Integrand size = 24, antiderivative size = 907 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=-\frac {15 b^3 d^4 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^6}+\frac {40 b^3 d^3 n^3 \left (d+e \sqrt {x}\right )^3}{27 e^6}-\frac {15 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^4}{32 e^6}+\frac {12 b^3 d n^3 \left (d+e \sqrt {x}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e \sqrt {x}\right )^6}{108 e^6}-\frac {12 a b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {12 b^3 d^5 n^3 \sqrt {x}}{e^5}-\frac {12 b^3 d^5 n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^6}+\frac {15 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^6}-\frac {40 b^2 d^3 n^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 e^6}+\frac {15 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{8 e^6}-\frac {12 b^2 d n^2 \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{18 e^6}+\frac {6 b d^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^6}-\frac {15 b d^4 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^6}+\frac {20 b d^3 n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 e^6}-\frac {15 b d^2 n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 e^6}+\frac {6 b d n \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{5 e^6}-\frac {b n \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{6 e^6}-\frac {2 d^5 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {5 d^4 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {20 d^3 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}+\frac {5 d^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {2 d \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6} \]
[Out]
Time = 0.67 (sec) , antiderivative size = 907, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=-\frac {b^3 n^3 \left (d+e \sqrt {x}\right )^6}{108 e^6}+\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \left (d+e \sqrt {x}\right )^6}{3 e^6}-\frac {b n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (d+e \sqrt {x}\right )^6}{6 e^6}+\frac {b^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (d+e \sqrt {x}\right )^6}{18 e^6}+\frac {12 b^3 d n^3 \left (d+e \sqrt {x}\right )^5}{125 e^6}-\frac {2 d \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \left (d+e \sqrt {x}\right )^5}{e^6}+\frac {6 b d n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (d+e \sqrt {x}\right )^5}{5 e^6}-\frac {12 b^2 d n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (d+e \sqrt {x}\right )^5}{25 e^6}-\frac {15 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^4}{32 e^6}+\frac {5 d^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \left (d+e \sqrt {x}\right )^4}{e^6}-\frac {15 b d^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (d+e \sqrt {x}\right )^4}{4 e^6}+\frac {15 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (d+e \sqrt {x}\right )^4}{8 e^6}+\frac {40 b^3 d^3 n^3 \left (d+e \sqrt {x}\right )^3}{27 e^6}-\frac {20 d^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \left (d+e \sqrt {x}\right )^3}{3 e^6}+\frac {20 b d^3 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (d+e \sqrt {x}\right )^3}{3 e^6}-\frac {40 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (d+e \sqrt {x}\right )^3}{9 e^6}-\frac {15 b^3 d^4 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^6}+\frac {5 d^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \left (d+e \sqrt {x}\right )^2}{e^6}-\frac {15 b d^4 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (d+e \sqrt {x}\right )^2}{2 e^6}+\frac {15 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (d+e \sqrt {x}\right )^2}{2 e^6}-\frac {2 d^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \left (d+e \sqrt {x}\right )}{e^6}+\frac {6 b d^5 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (d+e \sqrt {x}\right )}{e^6}-\frac {12 b^3 d^5 n^2 \log \left (c \left (d+e \sqrt {x}\right )^n\right ) \left (d+e \sqrt {x}\right )}{e^6}+\frac {12 b^3 d^5 n^3 \sqrt {x}}{e^5}-\frac {12 a b^2 d^5 n^2 \sqrt {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}& = 2 \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right ) \\ & = 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,\sqrt {x}\right ) \\ & = \frac {2 \text {Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e^5}-\frac {(10 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 {x}\right )}{e^5}+\frac {\left (20 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 {x}\right )}{e^5}-\frac {\left (20 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 {x}\right )}{e^5}+\frac {\left (10 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 {x}\right )}{e^5}-\frac {\left (2 d^5\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e^5} \\ & = \frac {2 \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {(10 d) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^6}+\frac {\left (20 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 {x}\right )}{e^6}-\frac {\left (20 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 {x}\right )}{e^6}+\frac {\left (10 d^4\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {\left (2 d^5\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^6} \\ & = -\frac {2 d^5 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {5 d^4 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {20 d^3 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}+\frac {5 d^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {2 d \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}-\frac {(b n) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^6}+\frac {(6 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 {x}\right )}{e^6}-\frac {\left (15 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 {x}\right )}{e^6}+\frac {\left (20 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 {x}\right )}{e^6}-\frac {\left (15 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 {x}\right )}{e^6}+\frac {\left (6 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 {x}\right )}{e^6} \\ & = \frac {6 b d^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^6}-\frac {15 b d^4 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^6}+\frac {20 b d^3 n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 e^6}-\frac {15 b d^2 n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 e^6}+\frac {6 b d n \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{5 e^6}-\frac {b n \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{6 e^6}-\frac {2 d^5 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {5 d^4 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {20 d^3 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}+\frac {5 d^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {2 d \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 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 {x}\right )}{3 e^6}-\frac {\left (12 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 {x}\right )}{5 e^6}+\frac {\left (15 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 {x}\right )}{2 e^6}-\frac {\left (40 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 {x}\right )}{3 e^6}+\frac {\left (15 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 {x}\right )}{e^6}-\frac {\left (12 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 {x}\right )}{e^6} \\ & = -\frac {15 b^3 d^4 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^6}+\frac {40 b^3 d^3 n^3 \left (d+e \sqrt {x}\right )^3}{27 e^6}-\frac {15 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^4}{32 e^6}+\frac {12 b^3 d n^3 \left (d+e \sqrt {x}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e \sqrt {x}\right )^6}{108 e^6}-\frac {12 a b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {15 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^6}-\frac {40 b^2 d^3 n^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 e^6}+\frac {15 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{8 e^6}-\frac {12 b^2 d n^2 \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{18 e^6}+\frac {6 b d^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^6}-\frac {15 b d^4 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^6}+\frac {20 b d^3 n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 e^6}-\frac {15 b d^2 n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 e^6}+\frac {6 b d n \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{5 e^6}-\frac {b n \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{6 e^6}-\frac {2 d^5 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {5 d^4 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {20 d^3 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}+\frac {5 d^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {2 d \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}-\frac {\left (12 b^3 d^5 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt {x}\right )}{e^6} \\ & = -\frac {15 b^3 d^4 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^6}+\frac {40 b^3 d^3 n^3 \left (d+e \sqrt {x}\right )^3}{27 e^6}-\frac {15 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^4}{32 e^6}+\frac {12 b^3 d n^3 \left (d+e \sqrt {x}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e \sqrt {x}\right )^6}{108 e^6}-\frac {12 a b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {12 b^3 d^5 n^3 \sqrt {x}}{e^5}-\frac {12 b^3 d^5 n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^6}+\frac {15 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^6}-\frac {40 b^2 d^3 n^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 e^6}+\frac {15 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{8 e^6}-\frac {12 b^2 d n^2 \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{18 e^6}+\frac {6 b d^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^6}-\frac {15 b d^4 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^6}+\frac {20 b d^3 