Integrand size = 22, antiderivative size = 907 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=-\frac {45 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^2}{8 e^6}+\frac {20 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}-\frac {45 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^4}{64 e^6}+\frac {18 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{72 e^6}-\frac {18 a b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac {18 b^3 d^5 n^3 \sqrt [3]{x}}{e^5}-\frac {18 b^3 d^5 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^6}+\frac {45 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^6}-\frac {20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^6}+\frac {45 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{16 e^6}-\frac {18 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{12 e^6}+\frac {9 b d^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}{e^6}-\frac {45 b d^4 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 e^6}+\frac {10 b d^3 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^6}-\frac {45 b d^2 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 e^6}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{5 e^6}-\frac {b n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 e^6}-\frac {3 d^5 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {15 d^4 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^6}-\frac {10 d^3 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {15 d^2 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^6}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^6} \]
[Out]
Time = 0.65 (sec) , antiderivative size = 907, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=-\frac {b^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{72 e^6}+\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^6}{2 e^6}-\frac {b n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^6}{4 e^6}+\frac {b^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^6}{12 e^6}+\frac {18 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^6}-\frac {3 d \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^5}{e^6}+\frac {9 b d n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^5}{5 e^6}-\frac {18 b^2 d n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^5}{25 e^6}-\frac {45 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^4}{64 e^6}+\frac {15 d^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^4}{2 e^6}-\frac {45 b d^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^4}{8 e^6}+\frac {45 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^4}{16 e^6}+\frac {20 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}-\frac {10 d^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^3}{e^6}+\frac {10 b d^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^3}{e^6}-\frac {20 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^3}{3 e^6}-\frac {45 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^2}{8 e^6}+\frac {15 d^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^2}{2 e^6}-\frac {45 b d^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}+\frac {45 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^2}{4 e^6}-\frac {3 d^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )}{e^6}+\frac {9 b d^5 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )}{e^6}-\frac {18 b^3 d^5 n^2 \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \left (d+e \sqrt [3]{x}\right )}{e^6}+\frac {18 b^3 d^5 n^3 \sqrt [3]{x}}{e^5}-\frac {18 a b^2 d^5 n^2 \sqrt [3]{x}}{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}& = 3 \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \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,\sqrt [3]{x}\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,\sqrt [3]{x}\right )}{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,\sqrt [3]{x}\right )}{e^5}+\frac {\left (30 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,\sqrt [3]{x}\right )}{e^5}-\frac {\left (30 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,\sqrt [3]{x}\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,\sqrt [3]{x}\right )}{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,\sqrt [3]{x}\right )}{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 \sqrt [3]{x}\right )}{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 \sqrt [3]{x}\right )}{e^6}+\frac {\left (30 d^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^6}-\frac {\left (30 d^3\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\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 \sqrt [3]{x}\right )}{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 \sqrt [3]{x}\right )}{e^6} \\ & = -\frac {3 d^5 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {15 d^4 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^6}-\frac {10 d^3 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {15 d^2 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^6}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 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 \sqrt [3]{x}\right )}{2 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 \sqrt [3]{x}\right )}{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 \sqrt [3]{x}\right )}{2 e^6}+\frac {\left (30 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 \sqrt [3]{x}\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 \sqrt [3]{x}\right )}{2 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 \sqrt [3]{x}\right )}{e^6} \\ & = \frac {9 b d^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}{e^6}-\frac {45 b d^4 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 e^6}+\frac {10 b d^3 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^6}-\frac {45 b d^2 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 e^6}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{5 e^6}-\frac {b n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 e^6}-\frac {3 d^5 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {15 d^4 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^6}-\frac {10 d^3 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {15 d^2 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^6}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 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 \sqrt [3]{x}\right )}{2 e^6}-\frac {\left (18 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 \sqrt [3]{x}\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 \sqrt [3]{x}\right )}{4 e^6}-\frac {\left (20 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 \sqrt [3]{x}\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 \sqrt [3]{x}\right )}{2 e^6}-\frac {\left (18 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 \sqrt [3]{x}\right )}{e^6} \\ & = -\frac {45 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^2}{8 e^6}+\frac {20 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}-\frac {45 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^4}{64 e^6}+\frac {18 