Optimal. Leaf size=130 \[ -\frac{2 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{2 b^2 d n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{e^2}-\frac{d \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac{2 a b n x}{e}-\frac{2 b^2 n x \log \left (c x^n\right )}{e}+\frac{2 b^2 n^2 x}{e} \]
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Rubi [A] time = 0.155189, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2353, 2296, 2295, 2317, 2374, 6589} \[ -\frac{2 b d n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{2 b^2 d n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{e^2}-\frac{d \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac{2 a b n x}{e}-\frac{2 b^2 n x \log \left (c x^n\right )}{e}+\frac{2 b^2 n^2 x}{e} \]
Antiderivative was successfully verified.
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Rule 2353
Rule 2296
Rule 2295
Rule 2317
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx &=\int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)}\right ) \, dx\\ &=\frac{\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{e}-\frac{d \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac{d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^2}+\frac{(2 b d n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^2}-\frac{(2 b n) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}\\ &=-\frac{2 a b n x}{e}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac{d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^2}-\frac{2 b d n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e^2}-\frac{\left (2 b^2 n\right ) \int \log \left (c x^n\right ) \, dx}{e}+\frac{\left (2 b^2 d n^2\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{e^2}\\ &=-\frac{2 a b n x}{e}+\frac{2 b^2 n^2 x}{e}-\frac{2 b^2 n x \log \left (c x^n\right )}{e}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac{d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^2}-\frac{2 b d n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e^2}+\frac{2 b^2 d n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{e^2}\\ \end{align*}
Mathematica [A] time = 0.0648672, size = 103, normalized size = 0.79 \[ \frac{-2 b d n \left (\text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \text{PolyLog}\left (3,-\frac{e x}{d}\right )\right )-d \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+e x \left (a+b \log \left (c x^n\right )\right )^2-2 b e n x \left (a+b \log \left (c x^n\right )-b n\right )}{e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.718, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{ex+d}}\, 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{x}{e} - \frac{d \log \left (e x + d\right )}{e^{2}}\right )} + \int \frac{b^{2} x \log \left (x^{n}\right )^{2} + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} x \log \left (x^{n}\right ) +{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right )\right )} x}{e x + d}\,{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} x \log \left (c x^{n}\right )^{2} + 2 \, a b x \log \left (c x^{n}\right ) + a^{2} x}{e x + d}, 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{x \left (a + b \log{\left (c x^{n} \right )}\right )^{2}}{d + e x}\, 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} x}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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