Optimal. Leaf size=55 \[ 2 b n \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )-\text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2-2 b^2 n^2 \text{PolyLog}(4,-e x) \]
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Rubi [A] time = 0.060267, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2374, 2383, 6589} \[ 2 b n \text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )-\text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2-2 b^2 n^2 \text{PolyLog}(4,-e x) \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2383
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x} \, dx &=-\left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)+(2 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)}{x} \, dx\\ &=-\left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)+2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)-\left (2 b^2 n^2\right ) \int \frac{\text{Li}_3(-e x)}{x} \, dx\\ &=-\left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2(-e x)+2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3(-e x)-2 b^2 n^2 \text{Li}_4(-e x)\\ \end{align*}
Mathematica [A] time = 0.0753697, size = 53, normalized size = 0.96 \[ 2 b n \left (\text{PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )-b n \text{PolyLog}(4,-e x)\right )-\text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^2 \]
Antiderivative was successfully verified.
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Maple [F] time = 0.175, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\ln \left ( ex+1 \right ) }{x}}\, 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*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left (e x + 1\right )}{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^{2} \log \left (c x^{n}\right )^{2} \log \left (e x + 1\right ) + 2 \, a b \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a^{2} \log \left (e x + 1\right )}{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{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left (e x + 1\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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