n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{3 e^6}-\frac {15 b d^2 n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{4 e^6}+\frac {6 b d n \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{5 e^6}-\frac {b n \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{6 e^6}-\frac {2 d^5 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {5 d^4 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {20 d^3 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}+\frac {5 d^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {2 d \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 577, normalized size of antiderivative = 0.64 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {b^3 e n^3 \sqrt {x} \left (809340 d^5-140070 d^4 e \sqrt {x}+41180 d^3 e^2 x-13785 d^2 e^3 x^{3/2}+4368 d e^4 x^2-1000 e^5 x^{5/2}\right )+1800 a^2 b n \left (147 d^6+60 d^5 e \sqrt {x}-30 d^4 e^2 x+20 d^3 e^3 x^{3/2}-15 d^2 e^4 x^2+12 d e^5 x^{5/2}-10 e^6 x^3\right )-36000 a^3 \left (d^6-e^6 x^3\right )+60 a b^2 n^2 \left (8111 d^6-8820 d^5 e \sqrt {x}+2610 d^4 e^2 x-1140 d^3 e^3 x^{3/2}+555 d^2 e^4 x^2-264 d e^5 x^{5/2}+100 e^6 x^3\right )-60 b \left (b^2 n^2 \left (13489 d^6+8820 d^5 e \sqrt {x}-2610 d^4 e^2 x+1140 d^3 e^3 x^{3/2}-555 d^2 e^4 x^2+264 d e^5 x^{5/2}-100 e^6 x^3\right )-60 a b n \left (147 d^6+60 d^5 e \sqrt {x}-30 d^4 e^2 x+20 d^3 e^3 x^{3/2}-15 d^2 e^4 x^2+12 d e^5 x^{5/2}-10 e^6 x^3\right )+1800 a^2 \left (d^6-e^6 x^3\right )\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )-1800 b^2 \left (60 a \left (d^6-e^6 x^3\right )+b n \left (-147 d^6-60 d^5 e \sqrt {x}+30 d^4 e^2 x-20 d^3 e^3 x^{3/2}+15 d^2 e^4 x^2-12 d e^5 x^{5/2}+10 e^6 x^3\right )\right ) \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )-36000 b^3 \left (d^6-e^6 x^3\right ) \log ^3\left (c \left (d+e \sqrt {x}\right )^n\right )}{108000 e^6} \]
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\[\int x^{2} {\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{3}d x\]
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Time = 0.41 (sec) , antiderivative size = 1197, normalized size of antiderivative = 1.32 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]
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\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\int x^{2} \left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{3}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 666, normalized size of antiderivative = 0.73 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {1}{3} \, b^{3} x^{3} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{3} + a b^{2} x^{3} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} + a^{2} b x^{3} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + \frac {1}{3} \, a^{3} x^{3} - \frac {1}{60} \, a^{2} b e n {\left (\frac {60 \, d^{6} \log \left (e \sqrt {x} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{3} - 12 \, d e^{4} x^{\frac {5}{2}} + 15 \, d^{2} e^{3} x^{2} - 20 \, d^{3} e^{2} x^{\frac {3}{2}} + 30 \, d^{4} e x - 60 \, d^{5} \sqrt {x}}{e^{6}}\right )} - \frac {1}{1800} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (e \sqrt {x} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{3} - 12 \, d e^{4} x^{\frac {5}{2}} + 15 \, d^{2} e^{3} x^{2} - 20 \, d^{3} e^{2} x^{\frac {3}{2}} + 30 \, d^{4} e x - 60 \, d^{5} \sqrt {x}}{e^{6}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) - \frac {{\left (100 \, e^{6} x^{3} - 264 \, d e^{5} x^{\frac {5}{2}} + 555 \, d^{2} e^{4} x^{2} + 1800 \, d^{6} \log \left (e \sqrt {x} + d\right )^{2} - 1140 \, d^{3} e^{3} x^{\frac {3}{2}} + 2610 \, d^{4} e^{2} x + 8820 \, d^{6} \log \left (e \sqrt {x} + d\right ) - 8820 \, d^{5} e \sqrt {x}\right )} n^{2}}{e^{6}}\right )} a b^{2} - \frac {1}{108000} \, {\left (1800 \, e n {\left (\frac {60 \, d^{6} \log \left (e \sqrt {x} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{3} - 12 \, d e^{4} x^{\frac {5}{2}} + 15 \, d^{2} e^{3} x^{2} - 20 \, d^{3} e^{2} x^{\frac {3}{2}} + 30 \, d^{4} e x - 60 \, d^{5} \sqrt {x}}{e^{6}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} + e n {\left (\frac {{\left (1000 \, e^{6} x^{3} + 36000 \, d^{6} \log \left (e \sqrt {x} + d\right )^{3} - 4368 \, d e^{5} x^{\frac {5}{2}} + 13785 \, d^{2} e^{4} x^{2} + 264600 \, d^{6} \log \left (e \sqrt {x} + d\right )^{2} - 41180 \, d^{3} e^{3} x^{\frac {3}{2}} + 140070 \, d^{4} e^{2} x + 809340 \, d^{6} \log \left (e \sqrt {x} + d\right ) - 809340 \, d^{5} e \sqrt {x}\right )} n^{2}}{e^{7}} - \frac {60 \, {\left (100 \, e^{6} x^{3} - 264 \, d e^{5} x^{\frac {5}{2}} + 555 \, d^{2} e^{4} x^{2} + 1800 \, d^{6} \log \left (e \sqrt {x} + d\right )^{2} - 1140 \, d^{3} e^{3} x^{\frac {3}{2}} + 2610 \, d^{4} e^{2} x + 8820 \, d^{6} \log \left (e \sqrt {x} + d\right ) - 8820 \, d^{5} e \sqrt {x}\right )} n \log \left ({\left (e \sqrt {x} + 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. 2160 vs. \(2 (787) = 1574\).
