Optimal. Leaf size=412 \[ \frac {b e^6 n \log \left (1-\frac {d}{d+\frac {e}{x^{2/3}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 d^6}+\frac {b e^5 n x^{2/3} \left (d+\frac {e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 d^6}-\frac {b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{6 d^3}-\frac {b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{8 d^2}+\frac {b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{10 d}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2-\frac {b^2 e^6 n^2 \text {Li}_2\left (\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{2 d^6}+\frac {77 b^2 e^6 n^2 \log \left (d+\frac {e}{x^{2/3}}\right )}{120 d^6}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}-\frac {77 b^2 e^5 n^2 x^{2/3}}{120 d^5}+\frac {47 b^2 e^4 n^2 x^{4/3}}{240 d^4}-\frac {3 b^2 e^3 n^2 x^2}{40 d^3}+\frac {b^2 e^2 n^2 x^{8/3}}{40 d^2} \]
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Rubi [A] time = 1.02, antiderivative size = 436, normalized size of antiderivative = 1.06, number of steps used = 26, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 31
Rule 44
Rule 2301
Rule 2314
Rule 2317
Rule 2319
Rule 2344
Rule 2347
Rule 2391
Rule 2398
Rule 2411
Rule 2454
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*}
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Mathematica [B] time = 0.45, 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 {Li}_2\left (1-\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )+360 b^2 e^6 n^2 \text {Li}_2\left (\frac {1}{2}-\frac {\sqrt {-d} \sqrt [3]{x}}{2 \sqrt {e}}\right )+360 b^2 e^6 n^2 \text {Li}_2\left (\frac {1}{2} \left (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\right )\right )-720 b^2 e^6 n^2 \text {Li}_2\left (\frac {\sqrt [3]{x} \sqrt {-d}}{\sqrt {e}}+1\right )}{720 d^6} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{3} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right )^{2} + 2 \, a b x^{3} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right ) + a^{2} x^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{2} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )+a \right )^{2} x^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, b^{2} n^{2} x^{4} \log \left (d x^{\frac {2}{3}} + e\right )^{2} - \int -\frac {3 \, {\left (b^{2} d \log \relax (c)^{2} + 2 \, a b d \log \relax (c) + a^{2} d\right )} x^{4} + 3 \, {\left (b^{2} e \log \relax (c)^{2} + 2 \, a b e \log \relax (c) + a^{2} e\right )} x^{\frac {10}{3}} - {\left (b^{2} d n x^{4} - 6 \, {\left (b^{2} d \log \relax (c) + a b d\right )} x^{4} - 6 \, {\left (b^{2} e \log \relax (c) + a b e\right )} x^{\frac {10}{3}} + 12 \, {\left (b^{2} d x^{4} + b^{2} e x^{\frac {10}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )\right )} n \log \left (d x^{\frac {2}{3}} + e\right ) + 12 \, {\left (b^{2} d x^{4} + b^{2} e x^{\frac {10}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} - 12 \, {\left ({\left (b^{2} d \log \relax (c) + a b d\right )} x^{4} + {\left (b^{2} e \log \relax (c) + a b e\right )} x^{\frac {10}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )}{3 \, {\left (d x + e x^{\frac {1}{3}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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