Optimal. Leaf size=412 \[ -\frac {77 b^2 e^5 n^2 x^{2/3}}{120 d^5}+\frac {47 b^2 e^4 n^2 x^{4/3}}{240 d^4}-\frac {3 b^2 e^3 n^2 x^2}{40 d^3}+\frac {b^2 e^2 n^2 x^{8/3}}{40 d^2}+\frac {77 b^2 e^6 n^2 \log \left (d+\frac {e}{x^{2/3}}\right )}{120 d^6}+\frac {b e^5 n \left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 d^6}-\frac {b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{6 d^3}-\frac {b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{8 d^2}+\frac {b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{10 d}+\frac {b e^6 n \log \left (1-\frac {d}{d+\frac {e}{x^{2/3}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 d^6}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}-\frac {b^2 e^6 n^2 \text {Li}_2\left (\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{2 d^6} \]
[Out]
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Rubi [A]
time = 0.60, antiderivative size = 412, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2504, 2445,
2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \begin {gather*} -\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^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} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 46
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rule 2504
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} \text {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) \text {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) \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Leaf count is larger than twice the leaf count of optimal. \(968\) vs. \(2(412)=824\).
time = 0.32, size = 968, normalized size = 2.35 \begin {gather*} \frac {360 a b d e^5 n x^{2/3}-462 b^2 d e^5 n^2 x^{2/3}-180 a b d^2 e^4 n x^{4/3}+141 b^2 d^2 e^4 n^2 x^{4/3}+120 a b d^3 e^3 n x^2-54 b^2 d^3 e^3 n^2 x^2-90 a b d^4 e^2 n x^{8/3}+18 b^2 d^4 e^2 n^2 x^{8/3}+72 a b d^5 e n x^{10/3}+180 a^2 d^6 x^4+822 b^2 e^6 n^2 \log \left (d+\frac {e}{x^{2/3}}\right )+360 b^2 d e^5 n x^{2/3} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )-180 b^2 d^2 e^4 n x^{4/3} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )+120 b^2 d^3 e^3 n x^2 \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )-90 b^2 d^4 e^2 n x^{8/3} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )+72 b^2 d^5 e n x^{10/3} \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )+360 a b d^6 x^4 \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )+180 b^2 d^6 x^4 \log ^2\left (c \left (d+\frac {e}{x^{2/3}}\right )^n\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 )+180 b^2 e^6 n^2 \log ^2\left (\sqrt {e}-\sqrt {-d} \sqrt [3]{x}\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 )+180 b^2 e^6 n^2 \log ^2\left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\right )+360 b^2 e^6 n^2 \log \left (\sqrt {e}+\sqrt {-d} \sqrt [3]{x}\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 (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\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 )-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 (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )\right )-720 b^2 e^6 n^2 \text {Li}_2\left (1+\frac {\sqrt {-d} \sqrt [3]{x}}{\sqrt {e}}\right )}{720 d^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )^{2}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^3\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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