Optimal. Leaf size=98 \[ \frac{b e^2 \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d^3}+\frac{e^2 \log \left (\frac{d x}{e}+1\right ) (a+b \log (c x))}{d^3}+\frac{x^2 (a+b \log (c x))}{2 d}-\frac{a e x}{d^2}-\frac{b e x \log (c x)}{d^2}+\frac{b e x}{d^2}-\frac{b x^2}{4 d} \]
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Rubi [A] time = 0.110443, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {263, 43, 2351, 2295, 2304, 2317, 2391} \[ \frac{b e^2 \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d^3}+\frac{e^2 \log \left (\frac{d x}{e}+1\right ) (a+b \log (c x))}{d^3}+\frac{x^2 (a+b \log (c x))}{2 d}-\frac{a e x}{d^2}-\frac{b e x \log (c x)}{d^2}+\frac{b e x}{d^2}-\frac{b x^2}{4 d} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rule 2351
Rule 2295
Rule 2304
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{x (a+b \log (c x))}{d+\frac{e}{x}} \, dx &=\int \left (-\frac{e (a+b \log (c x))}{d^2}+\frac{x (a+b \log (c x))}{d}+\frac{e^2 (a+b \log (c x))}{d^2 (e+d x)}\right ) \, dx\\ &=\frac{\int x (a+b \log (c x)) \, dx}{d}-\frac{e \int (a+b \log (c x)) \, dx}{d^2}+\frac{e^2 \int \frac{a+b \log (c x)}{e+d x} \, dx}{d^2}\\ &=-\frac{a e x}{d^2}-\frac{b x^2}{4 d}+\frac{x^2 (a+b \log (c x))}{2 d}+\frac{e^2 (a+b \log (c x)) \log \left (1+\frac{d x}{e}\right )}{d^3}-\frac{(b e) \int \log (c x) \, dx}{d^2}-\frac{\left (b e^2\right ) \int \frac{\log \left (1+\frac{d x}{e}\right )}{x} \, dx}{d^3}\\ &=-\frac{a e x}{d^2}+\frac{b e x}{d^2}-\frac{b x^2}{4 d}-\frac{b e x \log (c x)}{d^2}+\frac{x^2 (a+b \log (c x))}{2 d}+\frac{e^2 (a+b \log (c x)) \log \left (1+\frac{d x}{e}\right )}{d^3}+\frac{b e^2 \text{Li}_2\left (-\frac{d x}{e}\right )}{d^3}\\ \end{align*}
Mathematica [A] time = 0.0439753, size = 99, normalized size = 1.01 \[ \frac{b e^2 \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{d^3}+\frac{e^2 \log \left (\frac{d x+e}{e}\right ) (a+b \log (c x))}{d^3}+\frac{x^2 (a+b \log (c x))}{2 d}-\frac{a e x}{d^2}-\frac{b e x \log (c x)}{d^2}+\frac{b e x}{d^2}-\frac{b x^2}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 129, normalized size = 1.3 \begin{align*}{\frac{a{x}^{2}}{2\,d}}-{\frac{aex}{{d}^{2}}}+{\frac{a{e}^{2}\ln \left ( cdx+ce \right ) }{{d}^{3}}}+{\frac{b{x}^{2}\ln \left ( cx \right ) }{2\,d}}-{\frac{b{x}^{2}}{4\,d}}-{\frac{bex\ln \left ( cx \right ) }{{d}^{2}}}+{\frac{bex}{{d}^{2}}}+{\frac{b{e}^{2}}{{d}^{3}}{\it dilog} \left ({\frac{cdx+ce}{ce}} \right ) }+{\frac{b{e}^{2}\ln \left ( cx \right ) }{{d}^{3}}\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 = 1.55321, size = 151, normalized size = 1.54 \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^{2}}{d^{3}} + \frac{{\left ({\left (2 \, d \log \left (c\right ) - d\right )} b + 2 \, a d\right )} x^{2} - 4 \,{\left ({\left (e \log \left (c\right ) - e\right )} b + a e\right )} x + 2 \,{\left (b d x^{2} - 2 \, b e x\right )} \log \left (x\right )}{4 \, d^{2}} + \frac{{\left (b e^{2} \log \left (c\right ) + a e^{2}\right )} \log \left (d x + e\right )}{d^{3}} \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^{2} \log \left (c x\right ) + a x^{2}}{d x + e}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 86.6934, size = 189, normalized size = 1.93 \begin{align*} \frac{a x^{2}}{2 d} + \frac{a e^{2} \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\log{\left (d x + e \right )}}{d} & \text{otherwise} \end{cases}\right )}{d^{2}} - \frac{a e x}{d^{2}} + \frac{b x^{2} \log{\left (c x \right )}}{2 d} - \frac{b x^{2}}{4 d} - \frac{b e^{2} \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^{2}} + \frac{b e^{2} \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^{2}} - \frac{b e x \log{\left (c x \right )}}{d^{2}} + \frac{b e x}{d^{2}} \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{{\left (b \log \left (c x\right ) + a\right )} x}{d + \frac{e}{x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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