Optimal. Leaf size=77 \[ \frac{b n x^{1-m} (f x)^{m-1} \text{PolyLog}\left (2,-\frac{e x^m}{d}\right )}{e m^2}+\frac{x^{1-m} (f x)^{m-1} \log \left (\frac{e x^m}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e m} \]
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Rubi [A] time = 0.191445, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2339, 2337, 2391} \[ \frac{b n x^{1-m} (f x)^{m-1} \text{PolyLog}\left (2,-\frac{e x^m}{d}\right )}{e m^2}+\frac{x^{1-m} (f x)^{m-1} \log \left (\frac{e x^m}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e m} \]
Antiderivative was successfully verified.
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Rule 2339
Rule 2337
Rule 2391
Rubi steps
\begin{align*} \int \frac{(f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{d+e x^m} \, dx &=\left (x^{1-m} (f x)^{-1+m}\right ) \int \frac{x^{-1+m} \left (a+b \log \left (c x^n\right )\right )}{d+e x^m} \, dx\\ &=\frac{x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x^m}{d}\right )}{e m}-\frac{\left (b n x^{1-m} (f x)^{-1+m}\right ) \int \frac{\log \left (1+\frac{e x^m}{d}\right )}{x} \, dx}{e m}\\ &=\frac{x^{1-m} (f x)^{-1+m} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x^m}{d}\right )}{e m}+\frac{b n x^{1-m} (f x)^{-1+m} \text{Li}_2\left (-\frac{e x^m}{d}\right )}{e m^2}\\ \end{align*}
Mathematica [A] time = 0.141644, size = 141, normalized size = 1.83 \[ \frac{x^{-m} (f x)^m \left (-b n \text{PolyLog}\left (2,\frac{e x^m}{d}+1\right )+m \log (x) \left (a m+b m \log \left (c x^n\right )+b n \log \left (d+e x^m\right )-b n \log \left (d-d x^m\right )\right )+a m \log \left (d-d x^m\right )+b m \log \left (c x^n\right ) \log \left (d-d x^m\right )-b n \log \left (-\frac{e x^m}{d}\right ) \log \left (d+e x^m\right )-b m^2 n \log ^2(x)\right )}{e f m^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.98, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{-1+m} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{d+e{x}^{m}}}\, dx \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{f^{m} x^{m} \log \left (c\right ) + f^{m} x^{m} \log \left (x^{n}\right )}{e f x x^{m} + d f x}\,{d x} + \frac{a f^{m - 1} \log \left (\frac{e x^{m} + d}{e}\right )}{e m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34501, size = 190, normalized size = 2.47 \begin{align*} \frac{b f^{m - 1} m n \log \left (x\right ) \log \left (\frac{e x^{m} + d}{d}\right ) + b f^{m - 1} n{\rm Li}_2\left (-\frac{e x^{m} + d}{d} + 1\right ) +{\left (b m \log \left (c\right ) + a m\right )} f^{m - 1} \log \left (e x^{m} + d\right )}{e m^{2}} \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{{\left (b \log \left (c x^{n}\right ) + a\right )} \left (f x\right )^{m - 1}}{e x^{m} + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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