Optimal. Leaf size=102 \[ \frac{b n \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right )}{d^2 r^2}-\frac{e x^r \left (a+b \log \left (c x^n\right )\right )}{d^2 r \left (d+e x^r\right )}-\frac{\log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 r}+\frac{b n \log \left (d+e x^r\right )}{d^2 r^2} \]
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Rubi [A] time = 0.229273, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2349, 2345, 2391, 2335, 260} \[ \frac{b n \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right )}{d^2 r^2}-\frac{e x^r \left (a+b \log \left (c x^n\right )\right )}{d^2 r \left (d+e x^r\right )}-\frac{\log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 r}+\frac{b n \log \left (d+e x^r\right )}{d^2 r^2} \]
Antiderivative was successfully verified.
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Rule 2349
Rule 2345
Rule 2391
Rule 2335
Rule 260
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^2} \, dx &=\frac{\int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx}{d}-\frac{e \int \frac{x^{-1+r} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx}{d}\\ &=-\frac{e x^r \left (a+b \log \left (c x^n\right )\right )}{d^2 r \left (d+e x^r\right )}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x^{-r}}{e}\right )}{d^2 r}+\frac{(b n) \int \frac{\log \left (1+\frac{d x^{-r}}{e}\right )}{x} \, dx}{d^2 r}+\frac{(b e n) \int \frac{x^{-1+r}}{d+e x^r} \, dx}{d^2 r}\\ &=-\frac{e x^r \left (a+b \log \left (c x^n\right )\right )}{d^2 r \left (d+e x^r\right )}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x^{-r}}{e}\right )}{d^2 r}+\frac{b n \log \left (d+e x^r\right )}{d^2 r^2}+\frac{b n \text{Li}_2\left (-\frac{d x^{-r}}{e}\right )}{d^2 r^2}\\ \end{align*}
Mathematica [A] time = 0.313321, size = 132, normalized size = 1.29 \[ \frac{b n \left (\text{PolyLog}\left (2,\frac{e x^r}{d}+1\right )+\left (\log \left (-\frac{e x^r}{d}\right )-r \log (x)\right ) \log \left (d+e x^r\right )+\frac{1}{2} r^2 \log ^2(x)\right )+\frac{d r \left (a+b \log \left (c x^n\right )\right )}{d+e x^r}-a r \log \left (d-d x^r\right )+b r \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+b n \log \left (d-d x^r\right )}{d^2 r^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.066, size = 715, normalized size = 7. \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{1}{d e r x^{r} + d^{2} r} + \frac{\log \left (x\right )}{d^{2}} - \frac{\log \left (\frac{e x^{r} + d}{e}\right )}{d^{2} r}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e^{2} x x^{2 \, r} + 2 \, d e x x^{r} + d^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.38566, size = 527, normalized size = 5.17 \begin{align*} \frac{b d n r^{2} \log \left (x\right )^{2} + 2 \, b d r \log \left (c\right ) + 2 \, a d r +{\left (b e n r^{2} \log \left (x\right )^{2} + 2 \,{\left (b e r^{2} \log \left (c\right ) - b e n r + a e r^{2}\right )} \log \left (x\right )\right )} x^{r} - 2 \,{\left (b e n x^{r} + b d n\right )}{\rm Li}_2\left (-\frac{e x^{r} + d}{d} + 1\right ) - 2 \,{\left (b d r \log \left (c\right ) - b d n + a d r +{\left (b e r \log \left (c\right ) - b e n + a e r\right )} x^{r}\right )} \log \left (e x^{r} + d\right ) + 2 \,{\left (b d r^{2} \log \left (c\right ) + a d r^{2}\right )} \log \left (x\right ) - 2 \,{\left (b e n r x^{r} \log \left (x\right ) + b d n r \log \left (x\right )\right )} \log \left (\frac{e x^{r} + d}{d}\right )}{2 \,{\left (d^{2} e r^{2} x^{r} + d^{3} r^{2}\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 (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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