Optimal. Leaf size=68 \[ \frac{(b B-A c) \log (b+c x)}{b (c d-b e)}-\frac{(B d-A e) \log (d+e x)}{d (c d-b e)}+\frac{A \log (x)}{b d} \]
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Rubi [A] time = 0.0727457, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{(b B-A c) \log (b+c x)}{b (c d-b e)}-\frac{(B d-A e) \log (d+e x)}{d (c d-b e)}+\frac{A \log (x)}{b d} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{A+B x}{(d+e x) \left (b x+c x^2\right )} \, dx &=\int \left (\frac{A}{b d x}-\frac{c (b B-A c)}{b (-c d+b e) (b+c x)}-\frac{e (B d-A e)}{d (c d-b e) (d+e x)}\right ) \, dx\\ &=\frac{A \log (x)}{b d}+\frac{(b B-A c) \log (b+c x)}{b (c d-b e)}-\frac{(B d-A e) \log (d+e x)}{d (c d-b e)}\\ \end{align*}
Mathematica [A] time = 0.0445822, size = 63, normalized size = 0.93 \[ \frac{\log (b+c x) (A c d-b B d)+b (B d-A e) \log (d+e x)+A \log (x) (b e-c d)}{b d (b e-c d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 94, normalized size = 1.4 \begin{align*}{\frac{A\ln \left ( x \right ) }{bd}}-{\frac{\ln \left ( ex+d \right ) Ae}{d \left ( be-cd \right ) }}+{\frac{\ln \left ( ex+d \right ) B}{be-cd}}+{\frac{\ln \left ( cx+b \right ) Ac}{b \left ( be-cd \right ) }}-{\frac{\ln \left ( cx+b \right ) B}{be-cd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14539, size = 92, normalized size = 1.35 \begin{align*} \frac{{\left (B b - A c\right )} \log \left (c x + b\right )}{b c d - b^{2} e} - \frac{{\left (B d - A e\right )} \log \left (e x + d\right )}{c d^{2} - b d e} + \frac{A \log \left (x\right )}{b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 8.18057, size = 143, normalized size = 2.1 \begin{align*} \frac{{\left (B b - A c\right )} d \log \left (c x + b\right ) -{\left (B b d - A b e\right )} \log \left (e x + d\right ) +{\left (A c d - A b e\right )} \log \left (x\right )}{b c d^{2} - b^{2} d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33476, size = 184, normalized size = 2.71 \begin{align*} -\frac{A \log \left ({\left | c x^{2} e + c d x + b x e + b d \right |}\right )}{2 \, b d} + \frac{A \log \left ({\left | x \right |}\right )}{b d} + \frac{{\left (2 \, B b d - A c d - A b e\right )} \log \left (\frac{{\left | 2 \, c x e + c d + b e -{\left | c d - b e \right |} \right |}}{{\left | 2 \, c x e + c d + b e +{\left | c d - b e \right |} \right |}}\right )}{2 \, b d{\left | c d - b e \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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