Optimal. Leaf size=143 \[ \frac{2 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{2 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^2}-\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{e^2}+\frac{2 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)} \]
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Rubi [A] time = 0.201671, antiderivative size = 143, 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, 2318, 2317, 2391, 2374, 6589} \[ \frac{2 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{2 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^2}-\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{e^2}+\frac{2 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 2353
Rule 2318
Rule 2317
Rule 2391
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)^2} \, dx &=\int \left (-\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{e}-\frac{d \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{e}\\ &=-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^2}-\frac{(2 b 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 \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e}\\ &=-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^2}+\frac{2 b 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^2\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^2}-\frac{\left (2 b^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{e^2}\\ &=-\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{e (d+e x)}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e^2}+\frac{2 b^2 n^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{e^2}+\frac{2 b 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 n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{e^2}\\ \end{align*}
Mathematica [A] time = 0.123893, size = 142, normalized size = 0.99 \[ \frac{2 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )+2 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )-2 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )+2 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\left (a+b \log \left (c x^n\right )\right )^2}{e^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.7, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{ \left ( ex+d \right ) ^{2}}}\, 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{d}{e^{3} x + d e^{2}} + \frac{\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^{2} x^{2} + 2 \, d e x + d^{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} x \log \left (c x^{n}\right )^{2} + 2 \, a b x \log \left (c x^{n}\right ) + a^{2} x}{e^{2} x^{2} + 2 \, d e x + d^{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{x \left (a + b \log{\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{2}}\, 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}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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