -\frac{1}{(a+b x) (b d-a e)}-\frac{e \log (a+b x)}{(b d-a e)^2}+\frac{e \log (d+e x)}{(b d-a e)^2} \]
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 )} \, dx &=\int \frac{1}{(a+b x)^2 (d+e x)} \, dx\\ &=\int \left (\frac{b}{(b d-a e) (a+b x)^2}-\frac{b e}{(b d-a e)^2 (a+b x)}+\frac{e^2}{(b d-a e)^2 (d+e x)}\right ) \, dx\\ &=-\frac{1}{(b d-a e) (a+b x)}-\frac{e \log (a+b x)}{(b d-a e)^2}+\frac{e \log (d+e x)}{(b d-a e)^2}\\ \end{align*}
Mathematica [A] time = 0.0249867, size = 53, normalized size = 0.93 \[ \frac{e (a+b x) \log (d+e x)-e (a+b x) \log (a+b x)+a e-b d}{(a+b x) (b d-a e)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 57, normalized size = 1. \begin{align*}{\frac{e\ln \left ( ex+d \right ) }{ \left ( ae-bd \right ) ^{2}}}+{\frac{1}{ \left ( ae-bd \right ) \left ( bx+a \right ) }}-{\frac{e\ln \left ( bx+a \right ) }{ \left ( ae-bd \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11861, size = 124, normalized size = 2.18 \begin{align*} -\frac{e \log \left (b x + a\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} + \frac{e \log \left (e x + d\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} - \frac{1}{a b d - a^{2} e +{\left (b^{2} d - a b e\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76115, size = 200, normalized size = 3.51 \begin{align*} -\frac{b d - a e +{\left (b e x + a e\right )} \log \left (b x + a\right ) -{\left (b e x + a e\right )} \log \left (e x + d\right )}{a b^{2} d^{2} - 2 \, a^{2} b d e + a^{3} e^{2} +{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.829572, size = 233, normalized size = 4.09 \begin{align*} \frac{e \log{\left (x + \frac{- \frac{a^{3} e^{4}}{\left (a e - b d\right )^{2}} + \frac{3 a^{2} b d e^{3}}{\left (a e - b d\right )^{2}} - \frac{3 a b^{2} d^{2} e^{2}}{\left (a e - b d\right )^{2}} + a e^{2} + \frac{b^{3} d^{3} e}{\left (a e - b d\right )^{2}} + b d e}{2 b e^{2}} \right )}}{\left (a e - b d\right )^{2}} - \frac{e \log{\left (x + \frac{\frac{a^{3} e^{4}}{\left (a e - b d\right )^{2}} - \frac{3 a^{2} b d e^{3}}{\left (a e - b d\right )^{2}} + \frac{3 a b^{2} d^{2} e^{2}}{\left (a e - b d\right )^{2}} + a e^{2} - \frac{b^{3} d^{3} e}{\left (a e - b d\right )^{2}} + b d e}{2 b e^{2}} \right )}}{\left (a e - b d\right )^{2}} + \frac{1}{a^{2} e - a b d + x \left (a b e - b^{2} d\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14648, size = 128, normalized size = 2.25 \begin{align*} -\frac{b e \log \left ({\left | b x + a \right |}\right )}{b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}} + \frac{e^{2} \log \left ({\left | x e + d \right |}\right )}{b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}} - \frac{1}{{\left (b d - a e\right )}{\left (b x + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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