Optimal. Leaf size=412 \[ -\frac{b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{d}{d+\frac{e}{x^{2/3}}}\right )}{2 d^6}+\frac{b e^6 n \log \left (1-\frac{d}{d+\frac{e}{x^{2/3}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^6}+\frac{b e^5 n x^{2/3} \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^6}-\frac{b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{4 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{6 d^3}-\frac{b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{8 d^2}+\frac{b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{10 d}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{77 b^2 e^5 n^2 x^{2/3}}{120 d^5}+\frac{47 b^2 e^4 n^2 x^{4/3}}{240 d^4}-\frac{3 b^2 e^3 n^2 x^2}{40 d^3}+\frac{b^2 e^2 n^2 x^{8/3}}{40 d^2}+\frac{77 b^2 e^6 n^2 \log \left (d+\frac{e}{x^{2/3}}\right )}{120 d^6}+\frac{137 b^2 e^6 n^2 \log (x)}{180 d^6} \]
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Rubi [A] time = 1.01667, antiderivative size = 436, normalized size of antiderivative = 1.06, number of steps used = 26, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2319, 44} \[ \frac{b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{e}{d x^{2/3}}+1\right )}{2 d^6}-\frac{e^6 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 d^6}+\frac{b e^6 n \log \left (-\frac{e}{d x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^6}+\frac{b e^5 n x^{2/3} \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^6}-\frac{b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{4 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{6 d^3}-\frac{b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{8 d^2}+\frac{b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{10 d}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{77 b^2 e^5 n^2 x^{2/3}}{120 d^5}+\frac{47 b^2 e^4 n^2 x^{4/3}}{240 d^4}-\frac{3 b^2 e^3 n^2 x^2}{40 d^3}+\frac{b^2 e^2 n^2 x^{8/3}}{40 d^2}+\frac{77 b^2 e^6 n^2 \log \left (d+\frac{e}{x^{2/3}}\right )}{120 d^6}+\frac{137 b^2 e^6 n^2 \log (x)}{180 d^6} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2398
Rule 2411
Rule 2347
Rule 2344
Rule 2301
Rule 2317
Rule 2391
Rule 2314
Rule 31
Rule 2319
Rule 44
Rubi steps
\begin{align*} \int x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 \, dx &=-\left (\frac{3}{2} \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^7} \, dx,x,\frac{1}{x^{2/3}}\right )\right )\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{1}{2} (b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^6 (d+e x)} \, dx,x,\frac{1}{x^{2/3}}\right )\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{1}{2} (b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^6} \, dx,x,d+\frac{e}{x^{2/3}}\right )\\ &=\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{(b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^6} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d}+\frac{(b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^5} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d}\\ &=\frac{b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{10 d}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{(b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^5} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d^2}-\frac{\left (b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d^2}-\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^5} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{10 d}\\ &=-\frac{b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{8 d^2}+\frac{b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{10 d}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2-\frac{\left (b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d^3}+\frac{\left (b e^3 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d^3}-\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{e^5}{d (d-x)^5}-\frac{e^5}{d^2 (d-x)^4}-\frac{e^5}{d^3 (d-x)^3}-\frac{e^5}{d^4 (d-x)^2}-\frac{e^5}{d^5 (d-x)}-\frac{e^5}{d^5 x}\right ) \, dx,x,d+\frac{e}{x^{2/3}}\right )}{10 d}+\frac{\left (b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{8 d^2}\\ &=-\frac{b^2 e^5 n^2 x^{2/3}}{10 d^5}+\frac{b^2 e^4 n^2 x^{4/3}}{20 d^4}-\frac{b^2 e^3 n^2 x^2}{30 d^3}+\frac{b^2 e^2 n^2 x^{8/3}}{40 d^2}+\frac{b^2 e^6 n^2 \log \left (d+\frac{e}{x^{2/3}}\right )}{10 