Optimal. Leaf size=135 \[ -\frac{2 b e n \text{PolyLog}\left (2,-\frac{d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac{2 b^2 e n^2 \text{PolyLog}\left (3,-\frac{d}{e x}\right )}{d^2}+\frac{e \log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d x}-\frac{2 b^2 n^2}{d x} \]
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Rubi [A] time = 0.241545, antiderivative size = 155, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {2353, 2305, 2304, 2302, 30, 2317, 2374, 6589} \[ \frac{2 b e n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac{2 b^2 e n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^2 n}+\frac{e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d x}-\frac{2 b^2 n^2}{d x} \]
Antiderivative was successfully verified.
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Rule 2353
Rule 2305
Rule 2304
Rule 2302
Rule 30
Rule 2317
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)} \, dx &=\int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d x^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{d^2 x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{d^2 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{d}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{d^2}+\frac{e^2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{d^2}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d x}+\frac{e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^2}-\frac{e \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{b d^2 n}+\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{d}-\frac{(2 b e n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^2}\\ &=-\frac{2 b^2 n^2}{d x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d x}-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^2 n}+\frac{e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^2}+\frac{2 b e n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^2}-\frac{\left (2 b^2 e n^2\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{d^2}\\ &=-\frac{2 b^2 n^2}{d x}-\frac{2 b n \left (a+b \log \left (c x^n\right )\right )}{d x}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d x}-\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{3 b d^2 n}+\frac{e \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^2}+\frac{2 b e n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^2}-\frac{2 b^2 e n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.100397, size = 130, normalized size = 0.96 \[ -\frac{-6 b e n \left (\text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text{PolyLog}\left (3,-\frac{e x}{d}\right )\right )-3 e \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{3 d \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac{6 b d n \left (a+b \log \left (c x^n\right )+b n\right )}{x}+\frac{e \left (a+b \log \left (c x^n\right )\right )^3}{b n}}{3 d^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.683, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{{x}^{2} \left ( ex+d \right ) }}\, 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*} a^{2}{\left (\frac{e \log \left (e x + d\right )}{d^{2}} - \frac{e \log \left (x\right )}{d^{2}} - \frac{1}{d x}\right )} + \int \frac{b^{2} \log \left (c\right )^{2} + b^{2} \log \left (x^{n}\right )^{2} + 2 \, a b \log \left (c\right ) + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (x^{n}\right )}{e x^{3} + d x^{2}}\,{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^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}}{e x^{3} + d x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c x^{n} \right )}\right )^{2}}{x^{2} \left (d + e x\right )}\, dx \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 )}^{2}}{{\left (e x + d\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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