Optimal. Leaf size=84 \[ \frac{b d \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{e^2}-\frac{d (a+b \log (c x))^2}{2 b e^2}+\frac{d \log \left (\frac{d x}{e}+1\right ) (a+b \log (c x))}{e^2}-\frac{a+b \log (c x)}{e x}-\frac{b}{e x} \]
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Rubi [A] time = 0.131292, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {263, 44, 2351, 2304, 2301, 2317, 2391} \[ \frac{b d \text{PolyLog}\left (2,-\frac{d x}{e}\right )}{e^2}-\frac{d (a+b \log (c x))^2}{2 b e^2}+\frac{d \log \left (\frac{d x}{e}+1\right ) (a+b \log (c x))}{e^2}-\frac{a+b \log (c x)}{e x}-\frac{b}{e x} \]
Antiderivative was successfully verified.
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Rule 263
Rule 44
Rule 2351
Rule 2304
Rule 2301
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log (c x)}{\left (d+\frac{e}{x}\right ) x^3} \, dx &=\int \left (\frac{a+b \log (c x)}{e x^2}-\frac{d (a+b \log (c x))}{e^2 x}+\frac{d^2 (a+b \log (c x))}{e^2 (e+d x)}\right ) \, dx\\ &=-\frac{d \int \frac{a+b \log (c x)}{x} \, dx}{e^2}+\frac{d^2 \int \frac{a+b \log (c x)}{e+d x} \, dx}{e^2}+\frac{\int \frac{a+b \log (c x)}{x^2} \, dx}{e}\\ &=-\frac{b}{e x}-\frac{a+b \log (c x)}{e x}-\frac{d (a+b \log (c x))^2}{2 b e^2}+\frac{d (a+b \log (c x)) \log \left (1+\frac{d x}{e}\right )}{e^2}-\frac{(b d) \int \frac{\log \left (1+\frac{d x}{e}\right )}{x} \, dx}{e^2}\\ &=-\frac{b}{e x}-\frac{a+b \log (c x)}{e x}-\frac{d (a+b \log (c x))^2}{2 b e^2}+\frac{d (a+b \log (c x)) \log \left (1+\frac{d x}{e}\right )}{e^2}+\frac{b d \text{Li}_2\left (-\frac{d x}{e}\right )}{e^2}\\ \end{align*}
Mathematica [A] time = 0.0883366, size = 77, normalized size = 0.92 \[ -\frac{-2 b d \text{PolyLog}\left (2,-\frac{d x}{e}\right )-2 d \log \left (\frac{d x}{e}+1\right ) (a+b \log (c x))+\frac{d (a+b \log (c x))^2}{b}+\frac{2 e (a+b \log (c x))}{x}+\frac{2 b e}{x}}{2 e^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 120, normalized size = 1.4 \begin{align*}{\frac{ad\ln \left ( cdx+ce \right ) }{{e}^{2}}}-{\frac{a}{ex}}-{\frac{ad\ln \left ( cx \right ) }{{e}^{2}}}+{\frac{bd}{{e}^{2}}{\it dilog} \left ({\frac{cdx+ce}{ce}} \right ) }+{\frac{bd\ln \left ( cx \right ) }{{e}^{2}}\ln \left ({\frac{cdx+ce}{ce}} \right ) }-{\frac{bd \left ( \ln \left ( cx \right ) \right ) ^{2}}{2\,{e}^{2}}}-{\frac{b\ln \left ( cx \right ) }{ex}}-{\frac{b}{ex}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.36649, size = 130, normalized size = 1.55 \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 d}{e^{2}} + \frac{{\left (b d \log \left (c\right ) + a d\right )} \log \left (d x + e\right )}{e^{2}} - \frac{b d x \log \left (x\right )^{2} + 2 \,{\left (e \log \left (c\right ) + e\right )} b + 2 \, a e + 2 \,{\left (b e +{\left (b d \log \left (c\right ) + a d\right )} x\right )} \log \left (x\right )}{2 \, e^{2} x} \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 \log \left (c x\right ) + a}{d x^{3} + e x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 52.3139, size = 187, normalized size = 2.23 \begin{align*} \frac{a d^{2} \left (\begin{cases} \frac{x}{e} & \text{for}\: d = 0 \\\frac{\log{\left (d x + e \right )}}{d} & \text{otherwise} \end{cases}\right )}{e^{2}} - \frac{a d \log{\left (x \right )}}{e^{2}} - \frac{a}{e x} - \frac{b d^{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 )}{e^{2}} + \frac{b d^{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 )}}{e^{2}} + \frac{b d \log{\left (x \right )}^{2}}{2 e^{2}} - \frac{b d \log{\left (x \right )} \log{\left (c x \right )}}{e^{2}} - \frac{b \log{\left (c x \right )}}{e x} - \frac{b}{e x} \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}{{\left (d + \frac{e}{x}\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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