Integrand size = 24, antiderivative size = 913 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=-\frac {45 b^3 d^4 n^3 \left (d+e x^{2/3}\right )^2}{16 e^6}+\frac {10 b^3 d^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^6}-\frac {45 b^3 d^2 n^3 \left (d+e x^{2/3}\right )^4}{128 e^6}+\frac {9 b^3 d n^3 \left (d+e x^{2/3}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e x^{2/3}\right )^6}{144 e^6}-\frac {9 a b^2 d^5 n^2 x^{2/3}}{e^5}+\frac {9 b^3 d^5 n^3 x^{2/3}}{e^5}-\frac {9 b^3 d^5 n^2 \left (d+e x^{2/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e^6}+\frac {45 b^2 d^4 n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 e^6}-\frac {10 b^2 d^3 n^2 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^6}+\frac {45 b^2 d^2 n^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{32 e^6}-\frac {9 b^2 d n^2 \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{24 e^6}+\frac {9 b d^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^6}-\frac {45 b d^4 n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{8 e^6}+\frac {5 b d^3 n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^6}-\frac {45 b d^2 n \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{16 e^6}+\frac {9 b d n \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{10 e^6}-\frac {b n \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{8 e^6}-\frac {3 d^5 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {15 d^4 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {5 d^3 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{e^6}+\frac {15 d^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {3 d \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {\left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6} \]
[Out]
Time = 0.69 (sec) , antiderivative size = 913, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=-\frac {b^3 n^3 \left (d+e x^{2/3}\right )^6}{144 e^6}+\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^6}{4 e^6}-\frac {b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^6}{8 e^6}+\frac {b^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^6}{24 e^6}+\frac {9 b^3 d n^3 \left (d+e x^{2/3}\right )^5}{125 e^6}-\frac {3 d \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^5}{2 e^6}+\frac {9 b d n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^5}{10 e^6}-\frac {9 b^2 d n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^5}{25 e^6}-\frac {45 b^3 d^2 n^3 \left (d+e x^{2/3}\right )^4}{128 e^6}+\frac {15 d^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^4}{4 e^6}-\frac {45 b d^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^4}{16 e^6}+\frac {45 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^4}{32 e^6}+\frac {10 b^3 d^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^6}-\frac {5 d^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^3}{e^6}+\frac {5 b d^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^3}{e^6}-\frac {10 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^3}{3 e^6}-\frac {45 b^3 d^4 n^3 \left (d+e x^{2/3}\right )^2}{16 e^6}+\frac {15 d^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^2}{4 e^6}-\frac {45 b d^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^2}{8 e^6}+\frac {45 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^2}{8 e^6}-\frac {3 d^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )}{2 e^6}+\frac {9 b d^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )}{2 e^6}-\frac {9 b^3 d^5 n^2 \log \left (c \left (d+e x^{2/3}\right )^n\right ) \left (d+e x^{2/3}\right )}{e^6}+\frac {9 b^3 d^5 n^3 x^{2/3}}{e^5}-\frac {9 a b^2 d^5 n^2 x^{2/3}}{e^5} \]
[In]
[Out]
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2436
Rule 2437
Rule 2448
Rule 2504
Rubi steps \begin{align*} \text {integral}& = \frac {3}{2} \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right ) \\ & = \frac {3}{2} \text {Subst}\left (\int \left (-\frac {d^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {5 d^4 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}-\frac {10 d^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {10 d^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}-\frac {5 d (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {(d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}\right ) \, dx,x,x^{2/3}\right ) \\ & = \frac {3 \text {Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{2 e^5}-\frac {(15 d) \text {Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{2 e^5}+\frac {\left (15 d^2\right ) \text {Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{e^5}-\frac {\left (15 d^3\right ) \text {Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{e^5}+\frac {\left (15 d^4\right ) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{2 e^5}-\frac {\left (3 d^5\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )}{2 e^5} \\ & = \frac {3 \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{2 e^6}-\frac {(15 d) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{2 e^6}+\frac {\left (15 d^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{e^6}-\frac {\left (15 d^3\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{e^6}+\frac {\left (15 d^4\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{2 