Optimal. Leaf size=63 \[ \frac{b n \text{PolyLog}\left (2,\frac{f (d+e x)}{d f-e g}\right )}{f}+\frac{\log \left (-\frac{e (f x+g)}{d f-e g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f} \]
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Rubi [A] time = 0.100408, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2412, 2394, 2393, 2391} \[ \frac{b n \text{PolyLog}\left (2,\frac{f (d+e x)}{d f-e g}\right )}{f}+\frac{\log \left (-\frac{e (f x+g)}{d f-e g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f} \]
Antiderivative was successfully verified.
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Rule 2412
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{\left (f+\frac{g}{x}\right ) x} \, dx &=\int \frac{a+b \log \left (c (d+e x)^n\right )}{g+f x} \, dx\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (-\frac{e (g+f x)}{d f-e g}\right )}{f}-\frac{(b e n) \int \frac{\log \left (\frac{e (g+f x)}{-d f+e g}\right )}{d+e x} \, dx}{f}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (-\frac{e (g+f x)}{d f-e g}\right )}{f}-\frac{(b n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{f x}{-d f+e g}\right )}{x} \, dx,x,d+e x\right )}{f}\\ &=\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (-\frac{e (g+f x)}{d f-e g}\right )}{f}+\frac{b n \text{Li}_2\left (\frac{f (d+e x)}{d f-e g}\right )}{f}\\ \end{align*}
Mathematica [A] time = 0.0134861, size = 62, normalized size = 0.98 \[ \frac{b n \text{PolyLog}\left (2,\frac{f (d+e x)}{d f-e g}\right )}{f}+\frac{\log \left (\frac{e (f x+g)}{e g-d f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.584, size = 261, normalized size = 4.1 \begin{align*}{\frac{b\ln \left ( fx+g \right ) \ln \left ( \left ( ex+d \right ) ^{n} \right ) }{f}}-{\frac{bn}{f}{\it dilog} \left ({\frac{ \left ( fx+g \right ) e+df-eg}{df-eg}} \right ) }-{\frac{bn\ln \left ( fx+g \right ) }{f}\ln \left ({\frac{ \left ( fx+g \right ) e+df-eg}{df-eg}} \right ) }-{\frac{{\frac{i}{2}}\ln \left ( fx+g \right ) b\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ){\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) }{f}}+{\frac{{\frac{i}{2}}\ln \left ( fx+g \right ) b\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{f}}+{\frac{{\frac{i}{2}}\ln \left ( fx+g \right ) b\pi \,{\it csgn} \left ( i \left ( ex+d \right ) ^{n} \right ) \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{f}}-{\frac{{\frac{i}{2}}\ln \left ( fx+g \right ) b\pi \, \left ({\it csgn} \left ( ic \left ( ex+d \right ) ^{n} \right ) \right ) ^{3}}{f}}+{\frac{b\ln \left ( fx+g \right ) \ln \left ( c \right ) }{f}}+{\frac{a\ln \left ( fx+g \right ) }{f}} \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*} b \int \frac{\log \left ({\left (e x + d\right )}^{n}\right ) + \log \left (c\right )}{f x + g}\,{d x} + \frac{a \log \left (f x + g\right )}{f} \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 ({\left (e x + d\right )}^{n} c\right ) + a}{f x + g}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \log{\left (c \left (d + e x\right )^{n} \right )}}{f x + g}\, dx \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 ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (f + \frac{g}{x}\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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