Optimal. Leaf size=63 \[ -\frac{b e \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d^2}-\frac{e \log \left (\frac{d x}{e}+1\right ) (a+b \log (c x))}{d^2}+\frac{a x}{d}+\frac{b x \log (c x)}{d}-\frac{b x}{d} \]
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Rubi [A] time = 0.0700568, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {193, 43, 2330, 2295, 2317, 2391} \[ -\frac{b e \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d^2}-\frac{e \log \left (\frac{d x}{e}+1\right ) (a+b \log (c x))}{d^2}+\frac{a x}{d}+\frac{b x \log (c x)}{d}-\frac{b x}{d} \]
Antiderivative was successfully verified.
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Rule 193
Rule 43
Rule 2330
Rule 2295
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log (c x)}{d+\frac{e}{x}} \, dx &=\int \left (\frac{a+b \log (c x)}{d}-\frac{e (a+b \log (c x))}{d (e+d x)}\right ) \, dx\\ &=\frac{\int (a+b \log (c x)) \, dx}{d}-\frac{e \int \frac{a+b \log (c x)}{e+d x} \, dx}{d}\\ &=\frac{a x}{d}-\frac{e (a+b \log (c x)) \log \left (1+\frac{d x}{e}\right )}{d^2}+\frac{b \int \log (c x) \, dx}{d}+\frac{(b e) \int \frac{\log \left (1+\frac{d x}{e}\right )}{x} \, dx}{d^2}\\ &=\frac{a x}{d}-\frac{b x}{d}+\frac{b x \log (c x)}{d}-\frac{e (a+b \log (c x)) \log \left (1+\frac{d x}{e}\right )}{d^2}-\frac{b e \text{Li}_2\left (-\frac{d x}{e}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.0283437, size = 64, normalized size = 1.02 \[ -\frac{b e \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d^2}-\frac{e \log \left (\frac{d x+e}{e}\right ) (a+b \log (c x))}{d^2}+\frac{a x}{d}+\frac{b x \log (c x)}{d}-\frac{b x}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 91, normalized size = 1.4 \begin{align*}{\frac{ax}{d}}-{\frac{ae\ln \left ( cdx+ce \right ) }{{d}^{2}}}+{\frac{bx\ln \left ( cx \right ) }{d}}-{\frac{bx}{d}}-{\frac{be}{{d}^{2}}{\it dilog} \left ({\frac{cdx+ce}{ce}} \right ) }-{\frac{\ln \left ( cx \right ) be}{{d}^{2}}\ln \left ({\frac{cdx+ce}{ce}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.35473, size = 93, normalized size = 1.48 \begin{align*} -\frac{{\left (\log \left (\frac{d x}{e} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{d x}{e}\right )\right )} b e}{d^{2}} + \frac{b x \log \left (x\right ) +{\left (b{\left (\log \left (c\right ) - 1\right )} + a\right )} x}{d} - \frac{{\left (b e \log \left (c\right ) + a e\right )} \log \left (d x + e\right )}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x \log \left (c x\right ) + a x}{d x + e}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 57.9305, size = 138, normalized size = 2.19 \begin{align*} - \frac{a e \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\log{\left (d x + e \right )}}{d} & \text{otherwise} \end{cases}\right )}{d} + \frac{a x}{d} + \frac{b e \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\begin{cases} \log{\left (e \right )} \log{\left (x \right )} - \operatorname{Li}_{2}\left (\frac{d x e^{i \pi }}{e}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (e \right )} \log{\left (\frac{1}{x} \right )} - \operatorname{Li}_{2}\left (\frac{d x e^{i \pi }}{e}\right ) & \text{for}\: \frac{1}{\left |{x}\right |} < 1 \\-{G_{2, 2}^{2, 0}\left (\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle |{x} \right )} \log{\left (e \right )} +{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x} \right )} \log{\left (e \right )} - \operatorname{Li}_{2}\left (\frac{d x e^{i \pi }}{e}\right ) & \text{otherwise} \end{cases}}{d} & \text{otherwise} \end{cases}\right )}{d} - \frac{b e \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\log{\left (d x + e \right )}}{d} & \text{otherwise} \end{cases}\right ) \log{\left (c x \right )}}{d} + \frac{b x \log{\left (c x \right )}}{d} - \frac{b x}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x\right ) + a}{d + \frac{e}{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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