Optimal. Leaf size=773 \[ -\frac {3 b^2 e^6 n^2 \text {Li}_2\left (\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 {77 b^2 e^6 n^2 \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 )}{40 d^6}-\frac {3 b^2 e^6 n^2 \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 {77 b^2 e^5 n^2 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 )}{40 d^6}+\frac {47 b^2 e^4 n^2 x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{80 d^4}-\frac {9 b^2 e^3 n^2 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^3}+\frac {3 b^2 e^2 n^2 x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^2}+\frac {3 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}{4 d^6}+\frac {3 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}{4 d^6}-\frac {3 b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{8 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 )^2}{4 d^3}-\frac {3 b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{16 d^2}+\frac {3 b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{20 d}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {77 b^3 e^6 n^3 \text {Li}_2\left (\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{40 d^6}-\frac {3 b^3 e^6 n^3 \text {Li}_2\left (\frac {e}{d x^{2/3}}+1\right )}{2 d^6}-\frac {3 b^3 e^6 n^3 \text {Li}_3\left (\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{2 d^6}-\frac {71 b^3 e^6 n^3 \log \left (d+\frac {e}{x^{2/3}}\right )}{80 d^6}-\frac {15 b^3 e^6 n^3 \log (x)}{8 d^6}+\frac {71 b^3 e^5 n^3 x^{2/3}}{80 d^5}-\frac {3 b^3 e^4 n^3 x^{4/3}}{20 d^4}+\frac {b^3 e^3 n^3 x^2}{40 d^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 3.02, antiderivative size = 746, normalized size of antiderivative = 0.97, number of steps used = 73, number of rules used = 17, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {2454, 2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391, 2319, 2301, 2314, 31, 44} \[ \frac {3 b^2 e^6 n^2 \text {PolyLog}\left (2,\frac {e}{d x^{2/3}}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{2 d^6}-\frac {137 b^3 e^6 n^3 \text {PolyLog}\left (2,\frac {e}{d x^{2/3}}+1\right )}{40 d^6}-\frac {3 b^3 e^6 n^3 \text {PolyLog}\left (3,\frac {e}{d x^{2/3}}+1\right )}{2 d^6}-\frac {137 b^2 e^6 n^2 \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 )}{40 d^6}-\frac {77 b^2 e^5 n^2 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 )}{40 d^6}+\frac {47 b^2 e^4 n^2 x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{80 d^4}-\frac {9 b^2 e^3 n^2 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^3}+\frac {3 b^2 e^2 n^2 x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^2}-\frac {e^6 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{4 d^6}+\frac {77 b e^6 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{80 d^6}+\frac {3 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}{4 d^6}+\frac {3 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}{4 d^6}-\frac {3 b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{8 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 )^2}{4 d^3}-\frac {3 b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{16 d^2}+\frac {3 b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{20 d}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\frac {71 b^3 e^5 n^3 x^{2/3}}{80 d^5}-\frac {3 b^3 e^4 n^3 x^{4/3}}{20 d^4}+\frac {b^3 e^3 n^3 x^2}{40 d^3}-\frac {71 b^3 e^6 n^3 \log \left (d+\frac {e}{x^{2/3}}\right )}{80 d^6}-\frac {15 b^3 e^6 n^3 \log (x)}{8 d^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 30
Rule 31
Rule 44
Rule 2301
Rule 2302
Rule 2314
Rule 2317
Rule 2318
Rule 2319
Rule 2344
Rule 2347
Rule 2374
Rule 2391
Rule 2398
Rule 2411
Rule 2454
Rule 6589
Rubi steps
\begin {align*} \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx &=-\left (\frac {3}{2} \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{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 )^3-\frac {1}{4} (3 b e n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{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 )^3-\frac {1}{4} (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 )^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 )^3-\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 )^6} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{4 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 )^5} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{4 d}\\ &=\frac {3 b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{20 