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{72 e^6}-\frac {18 a b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac {45 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^6}-\frac {20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^6}+\frac {45 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{16 e^6}-\frac {18 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{12 e^6}+\frac {9 b d^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}{e^6}-\frac {45 b d^4 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 e^6}+\frac {10 b d^3 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^6}-\frac {45 b d^2 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 e^6}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{5 e^6}-\frac {b n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 e^6}-\frac {3 d^5 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {15 d^4 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^6}-\frac {10 d^3 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {15 d^2 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^6}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^6}-\frac {\left (18 b^3 d^5 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^6} \\ & = -\frac {45 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^2}{8 e^6}+\frac {20 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}-\frac {45 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^4}{64 e^6}+\frac {18 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{72 e^6}-\frac {18 a b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac {18 b^3 d^5 n^3 \sqrt [3]{x}}{e^5}-\frac {18 b^3 d^5 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^6}+\frac {45 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^6}-\frac {20 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^6}+\frac {45 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{16 e^6}-\frac {18 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{12 e^6}+\frac {9 b d^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}{e^6}-\frac {45 b d^4 n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 e^6}+\frac {10 b d^3 n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^6}-\frac {45 b d^2 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 e^6}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{5 e^6}-\frac {b n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{4 e^6}-\frac {3 d^5 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {15 d^4 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^6}-\frac {10 d^3 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {15 d^2 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^6}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{2 e^6} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 589, normalized size of antiderivative = 0.65 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {b^3 e n^3 \sqrt [3]{x} \left (809340 d^5-140070 d^4 e \sqrt [3]{x}+41180 d^3 e^2 x^{2/3}-13785 d^2 e^3 x+4368 d e^4 x^{4/3}-1000 e^5 x^{5/3}\right )+1800 a^2 b n \left (147 d^6+60 d^5 e \sqrt [3]{x}-30 d^4 e^2 x^{2/3}+20 d^3 e^3 x-15 d^2 e^4 x^{4/3}+12 d e^5 x^{5/3}-10 e^6 x^2\right )-36000 a^3 \left (d^6-e^6 x^2\right )+60 a b^2 n^2 \left (8111 d^6-8820 d^5 e \sqrt [3]{x}+2610 d^4 e^2 x^{2/3}-1140 d^3 e^3 x+555 d^2 e^4 x^{4/3}-264 d e^5 x^{5/3}+100 e^6 x^2\right )-60 b \left (b^2 n^2 \left (13489 d^6+8820 d^5 e \sqrt [3]{x}-2610 d^4 e^2 x^{2/3}+1140 d^3 e^3 x-555 d^2 e^4 x^{4/3}+264 d e^5 x^{5/3}-100 e^6 x^2\right )-60 a b n \left (147 d^6+60 d^5 e \sqrt [3]{x}-30 d^4 e^2 x^{2/3}+20 d^3 e^3 x-15 d^2 e^4 x^{4/3}+12 d e^5 x^{5/3}-10 e^6 x^2\right )+1800 a^2 \left (d^6-e^6 x^2\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-1800 b^2 \left (60 a \left (d^6-e^6 x^2\right )+b n \left (-147 d^6-60 d^5 e \sqrt [3]{x}+30 d^4 e^2 x^{2/3}-20 d^3 e^3 x+15 d^2 e^4 x^{4/3}-12 d e^5 x^{5/3}+10 e^6 x^2\right )\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )-36000 b^3 \left (d^6-e^6 x^2\right ) \log ^3\left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{72000 e^6} \]
[In]
[Out]
\[\int x {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{3}d x\]
[In]
[Out]
none
Time = 0.39 (sec) , antiderivative size = 1190, normalized size of antiderivative = 1.31 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\int x \left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{3}\, dx \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 668, normalized size of antiderivative = 0.74 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {1}{2} \, b^{3} x^{2} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{3} + \frac {3}{2} \, a b^{2} x^{2} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} - \frac {1}{40} \, a^{2} b e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac {5}{3}} + 15 \, d^{2} e^{3} x^{\frac {4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac {2}{3}} - 60 \, d^{5} x^{\frac {1}{3}}}{e^{6}}\right )} + \frac {3}{2} \, a^{2} b x^{2} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + \frac {1}{2} \, a^{3} x^{2} - \frac {1}{1200} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac {5}{3}} + 15 \, d^{2} e^{3} x^{\frac {4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac {2}{3}} - 60 \, d^{5} x^{\frac {1}{3}}}{e^{6}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) - \frac {{\left (100 \, e^{6} x^{2} + 1800 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 264 \, d e^{5} x^{\frac {5}{3}} + 555 \, d^{2} e^{4} x^{\frac {4}{3}} - 1140 \, d^{3} e^{3} x + 8820 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right ) + 2610 \, d^{4} e^{2} x^{\frac {2}{3}} - 8820 \, d^{5} e x^{\frac {1}{3}}\right )} n^{2}}{e^{6}}\right )} a b^{2} - \frac {1}{72000} \, {\left (1800 \, e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac {5}{3}} + 15 \, d^{2} e^{3} x^{\frac {4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac {2}{3}} - 60 \, d^{5} x^{\frac {1}{3}}}{e^{6}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} + e n {\left (\frac {{\left (36000 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )^{3} + 1000 \, e^{6} x^{2} + 264600 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 4368 \, d e^{5} x^{\frac {5}{3}} + 13785 \, d^{2} e^{4} x^{\frac {4}{3}} - 41180 \, d^{3} e^{3} x + 809340 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right ) + 140070 \, d^{4} e^{2} x^{\frac {2}{3}} - 809340 \, d^{5} e x^{\frac {1}{3}}\right )} n^{2}}{e^{7}} - \frac {60 \, {\left (100 \, e^{6} x^{2} + 1800 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 264 \, d e^{5} x^{\frac {5}{3}} + 555 \, d^{2} e^{4} x^{\frac {4}{3}} - 1140 \, d^{3} e^{3} x + 8820 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right ) + 2610 \, d^{4} e^{2} x^{\frac {2}{3}} - 8820 \, d^{5} e x^{\frac {1}{3}}\right )} n \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )}{e^{7}}\right )}\right )} b^{3} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 2160 vs. \(2 (787) = 1574\).