e^6}-\frac {\left (3 d^5\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\right )}{2 e^6} \\ & = -\frac {3 d^5 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {15 d^4 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {5 d^3 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{e^6}+\frac {15 d^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {3 d \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {\left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {(3 b n) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{4 e^6}+\frac {(9 b d n) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{2 e^6}-\frac {\left (45 b d^2 n\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{4 e^6}+\frac {\left (15 b d^3 n\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{e^6}-\frac {\left (45 b d^4 n\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{4 e^6}+\frac {\left (9 b d^5 n\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x^{2/3}\right )}{2 e^6} \\ & = \frac {9 b d^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^6}-\frac {45 b d^4 n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{8 e^6}+\frac {5 b d^3 n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^6}-\frac {45 b d^2 n \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{16 e^6}+\frac {9 b d n \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{10 e^6}-\frac {b n \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{8 e^6}-\frac {3 d^5 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {15 d^4 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {5 d^3 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{e^6}+\frac {15 d^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {3 d \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {\left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{4 e^6}-\frac {\left (9 b^2 d n^2\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{5 e^6}+\frac {\left (45 b^2 d^2 n^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{8 e^6}-\frac {\left (10 b^2 d^3 n^2\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{e^6}+\frac {\left (45 b^2 d^4 n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{4 e^6}-\frac {\left (9 b^2 d^5 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x^{2/3}\right )}{e^6} \\ & = -\frac {45 b^3 d^4 n^3 \left (d+e x^{2/3}\right )^2}{16 e^6}+\frac {10 b^3 d^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^6}-\frac {45 b^3 d^2 n^3 \left (d+e x^{2/3}\right )^4}{128 e^6}+\frac {9 b^3 d n^3 \left (d+e x^{2/3}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e x^{2/3}\right )^6}{144 e^6}-\frac {9 a b^2 d^5 n^2 x^{2/3}}{e^5}+\frac {45 b^2 d^4 n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 e^6}-\frac {10 b^2 d^3 n^2 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^6}+\frac {45 b^2 d^2 n^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{32 e^6}-\frac {9 b^2 d n^2 \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{24 e^6}+\frac {9 b d^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^6}-\frac {45 b d^4 n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{8 e^6}+\frac {5 b d^3 n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^6}-\frac {45 b d^2 n \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{16 e^6}+\frac {9 b d n \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{10 e^6}-\frac {b n \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{8 e^6}-\frac {3 d^5 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {15 d^4 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {5 d^3 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{e^6}+\frac {15 d^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {3 d \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {\left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {\left (9 b^3 d^5 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x^{2/3}\right )}{e^6} \\ & = -\frac {45 b^3 d^4 n^3 \left (d+e x^{2/3}\right )^2}{16 e^6}+\frac {10 b^3 d^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^6}-\frac {45 b^3 d^2 n^3 \left (d+e x^{2/3}\right )^4}{128 e^6}+\frac {9 b^3 d n^3 \left (d+e x^{2/3}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e x^{2/3}\right )^6}{144 e^6}-\frac {9 a b^2 d^5 n^2 x^{2/3}}{e^5}+\frac {9 b^3 d^5 n^3 x^{2/3}}{e^5}-\frac {9 b^3 d^5 n^2 \left (d+e x^{2/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e^6}+\frac {45 b^2 d^4 n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 e^6}-\frac {10 b^2 d^3 n^2 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^6}+\frac {45 b^2 d^2 n^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{32 e^6}-\frac {9 b^2 d n^2 \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{24 e^6}+\frac {9 b d^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^6}-\frac {45 b d^4 n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{8 e^6}+\frac {5 b d^3 n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^6}-\frac {45 b d^2 n \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{16 e^6}+\frac {9 b d n \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{10 e^6}-\frac {b n \left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{8 e^6}-\frac {3 d^5 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {15 