Optimal. Leaf size=113 \[ \frac{6 b^2 n^2 \text{PolyLog}\left (3,-\frac{d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d}+\frac{3 b n \text{PolyLog}\left (2,-\frac{d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}+\frac{6 b^3 n^3 \text{PolyLog}\left (4,-\frac{d}{e x}\right )}{d}-\frac{\log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d} \]
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Rubi [A] time = 0.207856, antiderivative size = 130, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {2344, 2302, 30, 2317, 2374, 2383, 6589} \[ \frac{6 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d}-\frac{3 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}-\frac{6 b^3 n^3 \text{PolyLog}\left (4,-\frac{e x}{d}\right )}{d}-\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}+\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d n} \]
Antiderivative was successfully verified.
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Rule 2344
Rule 2302
Rule 30
Rule 2317
Rule 2374
Rule 2383
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx &=\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx}{d}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{d+e x} \, dx}{d}\\ &=-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{e x}{d}\right )}{d}+\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{b d n}+\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d n}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{e x}{d}\right )}{d}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d}+\frac{\left (6 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d n}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{e x}{d}\right )}{d}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d}+\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{e x}{d}\right )}{d}-\frac{\left (6 b^3 n^3\right ) \int \frac{\text{Li}_3\left (-\frac{e x}{d}\right )}{x} \, dx}{d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d n}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{e x}{d}\right )}{d}-\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{e x}{d}\right )}{d}+\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{e x}{d}\right )}{d}-\frac{6 b^3 n^3 \text{Li}_4\left (-\frac{e x}{d}\right )}{d}\\ \end{align*}
Mathematica [B] time = 0.175878, size = 243, normalized size = 2.15 \[ \frac{-4 b^2 n^2 \left (6 \text{PolyLog}\left (3,-\frac{e x}{d}\right )-6 \log (x) \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log ^2(x) \left (\log (x)-3 \log \left (\frac{e x}{d}+1\right )\right )\right ) \left (-a-b \log \left (c x^n\right )+b n \log (x)\right )+6 b n \left (\log ^2(x)-2 \left (\text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log (x) \log \left (\frac{e x}{d}+1\right )\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2+b^3 n^3 \left (-24 \text{PolyLog}\left (4,-\frac{e x}{d}\right )-12 \log ^2(x) \text{PolyLog}\left (2,-\frac{e x}{d}\right )+24 \log (x) \text{PolyLog}\left (3,-\frac{e x}{d}\right )-4 \log ^3(x) \log \left (\frac{e x}{d}+1\right )+\log ^4(x)\right )-4 \log (d+e x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3+4 \log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3}{4 d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.718, size = 9909, normalized size = 87.7 \begin{align*} \text{output 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*} -a^{3}{\left (\frac{\log \left (e x + d\right )}{d} - \frac{\log \left (x\right )}{d}\right )} + \int \frac{b^{3} \log \left (c\right )^{3} + b^{3} \log \left (x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c\right )^{2} + 3 \, a^{2} b \log \left (c\right ) + 3 \,{\left (b^{3} \log \left (c\right ) + a b^{2}\right )} \log \left (x^{n}\right )^{2} + 3 \,{\left (b^{3} \log \left (c\right )^{2} + 2 \, a b^{2} \log \left (c\right ) + a^{2} b\right )} \log \left (x^{n}\right )}{e x^{2} + d 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^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}}{e x^{2} + d x}, 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 )^{3}}{x \left (d + e x\right )}\, 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 )}^{3}}{{\left (e x + d\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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