Optimal. Leaf size=72 \[ \frac{2 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{e}+\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e} \]
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Rubi [A] time = 0.0610311, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2317, 2374, 6589} \[ \frac{2 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{e}+\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e} \]
Antiderivative was successfully verified.
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Rule 2317
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx &=\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e}-\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}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e}-\frac{\left (2 b^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{e}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{e}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{e}-\frac{2 b^2 n^2 \text{Li}_3\left (-\frac{e x}{d}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.0252999, size = 68, normalized size = 0.94 \[ \frac{\log \left (\frac{d+e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac{2 b n \left (b n \text{PolyLog}\left (3,-\frac{e x}{d}\right )-\text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{e} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.295, size = 1412, normalized size = 19.6 \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{a^{2} \log \left (e x + d\right )}{e} + \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 + 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} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}}{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{\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}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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