Integrand size = 24, antiderivative size = 773 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\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^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 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 {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 {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 {15 b^3 e^6 n^3 \log (x)}{8 d^6}+\frac {77 b^3 e^6 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{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 ) \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{2 d^6}-\frac {3 b^3 e^6 n^3 \operatorname {PolyLog}\left (2,1+\frac {e}{d x^{2/3}}\right )}{2 d^6}-\frac {3 b^3 e^6 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{2 d^6} \]
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Time = 1.76 (sec) , antiderivative size = 773, normalized size of antiderivative = 1.00, number of steps used = 62, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31, 46} \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=-\frac {3 b^2 e^6 n^2 \operatorname {PolyLog}\left (2,\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 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{40 d^6}-\frac {3 b^3 e^6 n^3 \operatorname {PolyLog}\left (2,\frac {e}{d x^{2/3}}+1\right )}{2 d^6}-\frac {3 b^3 e^6 n^3 \operatorname {PolyLog}\left (3,\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} \]
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Rule 31
Rule 46
Rule 2351
Rule 2354
Rule 2355
Rule 2356
Rule 2379
Rule 2389
Rule 2421
Rule 2438
Rule 2445
Rule 2458
Rule 2504
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {3}{2} \text {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) \text {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) \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \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 )}{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 ) \text {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 ) \text {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 ) \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 )}{10 d^2}+\frac {\left (3 b^2 e^2 n^2\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 )}{10 d^2}+\frac {\left (3 b^2 e^2 n^2\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 )}{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 ) \text {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 ) \text {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 ) \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 )}{10 d^3}+\frac {\left (3 b^2 e^2 n^2\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 )}{8 d^3}-\frac {\left (3 b^2 e^3 n^2\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 )}{10 d^3}-\frac {\left (3 b^2 e^3 n^2\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 )}{8 d^3}-\frac {\left (b^2 e^3 n^2\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 (3 b^3 e^2 n^3\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 )}{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 ) \text {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 ) \text {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 ) \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 )}{10 d^4}-\frac {\left (3 b^2 e^3 n^2\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 )}{8 d^4}-\frac {\left (b^2 e^3 n^2\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 (3 b^2 e^4 n^2\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 )}{10 d^4}+\frac {\left (3 b^2 e^4 n^2\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 )}{8 d^4}+\frac {\left (b^2 e^4 n^2\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 (3 b^2 e^4 n^2\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 )}{4 d^4}-\frac {\left (3 b^3 e^2 n^3\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 )}{40 d^2}+\frac {\left (b^3 e^3 n^3\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 )}{10 d^3}+\frac {\left (b^3 e^3 n^3\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 )}{8 d^3} \\ & = \text {Too large to display} \\ \end{align*}
Result contains complex when optimal does not.
Time = 18.53 (sec) , antiderivative size = 5557, normalized size of antiderivative = 7.19 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\text {Result too large to show} \]
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\[\int x^{3} {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )}^{3}d x\]
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\[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\text {Timed out} \]
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\[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} x^{3} \,d x } \]
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\[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int x^3\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^3 \,d x \]
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