d^6}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{6 d^3}-\frac{b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{8 d^2}+\frac{b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{10 d}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{b^2 e^6 n^2 \log (x)}{15 d^6}+\frac{\left (b e^3 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d^4}-\frac{\left (b e^4 n\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+\frac{e}{x^{2/3}}\right )}{2 d^4}+\frac{\left (b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{e^4}{d (d-x)^4}+\frac{e^4}{d^2 (d-x)^3}+\frac{e^4}{d^3 (d-x)^2}+\frac{e^4}{d^4 (d-x)}+\frac{e^4}{d^4 x}\right ) \, dx,x,d+\frac{e}{x^{2/3}}\right )}{8 d^2}-\frac{\left (b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{6 d^3}\\ &=-\frac{9 b^2 e^5 n^2 x^{2/3}}{40 d^5}+\frac{9 b^2 e^4 n^2 x^{4/3}}{80 d^4}-\frac{3 b^2 e^3 n^2 x^2}{40 d^3}+\frac{b^2 e^2 n^2 x^{8/3}}{40 d^2}+\frac{9 b^2 e^6 n^2 \log \left (d+\frac{e}{x^{2/3}}\right )}{40 d^6}-\frac{b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{4 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{6 d^3}-\frac{b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{8 d^2}+\frac{b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{10 d}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{3 b^2 e^6 n^2 \log (x)}{20 d^6}-\frac{\left (b e^4 n\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+\frac{e}{x^{2/3}}\right )}{2 d^5}+\frac{\left (b e^5 n\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+\frac{e}{x^{2/3}}\right )}{2 d^5}-\frac{\left (b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{e^3}{d (d-x)^3}-\frac{e^3}{d^2 (d-x)^2}-\frac{e^3}{d^3 (d-x)}-\frac{e^3}{d^3 x}\right ) \, dx,x,d+\frac{e}{x^{2/3}}\right )}{6 d^3}+\frac{\left (b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{4 d^4}\\ &=-\frac{47 b^2 e^5 n^2 x^{2/3}}{120 d^5}+\frac{47 b^2 e^4 n^2 x^{4/3}}{240 d^4}-\frac{3 b^2 e^3 n^2 x^2}{40 d^3}+\frac{b^2 e^2 n^2 x^{8/3}}{40 d^2}+\frac{47 b^2 e^6 n^2 \log \left (d+\frac{e}{x^{2/3}}\right )}{120 d^6}+\frac{b e^5 n \left (d+\frac{e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^6}-\frac{b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{4 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{6 d^3}-\frac{b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{8 d^2}+\frac{b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{10 d}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{47 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac{\left (b e^5 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d^6}-\frac{\left (b e^6 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d^6}+\frac{\left (b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{e^2}{d (d-x)^2}+\frac{e^2}{d^2 (d-x)}+\frac{e^2}{d^2 x}\right ) \, dx,x,d+\frac{e}{x^{2/3}}\right )}{4 d^4}-\frac{\left (b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d^6}\\ &=-\frac{77 b^2 e^5 n^2 x^{2/3}}{120 d^5}+\frac{47 b^2 e^4 n^2 x^{4/3}}{240 d^4}-\frac{3 b^2 e^3 n^2 x^2}{40 d^3}+\frac{b^2 e^2 n^2 x^{8/3}}{40 d^2}+\frac{77 b^2 e^6 n^2 \log \left (d+\frac{e}{x^{2/3}}\right )}{120 d^6}+\frac{b e^5 n \left (d+\frac{e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^6}-\frac{b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{4 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{6 d^3}-\frac{b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{8 d^2}+\frac{b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{10 d}-\frac{e^6 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 d^6}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{b e^6 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac{e}{d x^{2/3}}\right )}{2 d^6}+\frac{137 b^2 e^6 n^2 \log (x)}{180 d^6}-\frac{\left (b^2 e^6 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{x^{2/3}}\right )}{2 d^6}\\ &=-\frac{77 b^2 e^5 n^2 x^{2/3}}{120 d^5}+\frac{47 b^2 e^4 n^2 x^{4/3}}{240 d^4}-\frac{3 b^2 e^3 n^2 x^2}{40 d^3}+\frac{b^2 e^2 n^2 x^{8/3}}{40 d^2}+\frac{77 b^2 e^6 n^2 \log \left (d+\frac{e}{x^{2/3}}\right )}{120 d^6}+\frac{b e^5 n \left (d+\frac{e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{2 d^6}-\frac{b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{4 d^4}+\frac{b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{6 d^3}-\frac{b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{8 d^2}+\frac{b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{10 d}-\frac{e^6 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{4 d^6}+\frac{1}{4} x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2+\frac{b e^6 n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac{e}{d x^{2/3}}\right )}{2 d^6}+\frac{137 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac{b^2 e^6 n^2 \text{Li}_2\left (1+\frac{e}{d x^{2/3}}\right )}{2 d^6}\\ \end{align*}
