Optimal. Leaf size=44 \[ \frac{b n \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{d}-\frac{\log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d} \]
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Rubi [A] time = 0.0908774, antiderivative size = 66, normalized size of antiderivative = 1.5, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2344, 2301, 2317, 2391} \[ -\frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d}-\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b d n} \]
Antiderivative was successfully verified.
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Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x (d+e x)} \, dx &=\frac{\int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d}-\frac{e \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b d n}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d}+\frac{(b n) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b d n}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d}-\frac{b n \text{Li}_2\left (-\frac{e x}{d}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.03245, size = 63, normalized size = 1.43 \[ \frac{\left (a+b \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )-2 b n \log \left (\frac{e x}{d}+1\right )\right )}{2 b d n}-\frac{b n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.142, size = 336, normalized size = 7.6 \begin{align*} -{\frac{b\ln \left ({x}^{n} \right ) \ln \left ( ex+d \right ) }{d}}+{\frac{b\ln \left ({x}^{n} \right ) \ln \left ( x \right ) }{d}}-{\frac{bn \left ( \ln \left ( x \right ) \right ) ^{2}}{2\,d}}+{\frac{bn\ln \left ( ex+d \right ) }{d}\ln \left ( -{\frac{ex}{d}} \right ) }+{\frac{bn}{d}{\it dilog} \left ( -{\frac{ex}{d}} \right ) }+{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}\ln \left ( x \right ) }{d}}-{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}\ln \left ( ex+d \right ) }{d}}+{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) \ln \left ( ex+d \right ) }{d}}-{\frac{{\frac{i}{2}}b\pi \,{\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ){\it csgn} \left ( ic \right ) \ln \left ( x \right ) }{d}}-{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\ln \left ( x \right ) }{d}}-{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \ln \left ( ex+d \right ) }{d}}+{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\ln \left ( ex+d \right ) }{d}}+{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \ln \left ( x \right ) }{d}}-{\frac{b\ln \left ( c \right ) \ln \left ( ex+d \right ) }{d}}+{\frac{b\ln \left ( c \right ) \ln \left ( x \right ) }{d}}-{\frac{a\ln \left ( ex+d \right ) }{d}}+{\frac{a\ln \left ( x \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -a{\left (\frac{\log \left (e x + d\right )}{d} - \frac{\log \left (x\right )}{d}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e x^{2} + d x}\,{d 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^{n}\right ) + a}{e x^{2} + d x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 23.2241, size = 158, normalized size = 3.59 \begin{align*} - \frac{2 a e \left (\begin{cases} \frac{1}{2 e} + \frac{x}{d} & \text{for}\: e = 0 \\- \frac{\log{\left (- 2 e x \right )}}{2 e} & \text{otherwise} \end{cases}\right )}{d} - \frac{2 a e \left (\begin{cases} \frac{1}{2 e} + \frac{x}{d} & \text{for}\: e = 0 \\\frac{\log{\left (2 d + 2 e x \right )}}{2 e} & \text{otherwise} \end{cases}\right )}{d} + b n \left (\begin{cases} - \frac{1}{e x} & \text{for}\: d = 0 \\\frac{\begin{cases} \log{\left (e \right )} \log{\left (x \right )} + \operatorname{Li}_{2}\left (\frac{d e^{i \pi }}{e x}\right ) & \text{for}\: \left |{x}\right | < 1 \\- \log{\left (e \right )} \log{\left (\frac{1}{x} \right )} + \operatorname{Li}_{2}\left (\frac{d e^{i \pi }}{e x}\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 e^{i \pi }}{e x}\right ) & \text{otherwise} \end{cases}}{d} & \text{otherwise} \end{cases}\right ) - b \left (\begin{cases} \frac{1}{e x} & \text{for}\: d = 0 \\\frac{\log{\left (\frac{d}{x} + e \right )}}{d} & \text{otherwise} \end{cases}\right ) \log{\left (c x^{n} \right )} \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^{n}\right ) + a}{{\left (e x + d\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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