Optimal. Leaf size=54 \[ \frac{b n \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right )}{d r^2}-\frac{\log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d r} \]
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Rubi [A] time = 0.0777626, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2345, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right )}{d r^2}-\frac{\log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d r} \]
Antiderivative was successfully verified.
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Rule 2345
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx &=-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x^{-r}}{e}\right )}{d r}+\frac{(b n) \int \frac{\log \left (1+\frac{d x^{-r}}{e}\right )}{x} \, dx}{d r}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x^{-r}}{e}\right )}{d r}+\frac{b n \text{Li}_2\left (-\frac{d x^{-r}}{e}\right )}{d r^2}\\ \end{align*}
Mathematica [A] time = 0.113823, size = 108, normalized size = 2. \[ \frac{2 b n \text{PolyLog}\left (2,\frac{e x^r}{d}+1\right )-2 r \log \left (d-d x^r\right ) \left (a+b \log \left (c x^n\right )\right )+2 b n r \log (x) \left (\log \left (d-d x^r\right )-\log \left (d+e x^r\right )\right )+2 b n \log \left (-\frac{e x^r}{d}\right ) \log \left (d+e x^r\right )+b n r^2 \log ^2(x)}{2 d r^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.194, size = 451, normalized size = 8.4 \begin{align*}{\frac{b\ln \left ( d+e{x}^{r} \right ) n\ln \left ( x \right ) }{dr}}-{\frac{b\ln \left ( d+e{x}^{r} \right ) \ln \left ({x}^{n} \right ) }{dr}}-{\frac{b\ln \left ({x}^{r} \right ) n\ln \left ( x \right ) }{dr}}+{\frac{b\ln \left ({x}^{r} \right ) \ln \left ({x}^{n} \right ) }{dr}}+{\frac{bn \left ( \ln \left ( x \right ) \right ) ^{2}}{2\,d}}-{\frac{\ln \left ( x \right ) bn}{dr}\ln \left ( 1+{\frac{e{x}^{r}}{d}} \right ) }-{\frac{bn}{{r}^{2}d}{\it polylog} \left ( 2,-{\frac{e{x}^{r}}{d}} \right ) }+{\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 ( d+e{x}^{r} \right ) }{dr}}-{\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}^{r} \right ) }{dr}}+{\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}^{r} \right ) }{dr}}-{\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 ( d+e{x}^{r} \right ) }{dr}}-{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\ln \left ({x}^{r} \right ) }{dr}}+{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{3}\ln \left ( d+e{x}^{r} \right ) }{dr}}+{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \ln \left ({x}^{r} \right ) }{dr}}-{\frac{{\frac{i}{2}}b\pi \, \left ({\it csgn} \left ( ic{x}^{n} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) \ln \left ( d+e{x}^{r} \right ) }{dr}}-{\frac{b\ln \left ( c \right ) \ln \left ( d+e{x}^{r} \right ) }{dr}}+{\frac{b\ln \left ( c \right ) \ln \left ({x}^{r} \right ) }{dr}}-{\frac{a\ln \left ( d+e{x}^{r} \right ) }{dr}}+{\frac{a\ln \left ({x}^{r} \right ) }{dr}} \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 (x\right )}{d} - \frac{\log \left (\frac{e x^{r} + d}{e}\right )}{d r}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e x x^{r} + d x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32528, size = 235, normalized size = 4.35 \begin{align*} \frac{b n r^{2} \log \left (x\right )^{2} - 2 \, b n r \log \left (x\right ) \log \left (\frac{e x^{r} + d}{d}\right ) - 2 \, b n{\rm Li}_2\left (-\frac{e x^{r} + d}{d} + 1\right ) - 2 \,{\left (b r \log \left (c\right ) + a r\right )} \log \left (e x^{r} + d\right ) + 2 \,{\left (b r^{2} \log \left (c\right ) + a r^{2}\right )} \log \left (x\right )}{2 \, d r^{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{b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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