Optimal. Leaf size=287 \[ -b e^2 n \text{PolyLog}\left (2,-\frac{1}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} b^2 e^2 n^2 \text{PolyLog}\left (2,-\frac{1}{e x}\right )-b^2 e^2 n^2 \text{PolyLog}\left (3,-\frac{1}{e x}\right )+\frac{1}{2} e^2 \log \left (\frac{1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} b e^2 n \log \left (\frac{1}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 x}-\frac{3 b e n \left (a+b \log \left (c x^n\right )\right )}{2 x}-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac{b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{1}{4} b^2 e^2 n^2 \log (x)+\frac{1}{4} b^2 e^2 n^2 \log (e x+1)-\frac{b^2 n^2 \log (e x+1)}{4 x^2}-\frac{7 b^2 e n^2}{4 x} \]
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Rubi [A] time = 0.483553, antiderivative size = 310, normalized size of antiderivative = 1.08, number of steps used = 19, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591, Rules used = {2305, 2304, 2378, 44, 2351, 2301, 2317, 2391, 2353, 2302, 30, 2374, 6589} \[ b e^2 n \text{PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )+\frac{1}{2} b^2 e^2 n^2 \text{PolyLog}(2,-e x)-b^2 e^2 n^2 \text{PolyLog}(3,-e x)-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}-\frac{1}{4} e^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} e^2 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} b e^2 n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 x}-\frac{b n \log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{3 b e n \left (a+b \log \left (c x^n\right )\right )}{2 x}-\frac{1}{4} b^2 e^2 n^2 \log (x)+\frac{1}{4} b^2 e^2 n^2 \log (e x+1)-\frac{b^2 n^2 \log (e x+1)}{4 x^2}-\frac{7 b^2 e n^2}{4 x} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rule 2378
Rule 44
Rule 2351
Rule 2301
Rule 2317
Rule 2391
Rule 2353
Rule 2302
Rule 30
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{x^3} \, dx &=-\frac{b^2 n^2 \log (1+e x)}{4 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 x^2}-e \int \left (-\frac{b^2 n^2}{4 x^2 (1+e x)}-\frac{b n \left (a+b \log \left (c x^n\right )\right )}{2 x^2 (1+e x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 x^2 (1+e x)}\right ) \, dx\\ &=-\frac{b^2 n^2 \log (1+e x)}{4 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 x^2}+\frac{1}{2} e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (1+e x)} \, dx+\frac{1}{2} (b e n) \int \frac{a+b \log \left (c x^n\right )}{x^2 (1+e x)} \, dx+\frac{1}{4} \left (b^2 e n^2\right ) \int \frac{1}{x^2 (1+e x)} \, dx\\ &=-\frac{b^2 n^2 \log (1+e x)}{4 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 x^2}+\frac{1}{2} e \int \left (\frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^2}{1+e x}\right ) \, dx+\frac{1}{2} (b e n) \int \left (\frac{a+b \log \left (c x^n\right )}{x^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{1+e x}\right ) \, dx+\frac{1}{4} \left (b^2 e n^2\right ) \int \left (\frac{1}{x^2}-\frac{e}{x}+\frac{e^2}{1+e x}\right ) \, dx\\ &=-\frac{b^2 e n^2}{4 x}-\frac{1}{4} b^2 e^2 n^2 \log (x)+\frac{1}{4} b^2 e^2 n^2 \log (1+e x)-\frac{b^2 n^2 \log (1+e x)}{4 x^2}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 x^2}+\frac{1}{2} e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx-\frac{1}{2} e^2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\frac{1}{2} e^3 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{1+e x} \, dx+\frac{1}{2} (b e n) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx-\frac{1}{2} \left (b e^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx+\frac{1}{2} \left (b e^3 n\right ) \int \frac{a+b \log \left (c x^n\right )}{1+e x} \, dx\\ &=-\frac{3 b^2 e n^2}{4 x}-\frac{1}{4} b^2 e^2 n^2 \log (x)-\frac{b e n \left (a+b \log \left (c x^n\right )\right )}{2 x}-\frac{1}{4} e^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 x}+\frac{1}{4} b^2 