Time = 0.33 (sec) , antiderivative size = 2160, normalized size of antiderivative = 2.38 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 9.19 (sec) , antiderivative size = 979, normalized size of antiderivative = 1.08 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {a^3\,x^2}{2}+\frac {b^3\,x^2\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^3}{2}-\frac {b^3\,n^3\,x^2}{72}+\frac {3\,a\,b^2\,x^2\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2}-\frac {b^3\,n\,x^2\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{4}+\frac {b^3\,n^2\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{12}+\frac {a\,b^2\,n^2\,x^2}{12}-\frac {b^3\,d^6\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^3}{2\,e^6}+\frac {3\,a^2\,b\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2}-\frac {a^2\,b\,n\,x^2}{4}-\frac {a\,b^2\,n\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2}-\frac {13489\,b^3\,d^6\,n^3\,\ln \left (d+e\,x^{1/3}\right )}{1200\,e^6}-\frac {919\,b^3\,d^2\,n^3\,x^{4/3}}{4800\,e^2}-\frac {4669\,b^3\,d^4\,n^3\,x^{2/3}}{2400\,e^4}+\frac {13489\,b^3\,d^5\,n^3\,x^{1/3}}{1200\,e^5}-\frac {3\,a\,b^2\,d^6\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2\,e^6}+\frac {147\,b^3\,d^6\,n\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{40\,e^6}+\frac {2059\,b^3\,d^3\,n^3\,x}{3600\,e^3}+\frac {91\,b^3\,d\,n^3\,x^{5/3}}{1500\,e}-\frac {3\,a^2\,b\,d^6\,n\,\ln \left (d+e\,x^{1/3}\right )}{2\,e^6}+\frac {b^3\,d^3\,n\,x\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2\,e^3}-\frac {19\,b^3\,d^3\,n^2\,x\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{20\,e^3}+\frac {3\,b^3\,d\,n\,x^{5/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{10\,e}-\frac {11\,b^3\,d\,n^2\,x^{5/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{50\,e}-\frac {19\,a\,b^2\,d^3\,n^2\,x}{20\,e^3}-\frac {11\,a\,b^2\,d\,n^2\,x^{5/3}}{50\,e}-\frac {3\,a^2\,b\,d^2\,n\,x^{4/3}}{8\,e^2}-\frac {3\,a^2\,b\,d^4\,n\,x^{2/3}}{4\,e^4}+\frac {3\,a^2\,b\,d^5\,n\,x^{1/3}}{2\,e^5}+\frac {147\,a\,b^2\,d^6\,n^2\,\ln \left (d+e\,x^{1/3}\right )}{20\,e^6}-\frac {3\,b^3\,d^2\,n\,x^{4/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{8\,e^2}+\frac {37\,b^3\,d^2\,n^2\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{80\,e^2}-\frac {3\,b^3\,d^4\,n\,x^{2/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{4\,e^4}+\frac {87\,b^3\,d^4\,n^2\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{40\,e^4}+\frac {3\,b^3\,d^5\,n\,x^{1/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2\,e^5}-\frac {147\,b^3\,d^5\,n^2\,x^{1/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{20\,e^5}+\frac {37\,a\,b^2\,d^2\,n^2\,x^{4/3}}{80\,e^2}+\frac {87\,a\,b^2\,d^4\,n^2\,x^{2/3}}{40\,e^4}-\frac {147\,a\,b^2\,d^5\,n^2\,x^{1/3}}{20\,e^5}+\frac {a^2\,b\,d^3\,n\,x}{2\,e^3}+\frac {3\,a^2\,b\,d\,n\,x^{5/3}}{10\,e}+\frac {a\,b^2\,d^3\,n\,x\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{e^3}+\frac {3\,a\,b^2\,d\,n\,x^{5/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{5\,e}-\frac {3\,a\,b^2\,d^2\,n\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{4\,e^2}-\frac {3\,a\,b^2\,d^4\,n\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2\,e^4}+\frac {3\,a\,b^2\,d^5\,n\,x^{1/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{e^5} \]
[In]
[Out]