d}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3+\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 )^5} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{4 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 )^4} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{4 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 )^5} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{10 d}\\ &=-\frac {3 b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{16 d^2}+\frac {3 b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{20 d}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3-\frac {\left (3 b e^2 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{4 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 \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{4 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 )^5} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{10 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 )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{10 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 )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{8 d^2}\\ &=\frac {3 b^2 e^2 n^2 x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^2}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 d^3}-\frac {3 b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{16 d^2}+\frac {3 b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{20 d}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^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}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{4 d^4}-\frac {\left (3 b e^4 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 )^2} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{4 d^4}+\frac {\left (3 b^2 e^2 n^2\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 )}{10 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 )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{8 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 \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{10 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 \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{8 d^3}-\frac {\left (b^2 e^3 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 )^3} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{2 d^3}-\frac {\left (3 b^3 e^2 n^3\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 )}{40 d^2}\\ &=-\frac {9 b^2 e^3 n^2 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^3}+\frac {3 b^2 e^2 n^2 x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^2}-\frac {3 b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{8 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 )^2}{4 d^3}-\frac {3 b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{16 d^2}+\frac {3 b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{20 d}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3-\frac {\left (3 b e^4 n\right ) \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+\frac {e}{x^{2/3}}\right )}{4 d^5}+\frac {\left (3 b e^5 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+\frac {e}{x^{2/3}}\right )}{4 d^5}-\frac {\left (3 b^2 e^3 n^2\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 )}{10 d^4}-\frac {\left (3 b^2 e^3 n^2\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 )}{8 d^4}-\frac {\left (b^2 e^3 n^2\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 (3 b^2 e^4 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+\frac {e}{x^{2/3}}\right )}{10 d^4}+\frac {\left (3 b^2 e^4 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+\frac {e}{x^{2/3}}\right )}{8 d^4}+\frac {\left (b^2 e^4 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+\frac {e}{x^{2/3}}\right )}{2 d^4}+\frac {\left (3 b^2 e^4 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+\frac {e}{x^{2/3}}\right )}{4 d^4}-\frac {\left (3 b^3 e^2 n^3\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 )}{40 d^2}+\frac {\left (b^3 e^3 n^3\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 )}{10 d^3}+\frac {\left (b^3 e^3 n^3\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 )}{8 d^3}\\ &=\frac {3 b^3 e^5 n^3 x^{2/3}}{40 d^5}-\frac {3 b^3 e^4 n^3 x^{4/3}}{80 d^4}+\frac {b^3 e^3 n^3 x^2}{40 d^3}-\frac {3 b^3 e^6 n^3 \log \left (d+\frac {e}{x^{2/3}}\right )}{40 d^6}+\frac {47 b^2 e^4 n^2 x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{80 d^4}-\frac {9 b^2 e^3 n^2 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^3}+\frac {3 b^2 e^2 n^2 x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^2}+\frac {3 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}{4 d^6}-\frac {3 b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{8 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 )^2}{4 d^3}-\frac {3 b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{16 d^2}+\frac {3 b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{20 d}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3-\frac {b^3 e^6 n^3 \log (x)}{20 d^6}+\frac {\left (3 b e^5 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+\frac {e}{x^{2/3}}\right )}{4 d^6}-\frac {\left (3 b e^6 n\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{4 d^6}+\frac {\left (3 b^2 e^4 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+\frac {e}{x^{2/3}}\right )}{10 d^5}+\frac {\left (3 b^2 e^4 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+\frac {e}{x^{2/3}}\right )}{8 d^5}+\frac {\left (b^2 e^4 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+\frac {e}{x^{2/3}}\right )}{2 d^5}+\frac {\left (3 b^2 e^4 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+\frac {e}{x^{2/3}}\right )}{4 d^5}-\frac {\left (3 b^2 e^5 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+\frac {e}{x^{2/3}}\right )}{2 d^6}-\frac {\left (3 b^2 e^5 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+\frac {e}{x^{2/3}}\right )}{10 d^5}-\frac {\left (3 b^2 e^5 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+\frac {e}{x^{2/3}}\right )}{8 d^5}-\frac {\left (b^2 e^5 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+\frac {e}{x^{2/3}}\right )}{2 d^5}-\frac {\left (3 b^2 e^5 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+\frac {e}{x^{2/3}}\right )}{4 d^5}+\frac {\left (b^3 e^3 n^3\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 )}{10 d^3}+\frac {\left (b^3 e^3 n^3\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 )}{8 d^3}-\frac {\left (3 b^3 e^4 n^3\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 )}{20 d^4}-\frac {\left (3 b^3 e^4 n^3\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 )}{16 d^4}-\frac {\left (b^3 e^4 n^3\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 {3 b^3 e^5 n^3 x^{2/3}}{10 d^5}-\frac {3 b^3 e^4 n^3 x^{4/3}}{20 d^4}+\frac {b^3 e^3 n^3 x^2}{40 d^3}-\frac {3 b^3 e^6 n^3 \log \left (d+\frac {e}{x^{2/3}}\right )}{10 d^6}-\frac {77 b^2 e^5 n^2 \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 )}{40 d^6}+\frac {47 b^2 e^4 n^2 x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{80 d^4}-\frac {9 b^2 e^3 n^2 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^3}+\frac {3 b^2 e^2 n^2 x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^2}+\frac {3 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}{4 d^6}-\frac {3 b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{8 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 )^2}{4 d^3}-\frac {3 b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{16 d^2}+\frac {3 b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{20 d}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3-\frac {3 b^2 e^6 n^2 \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 {3 b e^6 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \log \left (-\frac {e}{d x^{2/3}}\right )}{4 d^6}-\frac {b^3 e^6 n^3 \log (x)}{5 d^6}-\frac {\left (3 e^6\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{4 d^6}-\frac {\left (3 b^2 e^5 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+\frac {e}{x^{2/3}}\right )}{10 d^6}-\frac {\left (3 b^2 e^5 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+\frac {e}{x^{2/3}}\right )}{8 d^6}-\frac {\left (b^2 e^5 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+\frac {e}{x^{2/3}}\right )}{2 d^6}-\frac {\left (3 b^2 e^5 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+\frac {e}{x^{2/3}}\right )}{4 d^6}+\frac {\left (3 b^2 e^6 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{10 d^6}+\frac {\left (3 b^2 e^6 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{8 d^6}+\frac {\left (b^2 e^6 n^2\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 (3 b^2 e^6 n^2\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{4 d^6}-\frac {\left (3 b^2 e^6 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+\frac {e}{x^{2/3}}\right )}{2 d^6}-\frac {\left (3 b^3 e^4 n^3\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 )}{20 d^4}-\frac {\left (3 b^3 e^4 n^3\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 )}{16 d^4}-\frac {\left (b^3 e^4 n^3\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 (3 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{10 d^6}+\frac {\left (3 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{8 d^6}+\frac {\left (b^3 e^5 n^3\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 {\left (3 b^3 e^5 n^3\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{4 d^6}+\frac {\left (3 b^3 e^6 n^3\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 {71 b^3 e^5 n^3 x^{2/3}}{80 d^5}-\frac {3 b^3 e^4 n^3 x^{4/3}}{20 d^4}+\frac {b^3 e^3 n^3 x^2}{40 d^3}-\frac {71 b^3 e^6 n^3 \log \left (d+\frac {e}{x^{2/3}}\right )}{80 d^6}-\frac {77 b^2 e^5 n^2 \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 )}{40 d^6}+\frac {47 b^2 e^4 n^2 x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{80 d^4}-\frac {9 b^2 e^3 n^2 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^3}+\frac {3 b^2 e^2 n^2 x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^2}+\frac {77 b e^6 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{80 d^6}+\frac {3 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}{4 d^6}-\frac {3 b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{8 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 )^2}{4 d^3}-\frac {3 b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{16 d^2}+\frac {3 b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{20 d}-\frac {e^6 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{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 )^3-\frac {137 b^2 e^6 n^2 \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 )}{40 d^6}+\frac {3 b e^6 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \log \left (-\frac {e}{d x^{2/3}}\right )}{4 d^6}-\frac {15 b^3 e^6 n^3 \log (x)}{8 d^6}-\frac {3 b^3 e^6 n^3 \text {Li}_2\left (1+\frac {e}{d x^{2/3}}\right )}{2 d^6}+\frac {3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e}{d x^{2/3}}\right )}{2 d^6}+\frac {\left (3 b^3 e^6 n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{10 d^6}+\frac {\left (3 b^3 e^6 n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{8 d^6}+\frac {\left (b^3 e^6 n^3\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 {\left (3 b^3 e^6 n^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{4 d^6}-\frac {\left (3 b^3 e^6 n^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{x^{2/3}}\right )}{2 d^6}\\ &=\frac {71 b^3 e^5 n^3 x^{2/3}}{80 d^5}-\frac {3 b^3 e^4 n^3 x^{4/3}}{20 d^4}+\frac {b^3 e^3 n^3 x^2}{40 d^3}-\frac {71 b^3 e^6 n^3 \log \left (d+\frac {e}{x^{2/3}}\right )}{80 d^6}-\frac {77 b^2 e^5 n^2 \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 )}{40 d^6}+\frac {47 b^2 e^4 n^2 x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{80 d^4}-\frac {9 b^2 e^3 n^2 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^3}+\frac {3 b^2 e^2 n^2 x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^2}+\frac {77 b e^6 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{80 d^6}+\frac {3 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}{4 d^6}-\frac {3 b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{8 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 )^2}{4 d^3}-\frac {3 b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{16 d^2}+\frac {3 b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{20 d}-\frac {e^6 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3}{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 )^3-\frac {137 b^2 e^6 n^2 \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 )}{40 d^6}+\frac {3 b e^6 n \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \log \left (-\frac {e}{d x^{2/3}}\right )}{4 d^6}-\frac {15 b^3 e^6 n^3 \log (x)}{8 d^6}-\frac {137 b^3 e^6 n^3 \text {Li}_2\left (1+\frac {e}{d x^{2/3}}\right )}{40 d^6}+\frac {3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e}{d x^{2/3}}\right )}{2 d^6}-\frac {3 b^3 e^6 n^3 \text {Li}_3\left (1+\frac {e}{d x^{2/3}}\right )}{2 d^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.59, size = 1014, normalized size = 1.31 \[ \frac {20 x^4 \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 d^6+60 b n x^4 \log \left (d+\frac {e}{x^{2/3}}\right ) \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 d^6+12 b e n x^{10/3} \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 d^5-15 b e^2 n x^{8/3} \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 d^4+20 b e^3 n x^2 \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 