Optimal. Leaf size=40 \[ \frac{b n \text{PolyLog}\left (3,-d f x^m\right )}{m^2}-\frac{\text{PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m} \]
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Rubi [A] time = 0.0482115, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2374, 6589} \[ \frac{b n \text{PolyLog}\left (3,-d f x^m\right )}{m^2}-\frac{\text{PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m} \]
Antiderivative was successfully verified.
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Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac{1}{d}+f x^m\right )\right )}{x} \, dx &=-\frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^m\right )}{m}+\frac{(b n) \int \frac{\text{Li}_2\left (-d f x^m\right )}{x} \, dx}{m}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f x^m\right )}{m}+\frac{b n \text{Li}_3\left (-d f x^m\right )}{m^2}\\ \end{align*}
Mathematica [A] time = 0.0090543, size = 52, normalized size = 1.3 \[ -\frac{a \text{PolyLog}\left (2,-d f x^m\right )}{m}-\frac{b \log \left (c x^n\right ) \text{PolyLog}\left (2,-d f x^m\right )}{m}+\frac{b n \text{PolyLog}\left (3,-d f x^m\right )}{m^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.043, size = 308, normalized size = 7.7 \begin{align*} -{\frac{b\ln \left ( d \left ({d}^{-1}+f{x}^{m} \right ) \right ) n \left ( \ln \left ( x \right ) \right ) ^{2}}{2}}+b\ln \left ( x \right ) \ln \left ( d \left ({d}^{-1}+f{x}^{m} \right ) \right ) \ln \left ({x}^{n} \right ) +{\frac{bn \left ( \ln \left ( x \right ) \right ) ^{2}\ln \left ( df{x}^{m}+1 \right ) }{2}}-{\frac{bn\ln \left ( x \right ){\it polylog} \left ( 2,-df{x}^{m} \right ) }{m}}+{\frac{bn{\it polylog} \left ( 3,-df{x}^{m} \right ) }{{m}^{2}}}+{\frac{b{\it dilog} \left ( df{x}^{m}+1 \right ) n\ln \left ( x \right ) }{m}}-{\frac{b{\it dilog} \left ( df{x}^{m}+1 \right ) \ln \left ({x}^{n} \right ) }{m}}-b\ln \left ( x \right ) \ln \left ({x}^{n} \right ) \ln \left ( df{x}^{m}+1 \right ) +{\frac{{\frac{i}{2}}{\it dilog} \left ( df{x}^{m}+1 \right ) b\pi \,{\it csgn} \left ( ic \right ){\it csgn} \left ( i{x}^{n} \right ){\it csgn} \left ( ic{x}^{n} \right ) }{m}}-{\frac{{\frac{i}{2}}{\it dilog} \left ( df{x}^{m}+1 \right ) b\pi \,{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}}{m}}-{\frac{{\frac{i}{2}}{\it dilog} \left ( df{x}^{m}+1 \right ) b\pi \,{\it csgn} \left ( i{x}^{n} \right ) \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}}{m}}+{\frac{{\frac{i}{2}}{\it dilog} \left ( df{x}^{m}+1 \right ) b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}}{m}}-{\frac{b{\it dilog} \left ( df{x}^{m}+1 \right ) \ln \left ( c \right ) }{m}}-{\frac{{\it dilog} \left ( df{x}^{m}+1 \right ) a}{m}} \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*} -\frac{1}{2} \,{\left (b n \log \left (x\right )^{2} - 2 \, b \log \left (x\right ) \log \left (x^{n}\right ) - 2 \,{\left (b \log \left (c\right ) + a\right )} \log \left (x\right )\right )} \log \left (d f x^{m} + 1\right ) - \int \frac{2 \, b d f m x^{m} \log \left (x\right ) \log \left (x^{n}\right ) -{\left (b d f m n \log \left (x\right )^{2} - 2 \,{\left (b d f m \log \left (c\right ) + a d f m\right )} \log \left (x\right )\right )} x^{m}}{2 \,{\left (d f x x^{m} + x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.34605, size = 113, normalized size = 2.82 \begin{align*} \frac{b n{\rm polylog}\left (3, -d f x^{m}\right ) -{\left (b m n \log \left (x\right ) + b m \log \left (c\right ) + a m\right )}{\rm Li}_2\left (-d f x^{m}\right )}{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 )} \log \left ({\left (f x^{m} + \frac{1}{d}\right )} d\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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