Optimal. Leaf size=361 \[ \frac{9 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac{6 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac{3 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}-\frac{3 b^3 n^3 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^3}-\frac{9 b^3 n^3 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^3}-\frac{6 b^3 n^3 \text{PolyLog}\left (4,-\frac{e x}{d}\right )}{d^3}-\frac{3 b^2 n^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^3}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}+\frac{9 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3}+\frac{3 b e n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d^3 n}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 d^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2} \]
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Rubi [A] time = 0.852127, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {2347, 2344, 2302, 30, 2317, 2374, 2383, 6589, 2318, 2319, 2391} \[ \frac{9 b^2 n^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}+\frac{6 b^2 n^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac{3 b n \text{PolyLog}\left (2,-\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}-\frac{3 b^3 n^3 \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^3}-\frac{9 b^3 n^3 \text{PolyLog}\left (3,-\frac{e x}{d}\right )}{d^3}-\frac{6 b^3 n^3 \text{PolyLog}\left (4,-\frac{e x}{d}\right )}{d^3}-\frac{3 b^2 n^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac{\log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^3}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}+\frac{9 b n \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3}+\frac{3 b e n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d^3 n}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 d^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2} \]
Antiderivative was successfully verified.
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Rule 2347
Rule 2344
Rule 2302
Rule 30
Rule 2317
Rule 2374
Rule 2383
Rule 6589
Rule 2318
Rule 2319
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx &=\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2} \, dx}{d}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^3} \, dx}{d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)} \, dx}{d^2}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^2} \, dx}{d^2}-\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2} \, dx}{2 d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx}{d^3}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{d+e x} \, dx}{d^3}-\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)} \, dx}{2 d^2}+\frac{(3 b e n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{d^3}+\frac{(3 b e n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2} \, dx}{2 d^2}\\ &=\frac{3 b e n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{d^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{e x}{d}\right )}{d^3}+\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{b d^3 n}-\frac{(3 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 d^3}+\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^3}+\frac{(3 b e n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx}{2 d^3}-\frac{\left (6 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^3}-\frac{\left (3 b^2 e n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^3}\\ &=\frac{3 b e n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d^3 n}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^3}+\frac{9 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{2 d^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{e x}{d}\right )}{d^3}+\frac{6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^3}-\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^3}-\frac{3 \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 d^3}-\frac{\left (3 b^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^3}+\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^3}+\frac{\left (3 b^3 n^3\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^3}-\frac{\left (6 b^3 n^3\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{d^3}\\ &=\frac{3 b e n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 (d+e x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 d^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d^3 n}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^3}+\frac{9 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{2 d^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{e x}{d}\right )}{d^3}-\frac{3 b^3 n^3 \text{Li}_2\left (-\frac{e x}{d}\right )}{d^3}+\frac{9 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^3}-\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^3}-\frac{6 b^3 n^3 \text{Li}_3\left (-\frac{e x}{d}\right )}{d^3}+\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^3}-\frac{\left (3 b^3 n^3\right ) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx}{d^3}-\frac{\left (6 b^3 n^3\right ) \int \frac{\text{Li}_3\left (-\frac{e x}{d}\right )}{x} \, dx}{d^3}\\ &=\frac{3 b e n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 (d+e x)}-\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 d^3}+\frac{\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}-\frac{e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}+\frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b d^3 n}-\frac{3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^3}+\frac{9 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{e x}{d}\right )}{2 d^3}-\frac{\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac{e x}{d}\right )}{d^3}-\frac{3 b^3 n^3 \text{Li}_2\left (-\frac{e x}{d}\right )}{d^3}+\frac{9 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{e x}{d}\right )}{d^3}-\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^3}-\frac{9 b^3 n^3 \text{Li}_3\left (-\frac{e x}{d}\right )}{d^3}+\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^3}-\frac{6 b^3 n^3 \text{Li}_4\left (-\frac{e x}{d}\right )}{d^3}\\ \end{align*}
Mathematica [A] time = 0.894021, size = 706, normalized size = 1.96 \[ \frac{2 b^2 n^2 \left (6 (d+e x)^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )-6 (d+e x)^2 \left (-2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )+2 \log (x) \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log ^2(x) \log \left (\frac{e x}{d}+1\right )\right )-6 (d+e x) \left (\log (x) \left (e x \log (x)-2 (d+e x) \log \left (\frac{e x}{d}+1\right )\right )-2 (d+e x) \text{PolyLog}\left (2,-\frac{e x}{d}\right )\right )+2 \log ^3(x) (d+e x)^2-3 e x \log ^2(x) (2 d+e x)+6 \log (x) (d+e x) \left ((d+e x) \log \left (\frac{e x}{d}+1\right )+e x\right )-6 (d+e x)^2 \log (d+e x)\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+6 b n \left (-2 (d+e x)^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log ^2(x) (d+e x)^2+(d+e x) (3 (d+e x) \log (d+e x)-d)-\log (x) \left (e x (4 d+3 e x)+2 (d+e x)^2 \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 (-4 (d+e x) \left (6 (d+e x) \text{PolyLog}\left (3,-\frac{e x}{d}\right )-6 \log (x) (d+e x) \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log ^2(x) \left (e x \log (x)-3 (d+e x) \log \left (\frac{e x}{d}+1\right )\right )\right )-2 \left (6 (d+e x)^2 \text{PolyLog}\left (3,-\frac{e x}{d}\right )-6 (\log (x)-1) (d+e x)^2 \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log (x) \left (e x \log ^2(x) (2 d+e x)+6 (d+e x)^2 \log \left (\frac{e x}{d}+1\right )-3 \log (x) (d+e x) \left ((d+e x) \log \left (\frac{e x}{d}+1\right )+e x\right )\right )\right )-4 (d+e x)^2 \left (6 \text{PolyLog}\left (4,-\frac{e x}{d}\right )+3 \log ^2(x) \text{PolyLog}\left (2,-\frac{e x}{d}\right )-6 \log (x) \text{PolyLog}\left (3,-\frac{e x}{d}\right )+\log ^3(x) \log \left (\frac{e x}{d}+1\right )\right )+\log ^4(x) (d+e x)^2\right )+2 d^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3+4 d (d+e x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3+4 \log (x) (d+e x)^2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3-4 (d+e x)^2 \log (d+e x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3}{4 d^3 (d+e x)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.187, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}{x \left ( ex+d \right ) ^{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{1}{2} \, a^{3}{\left (\frac{2 \, e x + 3 \, d}{d^{2} e^{2} x^{2} + 2 \, d^{3} e x + d^{4}} - \frac{2 \, \log \left (e x + d\right )}{d^{3}} + \frac{2 \, \log \left (x\right )}{d^{3}}\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^{3} x^{4} + 3 \, d e^{2} x^{3} + 3 \, d^{2} e x^{2} + d^{3} 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^{3} x^{4} + 3 \, d e^{2} x^{3} + 3 \, d^{2} e x^{2} + d^{3} 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 )^{3}}\, 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 )}^{3} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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