e^2 n^2 \log (1+e x)-\frac{b^2 n^2 \log (1+e x)}{4 x^2}+\frac{1}{2} b e^2 n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 x^2}+\frac{1}{2} e^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 x^2}-\frac{e^2 \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b n}+(b e n) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx-\left (b e^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{x} \, dx-\frac{1}{2} \left (b^2 e^2 n^2\right ) \int \frac{\log (1+e x)}{x} \, dx\\ &=-\frac{7 b^2 e n^2}{4 x}-\frac{1}{4} b^2 e^2 n^2 \log (x)-\frac{3 b e n \left (a+b \log \left (c x^n\right )\right )}{2 x}-\frac{1}{4} e^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 x}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+\frac{1}{4} b^2 e^2 n^2 \log (1+e x)-\frac{b^2 n^2 \log (1+e x)}{4 x^2}+\frac{1}{2} b e^2 n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 x^2}+\frac{1}{2} e^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 x^2}+\frac{1}{2} b^2 e^2 n^2 \text{Li}_2(-e x)+b e^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)-\left (b^2 e^2 n^2\right ) \int \frac{\text{Li}_2(-e x)}{x} \, dx\\ &=-\frac{7 b^2 e n^2}{4 x}-\frac{1}{4} b^2 e^2 n^2 \log (x)-\frac{3 b e n \left (a+b \log \left (c x^n\right )\right )}{2 x}-\frac{1}{4} e^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 x}-\frac{e^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+\frac{1}{4} b^2 e^2 n^2 \log (1+e x)-\frac{b^2 n^2 \log (1+e x)}{4 x^2}+\frac{1}{2} b e^2 n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{2 x^2}+\frac{1}{2} e^2 \left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log (1+e x)}{2 x^2}+\frac{1}{2} b^2 e^2 n^2 \text{Li}_2(-e x)+b e^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2(-e x)-b^2 e^2 n^2 \text{Li}_3(-e x)\\ \end{align*}
Mathematica [A] time = 0.189086, size = 513, normalized size = 1.79 \[ -\frac{-6 b e^2 n x^2 \text{PolyLog}(2,-e x) \left (2 a+2 b \log \left (c x^n\right )+b n\right )+12 b^2 e^2 n^2 x^2 \text{PolyLog}(3,-e x)+6 a^2 e^2 x^2 \log (x)-6 a^2 e^2 x^2 \log (e x+1)+6 a^2 e x+6 a^2 \log (e x+1)+12 a b e^2 x^2 \log (x) \log \left (c x^n\right )-12 a b e^2 x^2 \log (e x+1) \log \left (c x^n\right )+12 a b e x \log \left (c x^n\right )+12 a b \log (e x+1) \log \left (c x^n\right )-6 a b e^2 n x^2 \log ^2(x)+6 a b e^2 n x^2 \log (x)-6 a b e^2 n x^2 \log (e x+1)+18 a b e n x+6 a b n \log (e x+1)-6 b^2 e^2 n x^2 \log ^2(x) \log \left (c x^n\right )+6 b^2 e^2 x^2 \log (x) \log ^2\left (c x^n\right )-6 b^2 e^2 x^2 \log (e x+1) \log ^2\left (c x^n\right )+6 b^2 e^2 n x^2 \log (x) \log \left (c x^n\right )-6 b^2 e^2 n x^2 \log (e x+1) \log \left (c x^n\right )+6 b^2 e x \log ^2\left (c x^n\right )+6 b^2 \log (e x+1) \log ^2\left (c x^n\right )+18 b^2 e n x \log \left (c x^n\right )+6 b^2 n \log (e x+1) \log \left (c x^n\right )+2 b^2 e^2 n^2 x^2 \log ^3(x)-3 b^2 e^2 n^2 x^2 \log ^2(x)+3 b^2 e^2 n^2 x^2 \log (x)-3 b^2 e^2 n^2 x^2 \log (e x+1)+21 b^2 e n^2 x+3 b^2 n^2 \log (e x+1)}{12 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.15, 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}^{3}}}\, 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*} -\frac{{\left (b^{2} e^{2} x^{2} \log \left (x\right ) + b^{2} e x -{\left (b^{2} e^{2} x^{2} - b^{2}\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )^{2}}{2 \, x^{2}} - \int -\frac{{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2}\right )} \log \left (e x + 1\right ) +{\left (b^{2} e^{2} n x^{2} \log \left (x\right ) + b^{2} e n x -{\left (b^{2} e^{2} n x^{2} - b^{2}{\left (n + 2 \, \log \left (c\right )\right )} - 2 \, a b\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )}{x^{3}}\,{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^{3}}, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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