Mathematica [B] time = 0.434584, size = 968, normalized size = 2.35 \[ \frac{180 a^2 x^4 d^6+180 b^2 x^4 \log ^2\left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) d^6+360 a b x^4 \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) d^6+72 a b e n x^{10/3} d^5+72 b^2 e n x^{10/3} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) d^5+18 b^2 e^2 n^2 x^{8/3} d^4-90 a b e^2 n x^{8/3} d^4-90 b^2 e^2 n x^{8/3} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) d^4-54 b^2 e^3 n^2 x^2 d^3+120 a b e^3 n x^2 d^3+120 b^2 e^3 n x^2 \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) d^3+141 b^2 e^4 n^2 x^{4/3} d^2-180 a b e^4 n x^{4/3} d^2-180 b^2 e^4 n x^{4/3} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) d^2+360 b^2 e^5 n x^{2/3} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) d-462 b^2 e^5 n^2 x^{2/3} d+360 a b e^5 n x^{2/3} d+180 b^2 e^6 n^2 \log ^2\left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )+180 b^2 e^6 n^2 \log ^2\left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right )+822 b^2 e^6 n^2 \log \left (d+\frac{e}{x^{2/3}}\right )-360 a b e^6 n \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )-360 b^2 e^6 n \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )-360 a b e^6 n \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right )-360 b^2 e^6 n \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right ) \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right )+360 b^2 e^6 n^2 \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )+360 b^2 e^6 n^2 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2} \left (\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )\right )-720 b^2 e^6 n^2 \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )-720 b^2 e^6 n^2 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+548 b^2 e^6 n^2 \log (x)-720 b^2 e^6 n^2 \text{PolyLog}\left (2,1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+360 b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )+360 b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )\right )-720 b^2 e^6 n^2 \text{PolyLog}\left (2,\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )}{720 d^6} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.359, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{n} \right ) \right ) ^{2}\, 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}{4} \, b^{2} x^{4} \log \left ({\left (d x^{\frac{2}{3}} + e\right )}^{n}\right )^{2} - \int -\frac{3 \,{\left (b^{2} d \log \left (c\right )^{2} + 2 \, a b d \log \left (c\right ) + a^{2} d\right )} x^{4} + 3 \,{\left (b^{2} e \log \left (c\right )^{2} + 2 \, a b e \log \left (c\right ) + a^{2} e\right )} x^{\frac{10}{3}} + 12 \,{\left (b^{2} d x^{4} + b^{2} e x^{\frac{10}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )^{2} -{\left (b^{2} d n x^{4} - 6 \,{\left (b^{2} d \log \left (c\right ) + a b d\right )} x^{4} - 6 \,{\left (b^{2} e \log \left (c\right ) + a b e\right )} x^{\frac{10}{3}} + 12 \,{\left (b^{2} d x^{4} + b^{2} e x^{\frac{10}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )\right )} \log \left ({\left (d x^{\frac{2}{3}} + e\right )}^{n}\right ) - 12 \,{\left ({\left (b^{2} d \log \left (c\right ) + a b d\right )} x^{4} +{\left (b^{2} e \log \left (c\right ) + a b e\right )} x^{\frac{10}{3}}\right )} \log \left (x^{\frac{1}{3} \, n}\right )}{3 \,{\left (d x + e x^{\frac{1}{3}}\right )}}\,{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 (b^{2} x^{3} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right )^{2} + 2 \, a b x^{3} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right ) + a^{2} 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{\left (b \log \left (c{\left (d + \frac{e}{x^{\frac{2}{3}}}\right )}^{n}\right ) + a\right )}^{2} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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