d^3-30 b e^4 n x^{4/3} \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 d^2+60 b e^5 n x^{2/3} \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 d-60 b e^6 n \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2 \log \left (x^{2/3} d+e\right )+b^2 n^2 \left (a-b n \log \left (d+\frac {e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \left (-274 \log \left (-\frac {e}{d x^{2/3}}\right ) e^6+120 \text {Li}_2\left (\frac {e}{d x^{2/3}}+1\right ) e^6+d x^{2/3} \left (6 x^2 d^3-18 e x^{4/3} d^2+47 e^2 x^{2/3} d-154 e^3\right ) e^2+2 \log \left (d+\frac {e}{x^{2/3}}\right ) \left (12 x^{10/3} d^5-15 e x^{8/3} d^4+20 e^2 x^2 d^3-30 e^3 x^{4/3} d^2+60 e^4 x^{2/3} d+137 e^5+60 e^5 \log \left (-\frac {e}{d x^{2/3}}\right )\right ) e-60 \left (e^6-d^6 x^4\right ) \log ^2\left (d+\frac {e}{x^{2/3}}\right )\right )+b^3 n^3 \left (20 x^4 \log ^3\left (d+\frac {e}{x^{2/3}}\right ) d^6+12 e x^{10/3} \log ^2\left (d+\frac {e}{x^{2/3}}\right ) d^5+3 e^2 x^{8/3} \left (2-5 \log \left (d+\frac {e}{x^{2/3}}\right )\right ) \log \left (d+\frac {e}{x^{2/3}}\right ) d^4+2 e^3 x^2 \left (10 \log ^2\left (d+\frac {e}{x^{2/3}}\right )-9 \log \left (d+\frac {e}{x^{2/3}}\right )+1\right ) d^3-e^4 x^{4/3} \left (30 \log ^2\left (d+\frac {e}{x^{2/3}}\right )-47 \log \left (d+\frac {e}{x^{2/3}}\right )+12\right ) d^2+e^5 x^{2/3} \left (60 \log ^2\left (d+\frac {e}{x^{2/3}}\right )-154 \log \left (d+\frac {e}{x^{2/3}}\right )+71\right ) d+225 e^6 \left (\log \left (-\frac {e}{d x^{2/3}}\right )-\log \left (d+\frac {e}{x^{2/3}}\right )\right )+137 e^6 \left (\log \left (d+\frac {e}{x^{2/3}}\right ) \left (\log \left (d+\frac {e}{x^{2/3}}\right )-2 \log \left (-\frac {e}{d x^{2/3}}\right )\right )-2 \text {Li}_2\left (\frac {e}{d x^{2/3}}+1\right )\right )-20 e^6 \left (\left (\log \left (d+\frac {e}{x^{2/3}}\right )-3 \log \left (-\frac {e}{d x^{2/3}}\right )\right ) \log ^2\left (d+\frac {e}{x^{2/3}}\right )-6 \text {Li}_2\left (\frac {e}{d x^{2/3}}+1\right ) \log \left (d+\frac {e}{x^{2/3}}\right )+6 \text {Li}_3\left (\frac {e}{d x^{2/3}}+1\right )\right )\right )}{80 d^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{3} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right )^{3} + 3 \, a b^{2} x^{3} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right )^{2} + 3 \, a^{2} b x^{3} \log \left (c \left (\frac {d x + e x^{\frac {1}{3}}}{x}\right )^{n}\right ) + a^{3} x^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )+a \right )^{3} x^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, b^{3} n^{3} x^{4} \log \left (d x^{\frac {2}{3}} + e\right )^{3} - \int -\frac {2 \, {\left (b^{3} d \log \relax (c)^{3} + 3 \, a b^{2} d \log \relax (c)^{2} + 3 \, a^{2} b d \log \relax (c) + a^{3} d\right )} x^{4} - {\left (b^{3} d n x^{4} - 6 \, {\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x^{4} - 6 \, {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {10}{3}} + 12 \, {\left (b^{3} d x^{4} + b^{3} e x^{\frac {10}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )\right )} n^{2} \log \left (d x^{\frac {2}{3}} + e\right )^{2} + 2 \, {\left (b^{3} e \log \relax (c)^{3} + 3 \, a b^{2} e \log \relax (c)^{2} + 3 \, a^{2} b e \log \relax (c) + a^{3} e\right )} x^{\frac {10}{3}} - 16 \, {\left (b^{3} d x^{4} + b^{3} e x^{\frac {10}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{3} + 6 \, {\left ({\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d\right )} x^{4} + {\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x^{\frac {10}{3}} + 4 \, {\left (b^{3} d x^{4} + b^{3} e x^{\frac {10}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} - 4 \, {\left ({\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x^{4} + {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {10}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )\right )} n \log \left (d x^{\frac {2}{3}} + e\right ) + 24 \, {\left ({\left (b^{3} d \log \relax (c) + a b^{2} d\right )} x^{4} + {\left (b^{3} e \log \relax (c) + a b^{2} e\right )} x^{\frac {10}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )^{2} - 12 \, {\left ({\left (b^{3} d \log \relax (c)^{2} + 2 \, a b^{2} d \log \relax (c) + a^{2} b d\right )} x^{4} + {\left (b^{3} e \log \relax (c)^{2} + 2 \, a b^{2} e \log \relax (c) + a^{2} b e\right )} x^{\frac {10}{3}}\right )} \log \left (x^{\frac {1}{3} \, n}\right )}{2 \, {\left (d x + e x^{\frac {1}{3}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________