Time = 0.35 (sec) , antiderivative size = 2160, normalized size of antiderivative = 2.38 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]
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Time = 9.29 (sec) , antiderivative size = 976, normalized size of antiderivative = 1.08 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {a^3\,x^3}{3}+\frac {b^3\,x^3\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^3}{3}-\frac {b^3\,n^3\,x^3}{108}+a\,b^2\,x^3\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2-\frac {b^3\,n\,x^3\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{6}+\frac {b^3\,n^2\,x^3\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{18}+\frac {a\,b^2\,n^2\,x^3}{18}-\frac {b^3\,d^6\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^3}{3\,e^6}+a^2\,b\,x^3\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )-\frac {a^2\,b\,n\,x^3}{6}-\frac {a\,b^2\,n\,x^3\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3}-\frac {13489\,b^3\,d^6\,n^3\,\ln \left (d+e\,\sqrt {x}\right )}{1800\,e^6}-\frac {919\,b^3\,d^2\,n^3\,x^2}{7200\,e^2}+\frac {2059\,b^3\,d^3\,n^3\,x^{3/2}}{5400\,e^3}+\frac {13489\,b^3\,d^5\,n^3\,\sqrt {x}}{1800\,e^5}-\frac {a\,b^2\,d^6\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{e^6}+\frac {49\,b^3\,d^6\,n\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{20\,e^6}+\frac {91\,b^3\,d\,n^3\,x^{5/2}}{2250\,e}-\frac {4669\,b^3\,d^4\,n^3\,x}{3600\,e^4}-\frac {a^2\,b\,d^6\,n\,\ln \left (d+e\,\sqrt {x}\right )}{e^6}+\frac {b^3\,d\,n\,x^{5/2}\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{5\,e}-\frac {11\,b^3\,d\,n^2\,x^{5/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{75\,e}-\frac {b^3\,d^4\,n\,x\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{2\,e^4}+\frac {29\,b^3\,d^4\,n^2\,x\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{20\,e^4}-\frac {a^2\,b\,d^2\,n\,x^2}{4\,e^2}-\frac {11\,a\,b^2\,d\,n^2\,x^{5/2}}{75\,e}+\frac {29\,a\,b^2\,d^4\,n^2\,x}{20\,e^4}+\frac {a^2\,b\,d^3\,n\,x^{3/2}}{3\,e^3}+\frac {a^2\,b\,d^5\,n\,\sqrt {x}}{e^5}+\frac {49\,a\,b^2\,d^6\,n^2\,\ln \left (d+e\,\sqrt {x}\right )}{10\,e^6}-\frac {b^3\,d^2\,n\,x^2\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{4\,e^2}+\frac {37\,b^3\,d^2\,n^2\,x^2\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{120\,e^2}+\frac {b^3\,d^3\,n\,x^{3/2}\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{3\,e^3}-\frac {19\,b^3\,d^3\,n^2\,x^{3/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{30\,e^3}+\frac {b^3\,d^5\,n\,\sqrt {x}\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{e^5}-\frac {49\,b^3\,d^5\,n^2\,\sqrt {x}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{10\,e^5}+\frac {37\,a\,b^2\,d^2\,n^2\,x^2}{120\,e^2}-\frac {19\,a\,b^2\,d^3\,n^2\,x^{3/2}}{30\,e^3}-\frac {49\,a\,b^2\,d^5\,n^2\,\sqrt {x}}{10\,e^5}+\frac {a^2\,b\,d\,n\,x^{5/2}}{5\,e}-\frac {a^2\,b\,d^4\,n\,x}{2\,e^4}+\frac {2\,a\,b^2\,d\,n\,x^{5/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{5\,e}-\frac {a\,b^2\,d^4\,n\,x\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{e^4}-\frac {a\,b^2\,d^2\,n\,x^2\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{2\,e^2}+\frac {2\,a\,b^2\,d^3\,n\,x^{3/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3\,e^3}+\frac {2\,a\,b^2\,d^5\,n\,\sqrt {x}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{e^5} \]
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