Optimal. Leaf size=107 \[ -\frac{b n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{f}+\frac{b n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{f}-\frac{\log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}+\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f} \]
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Rubi [A] time = 0.1437, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {36, 29, 31, 2416, 2394, 2315, 2393, 2391} \[ -\frac{b n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{f}+\frac{b n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{f}-\frac{\log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}+\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f} \]
Antiderivative was successfully verified.
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Rule 36
Rule 29
Rule 31
Rule 2416
Rule 2394
Rule 2315
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{x (f+g x)} \, dx &=\int \left (\frac{a+b \log \left (c (d+e x)^n\right )}{f x}-\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f (f+g x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{f}-\frac{g \int \frac{a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{f}\\ &=\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{f}-\frac{(b e n) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{f}+\frac{(b e n) \int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{f}\\ &=\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{f}+\frac{b n \text{Li}_2\left (1+\frac{e x}{d}\right )}{f}+\frac{(b n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{f}\\ &=\frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{f}-\frac{b n \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{f}+\frac{b n \text{Li}_2\left (1+\frac{e x}{d}\right )}{f}\\ \end{align*}
Mathematica [A] time = 0.0373871, size = 85, normalized size = 0.79 \[ \frac{-b n \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )+b n \text{PolyLog}\left (2,\frac{e x}{d}+1\right )+\left (\log \left (-\frac{e x}{d}\right )-\log \left (\frac{e (f+g x)}{e f-d g}\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.515, size = 455, normalized size = 4.3 \begin{align*} \text{result too large to display} \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 (g x + f\right )}{f} - \frac{\log \left (x\right )}{f}\right )} + b \int \frac{\log \left ({\left (e x + d\right )}^{n}\right ) + \log \left (c\right )}{g x^{2} + f 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 ({\left (e x + d\right )}^{n} c\right ) + a}{g x^{2} + f x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (g x + f\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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