Optimal. Leaf size=765 \[ -\frac {b^3 e^3 n^3}{20 d^3 x}+\frac {3 b^3 e^4 n^3}{10 d^4 x^{2/3}}-\frac {71 b^3 e^5 n^3}{40 d^5 \sqrt [3]{x}}+\frac {71 b^3 e^6 n^3 \log \left (d+e \sqrt [3]{x}\right )}{40 d^6}-\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac {9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac {47 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{40 d^4 x^{2/3}}+\frac {77 b^2 e^5 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6 \sqrt [3]{x}}+\frac {77 b^2 e^6 n^2 \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac {3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac {3 b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6 \sqrt [3]{x}}-\frac {3 b e^6 n \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac {3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )}{d^6}-\frac {15 b^3 e^6 n^3 \log (x)}{8 d^6}-\frac {77 b^3 e^6 n^3 \text {Li}_2\left (\frac {d}{d+e \sqrt [3]{x}}\right )}{20 d^6}+\frac {3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+e \sqrt [3]{x}}\right )}{d^6}+\frac {3 b^3 e^6 n^3 \text {Li}_2\left (1+\frac {e \sqrt [3]{x}}{d}\right )}{d^6}+\frac {3 b^3 e^6 n^3 \text {Li}_3\left (\frac {d}{d+e \sqrt [3]{x}}\right )}{d^6} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.73, antiderivative size = 765, normalized size of antiderivative = 1.00, number of
steps used = 62, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used
= {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438, 2356, 2351, 31, 46}
\begin {gather*} \frac {3 b^2 e^6 n^2 \text {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^6}-\frac {77 b^3 e^6 n^3 \text {PolyLog}\left (2,\frac {d}{d+e \sqrt [3]{x}}\right )}{20 d^6}+\frac {3 b^3 e^6 n^3 \text {PolyLog}\left (2,\frac {e \sqrt [3]{x}}{d}+1\right )}{d^6}+\frac {3 b^3 e^6 n^3 \text {PolyLog}\left (3,\frac {d}{d+e \sqrt [3]{x}}\right )}{d^6}+\frac {77 b^2 e^6 n^2 \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6}+\frac {3 b^2 e^6 n^2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{d^6}+\frac {77 b^2 e^5 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6 \sqrt [3]{x}}-\frac {47 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{40 d^4 x^{2/3}}+\frac {9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}-\frac {3 b e^6 n \log \left (1-\frac {d}{d+e \sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6}-\frac {3 b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6 \sqrt [3]{x}}+\frac {3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac {71 b^3 e^6 n^3 \log \left (d+e \sqrt [3]{x}\right )}{40 d^6}-\frac {15 b^3 e^6 n^3 \log (x)}{8 d^6}-\frac {71 b^3 e^5 n^3}{40 d^5 \sqrt [3]{x}}+\frac {3 b^3 e^4 n^3}{10 d^4 x^{2/3}}-\frac {b^3 e^3 n^3}{20 d^3 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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*} \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx &=3 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^7} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac {1}{2} (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,\sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac {1}{2} (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+e \sqrt [3]{x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\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+e \sqrt [3]{x}\right )}{2 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+e \sqrt [3]{x}\right )}{2 d}\\ &=-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}-\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+e \sqrt [3]{x}\right )}{2 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+e \sqrt [3]{x}\right )}{2 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+e \sqrt [3]{x}\right )}{5 d}\\ &=-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^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}{\left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+e \sqrt [3]{x}\right )}{2 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+e \sqrt [3]{x}\right )}{2 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+e \sqrt [3]{x}\right )}{5 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+e \sqrt [3]{x}\right )}{5 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+e \sqrt [3]{x}\right )}{4 d^2}\\ &=-\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}-\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+e \sqrt [3]{x}\right )}{2 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+e \sqrt [3]{x}\right )}{2 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+e \sqrt [3]{x}\right )}{5 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+e \sqrt [3]{x}\right )}{4 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+e \sqrt [3]{x}\right )}{5 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+e \sqrt [3]{x}\right )}{4 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+e \sqrt [3]{x}\right )}{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+e \sqrt [3]{x}\right )}{20 d^2}\\ &=-\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac {9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac {3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\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+e \sqrt [3]{x}\right )}{2 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+e \sqrt [3]{x}\right )}{2 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+e \sqrt [3]{x}\right )}{5 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+e \sqrt [3]{x}\right )}{4 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+e \sqrt [3]{x}\right )}{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+e \sqrt [3]{x}\right )}{5 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+e \sqrt [3]{x}\right )}{4 