Optimal. Leaf size=451 \[ -\frac{3 b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3}+\frac{3 b^3 e^3 n^3 \text{PolyLog}\left (2,\frac{d}{d+e x^{2/3}}\right )}{2 d^3}-\frac{3 b^3 e^3 n^3 \text{PolyLog}\left (2,\frac{e x^{2/3}}{d}+1\right )}{d^3}-\frac{3 b^3 e^3 n^3 \text{PolyLog}\left (3,\frac{d}{d+e x^{2/3}}\right )}{d^3}-\frac{3 b^2 e^3 n^2 \log \left (1-\frac{d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3}-\frac{3 b^2 e^3 n^2 \log \left (-\frac{e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3}-\frac{3 b^2 e^2 n^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3 x^{2/3}}+\frac{3 b e^3 n \log \left (1-\frac{d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3}+\frac{3 b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3 x^{2/3}}-\frac{3 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d x^{4/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 x^2}+\frac{b^3 e^3 n^3 \log (x)}{d^3} \]
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Rubi [A] time = 1.00763, antiderivative size = 428, normalized size of antiderivative = 0.95, number of steps used = 22, number of rules used = 16, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {2454, 2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391, 2319, 2301, 2314, 31} \[ \frac{3 b^2 e^3 n^2 \text{PolyLog}\left (2,\frac{e x^{2/3}}{d}+1\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^3}-\frac{9 b^3 e^3 n^3 \text{PolyLog}\left (2,\frac{e x^{2/3}}{d}+1\right )}{2 d^3}-\frac{3 b^3 e^3 n^3 \text{PolyLog}\left (3,\frac{e x^{2/3}}{d}+1\right )}{d^3}-\frac{9 b^2 e^3 n^2 \log \left (-\frac{e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3}-\frac{3 b^2 e^2 n^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3 x^{2/3}}-\frac{e^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 d^3}+\frac{3 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d^3}+\frac{3 b e^3 n \log \left (-\frac{e x^{2/3}}{d}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3}+\frac{3 b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3 x^{2/3}}-\frac{3 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d x^{4/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 x^2}+\frac{b^3 e^3 n^3 \log (x)}{d^3} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2398
Rule 2411
Rule 2347
Rule 2344
Rule 2302
Rule 30
Rule 2317
Rule 2374
Rule 6589
Rule 2318
Rule 2391
Rule 2319
Rule 2301
Rule 2314
Rule 31
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^3} \, dx &=\frac{3}{2} \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^4} \, dx,x,x^{2/3}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 x^2}+\frac{1}{2} (3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 (d+e x)} \, dx,x,x^{2/3}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 x^2}+\frac{1}{2} (3 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e x^{2/3}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 x^2}+\frac{(3 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+e x^{2/3}\right )}{2 d}-\frac{(3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e x^{2/3}\right )}{2 d}\\ &=-\frac{3 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d x^{4/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 x^2}-\frac{(3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e x^{2/3}\right )}{2 d^2}+\frac{\left (3 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+e x^{2/3}\right )}{2 d^2}+\frac{\left (3 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e x^{2/3}\right )}{2 d}\\ &=-\frac{3 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d x^{4/3}}+\frac{3 b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3 x^{2/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 x^2}+\frac{\left (3 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x^{2/3}\right )}{2 d^3}-\frac{\left (3 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+e x^{2/3}\right )}{2 d^3}+\frac{\left (3 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e x^{2/3}\right )}{2 d^2}-\frac{\left (3 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x^{2/3}\right )}{d^3}-\frac{\left (3 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+e x^{2/3}\right )}{2 d^2}\\ &=-\frac{3 b^2 e^2 n^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3 x^{2/3}}-\frac{3 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d x^{4/3}}+\frac{3 b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3 x^{2/3}}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 x^2}-\frac{3 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \log \left (-\frac{e x^{2/3}}{d}\right )}{d^3}+\frac{3 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \log \left (-\frac{e x^{2/3}}{d}\right )}{2 d^3}-\frac{\left (3 e^3\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3}-\frac{\left (3 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x^{2/3}\right )}{2 d^3}+\frac{\left (3 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x^{2/3}\right )}{2 d^3}-\frac{\left (3 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x^{2/3}\right )}{d^3}+\frac{\left (3 b^3 e^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e x^{2/3}\right )}{2 d^3}+\frac{\left (3 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x^{2/3}\right )}{d^3}\\ &=-\frac{3 b^2 e^2 n^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3 x^{2/3}}+\frac{3 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d^3}-\frac{3 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d x^{4/3}}+\frac{3 b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3 x^{2/3}}-\frac{e^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 d^3}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 x^2}-\frac{9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \log \left (-\frac{e x^{2/3}}{d}\right )}{2 d^3}+\frac{3 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \log \left (-\frac{e x^{2/3}}{d}\right )}{2 d^3}+\frac{b^3 e^3 n^3 \log (x)}{d^3}-\frac{3 b^3 e^3 n^3 \text{Li}_2\left (1+\frac{e x^{2/3}}{d}\right )}{d^3}+\frac{3 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e x^{2/3}}{d}\right )}{d^3}+\frac{\left (3 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e x^{2/3}\right )}{2 d^3}-\frac{\left (3 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x^{2/3}\right )}{d^3}\\ &=-\frac{3 b^2 e^2 n^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3 x^{2/3}}+\frac{3 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d^3}-\frac{3 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d x^{4/3}}+\frac{3 b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3 x^{2/3}}-\frac{e^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 d^3}-\frac{\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 x^2}-\frac{9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \log \left (-\frac{e x^{2/3}}{d}\right )}{2 d^3}+\frac{3 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \log \left (-\frac{e x^{2/3}}{d}\right )}{2 d^3}+\frac{b^3 e^3 n^3 \log (x)}{d^3}-\frac{9 b^3 e^3 n^3 \text{Li}_2\left (1+\frac{e x^{2/3}}{d}\right )}{2 d^3}+\frac{3 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e x^{2/3}}{d}\right )}{d^3}-\frac{3 b^3 e^3 n^3 \text{Li}_3\left (1+\frac{e x^{2/3}}{d}\right )}{d^3}\\ \end{align*}
Mathematica [A] time = 0.781538, size = 764, normalized size = 1.69 \[ \frac{-6 b^2 n^2 \left (-2 e^3 x^2 \text{PolyLog}\left (2,\frac{e x^{2/3}}{d}+1\right )+\left (d^3+e^3 x^2\right ) \log ^2\left (d+e x^{2/3}\right )+\log \left (d+e x^{2/3}\right ) \left (d^2 e x^{2/3}-2 d e^2 x^{4/3}-2 e^3 x^2 \log \left (-\frac{e x^{2/3}}{d}\right )-3 e^3 x^2\right )+e^2 x^{4/3} \left (3 e x^{2/3} \log \left (-\frac{e x^{2/3}}{d}\right )+d\right )\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )+b^3 n^3 \left (-12 e^3 x^2 \text{PolyLog}\left (3,\frac{e x^{2/3}}{d}+1\right )+6 e^3 x^2 \left (2 \log \left (d+e x^{2/3}\right )-3\right ) \text{PolyLog}\left (2,\frac{e x^{2/3}}{d}+1\right )-3 d^2 e x^{2/3} \log ^2\left (d+e x^{2/3}\right )-2 d^3 \log ^3\left (d+e x^{2/3}\right )-2 e^3 x^2 \log ^3\left (d+e x^{2/3}\right )+9 e^3 x^2 \log ^2\left (d+e x^{2/3}\right )+6 e^3 x^2 \log ^2\left (d+e x^{2/3}\right ) \log \left (-\frac{e x^{2/3}}{d}\right )+6 d e^2 x^{4/3} \log ^2\left (d+e x^{2/3}\right )-6 e^3 x^2 \log \left (d+e x^{2/3}\right )+6 e^3 x^2 \log \left (-\frac{e x^{2/3}}{d}\right )-18 e^3 x^2 \log \left (d+e x^{2/3}\right ) \log \left (-\frac{e x^{2/3}}{d}\right )-6 d e^2 x^{4/3} \log \left (d+e x^{2/3}\right )\right )-2 d^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^3-6 b d^3 n \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2-3 b d^2 e n x^{2/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2+6 b d e^2 n x^{4/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2-6 b e^3 n x^2 \log \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2+4 b e^3 n x^2 \log (x) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )-b n \log \left (d+e x^{2/3}\right )\right )^2}{4 d^3 x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.399, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( a+b\ln \left ( c \left ( d+e{x}^{{\frac{2}{3}}} \right ) ^{n} \right ) \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{b^{3} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n}\right )^{3}}{2 \, x^{2}} + \int \frac{{\left (b^{3} e n x + 3 \,{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} x + 3 \,{\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x^{\frac{1}{3}}\right )} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n}\right )^{2} +{\left (b^{3} e \log \left (c\right )^{3} + 3 \, a b^{2} e \log \left (c\right )^{2} + 3 \, a^{2} b e \log \left (c\right ) + a^{3} e\right )} x + 3 \,{\left ({\left (b^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} e \log \left (c\right ) + a^{2} b e\right )} x +{\left (b^{3} d \log \left (c\right )^{2} + 2 \, a b^{2} d \log \left (c\right ) + a^{2} b d\right )} x^{\frac{1}{3}}\right )} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n}\right ) +{\left (b^{3} d \log \left (c\right )^{3} + 3 \, a b^{2} d \log \left (c\right )^{2} + 3 \, a^{2} b d \log \left (c\right ) + a^{3} d\right )} x^{\frac{1}{3}}}{e x^{4} + d x^{\frac{10}{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^{3} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + a^{3}}{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 ({\left (e x^{\frac{2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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