d^4 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {5 d^3 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{e^6}+\frac {15 d^2 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6}-\frac {3 d \left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^6}+\frac {\left (d+e x^{2/3}\right )^6 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{4 e^6} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 593, normalized size of antiderivative = 0.65 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\frac {e x^{2/3} \left (36000 a^3 e^5 x^{10/3}+b^3 n^3 \left (809340 d^5-140070 d^4 e x^{2/3}+41180 d^3 e^2 x^{4/3}-13785 d^2 e^3 x^2+4368 d e^4 x^{8/3}-1000 e^5 x^{10/3}\right )-60 a b^2 n^2 \left (8820 d^5-2610 d^4 e x^{2/3}+1140 d^3 e^2 x^{4/3}-555 d^2 e^3 x^2+264 d e^4 x^{8/3}-100 e^5 x^{10/3}\right )+1800 a^2 b n \left (60 d^5-30 d^4 e x^{2/3}+20 d^3 e^2 x^{4/3}-15 d^2 e^3 x^2+12 d e^4 x^{8/3}-10 e^5 x^{10/3}\right )\right )-60 b d^6 n \left (1800 a^2-8820 a b n+13489 b^2 n^2\right ) \log \left (d+e x^{2/3}\right )+60 b e x^{2/3} \left (1800 a^2 e^5 x^{10/3}+60 a b n \left (60 d^5-30 d^4 e x^{2/3}+20 d^3 e^2 x^{4/3}-15 d^2 e^3 x^2+12 d e^4 x^{8/3}-10 e^5 x^{10/3}\right )+b^2 n^2 \left (-8820 d^5+2610 d^4 e x^{2/3}-1140 d^3 e^2 x^{4/3}+555 d^2 e^3 x^2-264 d e^4 x^{8/3}+100 e^5 x^{10/3}\right )\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )+1800 b^2 \left (b n \left (147 d^6+60 d^5 e x^{2/3}-30 d^4 e^2 x^{4/3}+20 d^3 e^3 x^2-15 d^2 e^4 x^{8/3}+12 d e^5 x^{10/3}-10 e^6 x^4\right )-60 a \left (d^6-e^6 x^4\right )\right ) \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )-36000 b^3 \left (d^6-e^6 x^4\right ) \log ^3\left (c \left (d+e x^{2/3}\right )^n\right )}{144000 e^6} \]
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\[\int x^{3} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{3}d x\]
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Time = 0.49 (sec) , antiderivative size = 1241, normalized size of antiderivative = 1.36 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]
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Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 680, normalized size of antiderivative = 0.74 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\frac {1}{4} \, b^{3} x^{4} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{3} + \frac {3}{4} \, a b^{2} x^{4} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{2} + \frac {3}{4} \, a^{2} b x^{4} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + \frac {1}{4} \, a^{3} x^{4} - \frac {1}{80} \, a^{2} b e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{4} - 12 \, d e^{4} x^{\frac {10}{3}} + 15 \, d^{2} e^{3} x^{\frac {8}{3}} - 20 \, d^{3} e^{2} x^{2} + 30 \, d^{4} e x^{\frac {4}{3}} - 60 \, d^{5} x^{\frac {2}{3}}}{e^{6}}\right )} - \frac {1}{2400} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{4} - 12 \, d e^{4} x^{\frac {10}{3}} + 15 \, d^{2} e^{3} x^{\frac {8}{3}} - 20 \, d^{3} e^{2} x^{2} + 30 \, d^{4} e x^{\frac {4}{3}} - 60 \, d^{5} x^{\frac {2}{3}}}{e^{6}}\right )} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) - \frac {{\left (100 \, e^{6} x^{4} - 264 \, d e^{5} x^{\frac {10}{3}} + 555 \, d^{2} e^{4} x^{\frac {8}{3}} - 1140 \, d^{3} e^{3} x^{2} + 1800 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )^{2} + 2610 \, d^{4} e^{2} x^{\frac {4}{3}} + 8820 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right ) - 8820 \, d^{5} e x^{\frac {2}{3}}\right )} n^{2}}{e^{6}}\right )} a b^{2} - \frac {1}{144000} \, {\left (1800 \, e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{4} - 12 \, d e^{4} x^{\frac {10}{3}} + 15 \, d^{2} e^{3} x^{\frac {8}{3}} - 20 \, d^{3} e^{2} x^{2} + 30 \, d^{4} e x^{\frac {4}{3}} - 60 \, d^{5} x^{\frac {2}{3}}}{e^{6}}\right )} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{2} + e n {\left (\frac {{\left (1000 \, e^{6} x^{4} - 4368 \, d e^{5} x^{\frac {10}{3}} + 36000 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )^{3} + 13785 \, d^{2} e^{4} x^{\frac {8}{3}} - 41180 \, d^{3} e^{3} x^{2} + 264600 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )^{2} + 140070 \, d^{4} e^{2} x^{\frac {4}{3}} + 809340 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right ) - 809340 \, d^{5} e x^{\frac {2}{3}}\right )} n^{2}}{e^{7}} - \frac {60 \, {\left (100 \, e^{6} x^{4} - 264 \, d e^{5} x^{\frac {10}{3}} + 555 \, d^{2} e^{4} x^{\frac {8}{3}} - 1140 \, d^{3} e^{3} x^{2} + 1800 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right )^{2} + 2610 \, d^{4} e^{2} x^{\frac {4}{3}} + 8820 \, d^{6} \log \left (e x^{\frac {2}{3}} + d\right ) - 8820 \, d^{5} e x^{\frac {2}{3}}\right )} n \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )}{e^{7}}\right )}\right )} b^{3} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2104 vs. \(2 (787) = 1574\).
Time = 0.82 (sec) , antiderivative size = 2104, normalized size of antiderivative = 2.30 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]
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Time = 9.19 (sec) , antiderivative size = 992, normalized size of antiderivative = 1.09 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]
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