Optimal. Leaf size=82 \[ \frac{b n \text{PolyLog}\left (2,-\frac{d}{e x^2}\right )}{4 d^2}-\frac{\log \left (\frac{d}{e x^2}+1\right ) \left (2 a+2 b \log \left (c x^n\right )-b n\right )}{4 d^2}+\frac{a+b \log \left (c x^n\right )}{2 d \left (d+e x^2\right )} \]
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Rubi [A] time = 0.140375, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2340, 2345, 2391} \[ \frac{b n \text{PolyLog}\left (2,-\frac{d}{e x^2}\right )}{4 d^2}-\frac{\log \left (\frac{d}{e x^2}+1\right ) \left (2 a+2 b \log \left (c x^n\right )-b n\right )}{4 d^2}+\frac{a+b \log \left (c x^n\right )}{2 d \left (d+e x^2\right )} \]
Antiderivative was successfully verified.
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Rule 2340
Rule 2345
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^2\right )^2} \, dx &=\frac{a+b \log \left (c x^n\right )}{2 d \left (d+e x^2\right )}-\frac{\int \frac{-2 a+b n-2 b \log \left (c x^n\right )}{x \left (d+e x^2\right )} \, dx}{2 d}\\ &=\frac{a+b \log \left (c x^n\right )}{2 d \left (d+e x^2\right )}-\frac{\log \left (1+\frac{d}{e x^2}\right ) \left (2 a-b n+2 b \log \left (c x^n\right )\right )}{4 d^2}+\frac{(b n) \int \frac{\log \left (1+\frac{d}{e x^2}\right )}{x} \, dx}{2 d^2}\\ &=\frac{a+b \log \left (c x^n\right )}{2 d \left (d+e x^2\right )}-\frac{\log \left (1+\frac{d}{e x^2}\right ) \left (2 a-b n+2 b \log \left (c x^n\right )\right )}{4 d^2}+\frac{b n \text{Li}_2\left (-\frac{d}{e x^2}\right )}{4 d^2}\\ \end{align*}
Mathematica [C] time = 0.368261, size = 279, normalized size = 3.4 \[ \frac{b n \left (-2 \left (\text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )+\log (x) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )\right )-2 \left (\text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )+\log (x) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )\right )+\frac{\sqrt{e} x \log (x)}{-\sqrt{e} x+i \sqrt{d}}-\frac{\sqrt{e} x \log (x)}{\sqrt{e} x+i \sqrt{d}}+\log \left (-\sqrt{e} x+i \sqrt{d}\right )+\log \left (\sqrt{e} x+i \sqrt{d}\right )+2 \log ^2(x)\right )}{4 d^2}-\frac{\log \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{2 d^2}+\frac{a+b \log \left (c x^n\right )-b n \log (x)}{2 d^2+2 d e x^2}+\frac{\log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{d^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.157, size = 644, normalized size = 7.9 \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*} \frac{1}{2} \, a{\left (\frac{1}{d e x^{2} + d^{2}} - \frac{\log \left (e x^{2} + d\right )}{d^{2}} + \frac{2 \, \log \left (x\right )}{d^{2}}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e^{2} x^{5} + 2 \, d e x^{3} + d^{2} 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 (c x^{n}\right ) + a}{e^{2} x^{5} + 2 \, d e x^{3} + d^{2} 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 (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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