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+e \sqrt [3]{x}\right )}{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+e \sqrt [3]{x}\right )}{2 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+e \sqrt [3]{x}\right )}{20 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+e \sqrt [3]{x}\right )}{5 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+e \sqrt [3]{x}\right )}{4 d^3}\\ &=-\frac {b^3 e^3 n^3}{20 d^3 x}+\frac {3 b^3 e^4 n^3}{40 d^4 x^{2/3}}-\frac {3 b^3 e^5 n^3}{20 d^5 \sqrt [3]{x}}+\frac {3 b^3 e^6 n^3 \log \left (d+e \sqrt [3]{x}\right )}{20 d^6}-\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac {9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac {47 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{40 d^4 x^{2/3}}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac {3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac {3 b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}-\frac {b^3 e^6 n^3 \log (x)}{20 d^6}-\frac {\left (3 b e^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^6}+\frac {\left (3 b e^6 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^6}-\frac {\left (3 b^2 e^4 n^2\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+e \sqrt [3]{x}\right )}{5 d^5}-\frac {\left (3 b^2 e^4 n^2\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+e \sqrt [3]{x}\right )}{4 d^5}-\frac {\left (b^2 e^4 n^2\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+e \sqrt [3]{x}\right )}{d^5}-\frac {\left (3 b^2 e^4 n^2\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+e \sqrt [3]{x}\right )}{2 d^5}+\frac {\left (3 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}+\frac {\left (3 b^2 e^5 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 )} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^5}+\frac {\left (3 b^2 e^5 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 )} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^5}+\frac {\left (b^2 e^5 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 )} \, dx,x,d+e \sqrt [3]{x}\right )}{d^5}+\frac {\left (3 b^2 e^5 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 )} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^5}-\frac {\left (b^3 e^3 n^3\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+e \sqrt [3]{x}\right )}{5 d^3}-\frac {\left (b^3 e^3 n^3\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+e \sqrt [3]{x}\right )}{4 d^3}+\frac {\left (3 b^3 e^4 n^3\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{10 d^4}+\frac {\left (3 b^3 e^4 n^3\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{8 d^4}+\frac {\left (b^3 e^4 n^3\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^4}\\ &=-\frac {b^3 e^3 n^3}{20 d^3 x}+\frac {3 b^3 e^4 n^3}{10 d^4 x^{2/3}}-\frac {3 b^3 e^5 n^3}{5 d^5 \sqrt [3]{x}}+\frac {3 b^3 e^6 n^3 \log \left (d+e \sqrt [3]{x}\right )}{5 d^6}-\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac {9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac {47 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{40 d^4 x^{2/3}}+\frac {77 b^2 e^5 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6 \sqrt [3]{x}}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac {3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac {3 b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6 \sqrt [3]{x}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac {3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )}{d^6}-\frac {3 b e^6 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right )}{2 d^6}-\frac {b^3 e^6 n^3 \log (x)}{5 d^6}+\frac {\left (3 e^6\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 d^6}+\frac {\left (3 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^6}+\frac {\left (3 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^6}+\frac {\left (b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}+\frac {\left (3 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^6}-\frac {\left (3 b^2 e^6 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^6}-\frac {\left (3 b^2 e^6 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^6}-\frac {\left (b^2 e^6 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}-\frac {\left (3 b^2 e^6 n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^6}+\frac {\left (3 b^2 e^6 n^2\right ) \text {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+e \sqrt [3]{x}\right )}{d^6}+\frac {\left (3 b^3 e^4 n^3\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+e \sqrt [3]{x}\right )}{10 d^4}+\frac {\left (3 b^3 e^4 n^3\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+e \sqrt [3]{x}\right )}{8 d^4}+\frac {\left (b^3 e^4 n^3\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+e \sqrt [3]{x}\right )}{2 d^4}-\frac {\left (3 b^3 e^5 n^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^6}-\frac {\left (3 b^3 e^5 n^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^6}-\frac {\left (b^3 e^5 n^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}-\frac {\left (3 b^3 e^5 n^3\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^6}-\frac {\left (3 b^3 e^6 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}\\ &=-\frac {b^3 e^3 n^3}{20 d^3 x}+\frac {3 b^3 e^4 n^3}{10 d^4 x^{2/3}}-\frac {71 b^3 e^5 n^3}{40 d^5 \sqrt [3]{x}}+\frac {71 b^3 e^6 n^3 \log \left (d+e \sqrt [3]{x}\right )}{40 d^6}-\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac {9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac {47 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{40 d^4 x^{2/3}}+\frac {77 b^2 e^5 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6 \sqrt [3]{x}}-\frac {77 b e^6 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{40 d^6}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac {3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac {3 b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6 \sqrt [3]{x}}+\frac {e^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 d^6}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac {137 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )}{20 d^6}-\frac {3 b e^6 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right )}{2 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 \sqrt [3]{x}}{d}\right )}{d^6}-\frac {3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e \sqrt [3]{x}}{d}\right )}{d^6}-\frac {\left (3 b^3 e^6 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{5 d^6}-\frac {\left (3 b^3 e^6 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{4 d^6}-\frac {\left (b^3 e^6 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}-\frac {\left (3 b^3 e^6 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{2 d^6}+\frac {\left (3 b^3 e^6 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt [3]{x}\right )}{d^6}\\ &=-\frac {b^3 e^3 n^3}{20 d^3 x}+\frac {3 b^3 e^4 n^3}{10 d^4 x^{2/3}}-\frac {71 b^3 e^5 n^3}{40 d^5 \sqrt [3]{x}}+\frac {71 b^3 e^6 n^3 \log \left (d+e \sqrt [3]{x}\right )}{40 d^6}-\frac {3 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^2 x^{4/3}}+\frac {9 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^3 x}-\frac {47 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{40 d^4 x^{2/3}}+\frac {77 b^2 e^5 n^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{20 d^6 \sqrt [3]{x}}-\frac {77 b e^6 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{40 d^6}-\frac {3 b e n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{10 d x^{5/3}}+\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 d^2 x^{4/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^3 x}+\frac {3 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 d^4 x^{2/3}}-\frac {3 b e^5 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 d^6 \sqrt [3]{x}}+\frac {e^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 d^6}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 x^2}+\frac {137 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt [3]{x}}{d}\right )}{20 d^6}-\frac {3 b e^6 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right )}{2 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 \sqrt [3]{x}}{d}\right )}{20 d^6}-\frac {3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e \sqrt [3]{x}}{d}\right )}{d^6}+\frac {3 b^3 e^6 n^3 \text {Li}_3\left (1+\frac {e \sqrt [3]{x}}{d}\right )}{d^6}\\ \end {align*}
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Mathematica [A]
time = 1.15, size = 1074, normalized size = 1.40 \begin {gather*} -\frac {12 b d^5 e n \sqrt [3]{x} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-15 b d^4 e^2 n x^{2/3} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+20 b d^3 e^3 n x \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-30 b d^2 e^4 n x^{4/3} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+60 b d e^5 n x^{5/3} \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+60 b d^6 n \log \left (d+e \sqrt [3]{x}\right ) \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2-60 b e^6 n x^2 \log \left (d+e \sqrt [3]{x}\right ) \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2+20 d^6 \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3+20 b e^6 n x^2 \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \log (x)+b^2 n^2 \left (a-b n \log \left (d+e \sqrt [3]{x}\right )+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (6 d^4 e^2 x^{2/3}-18 d^3 e^3 x+47 d^2 e^4 x^{4/3}-154 d e^5 x^{5/3}+60 \left (d^6-e^6 x^2\right ) \log ^2\left (d+e \sqrt [3]{x}\right )-274 e^6 x^2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right )+2 \log \left (d+e \sqrt [3]{x}\right ) \left (12 d^5 e \sqrt [3]{x}-15 d^4 e^2 x^{2/3}+20 d^3 e^3 x-30 d^2 e^4 x^{4/3}+60 d e^5 x^{5/3}+137 e^6 x^2+60 e^6 x^2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right )\right )+120 e^6 x^2 \text {Li}_2\left (1+\frac {e \sqrt [3]{x}}{d}\right )\right )+b^3 n^3 \left (3 d^4 e^2 x^{2/3} \left (2-5 \log \left (d+e \sqrt [3]{x}\right )\right ) \log \left (d+e \sqrt [3]{x}\right )+12 d^5 e \sqrt [3]{x} \log ^2\left (d+e \sqrt [3]{x}\right )+20 d^6 \log ^3\left (d+e \sqrt [3]{x}\right )+2 d^3 e^3 x \left (1-9 \log \left (d+e \sqrt [3]{x}\right )+10 \log ^2\left (d+e \sqrt [3]{x}\right )\right )-d^2 e^4 x^{4/3} \left (12-47 \log \left (d+e \sqrt [3]{x}\right )+30 \log ^2\left (d+e \sqrt [3]{x}\right )\right )+d e^5 x^{5/3} \left (71-154 \log \left (d+e \sqrt [3]{x}\right )+60 \log ^2\left (d+e \sqrt [3]{x}\right )\right )+225 e^6 x^2 \left (-\log \left (d+e \sqrt [3]{x}\right )+\log \left (-\frac {e \sqrt [3]{x}}{d}\right )\right )+137 e^6 x^2 \left (\log \left (d+e \sqrt [3]{x}\right ) \left (\log \left (d+e \sqrt [3]{x}\right )-2 \log \left (-\frac {e \sqrt [3]{x}}{d}\right )\right )-2 \text {Li}_2\left (1+\frac {e \sqrt [3]{x}}{d}\right )\right )-20 e^6 x^2 \left (\log ^2\left (d+e \sqrt [3]{x}\right ) \left (\log \left (d+e \sqrt [3]{x}\right )-3 \log \left (-\frac {e \sqrt [3]{x}}{d}\right )\right )-6 \log \left (d+e \sqrt [3]{x}\right ) \text {Li}_2\left (1+\frac {e \sqrt [3]{x}}{d}\right )+6 \text {Li}_3\left (1+\frac {e \sqrt [3]{x}}{d}\right )\right )\right )}{40 d^6 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )^{3}}{x^{3}}\, dx\]
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 \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{3}}{x^{3}}\, dx \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 \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\right )}^3}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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