Integrand size = 24, antiderivative size = 595 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=\frac {9 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^4}-\frac {4 b^3 d n^3 \left (d+\frac {e}{\sqrt {x}}\right )^3}{9 e^4}+\frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{64 e^4}+\frac {12 a b^2 d^3 n^2}{e^3 \sqrt {x}}-\frac {12 b^3 d^3 n^3}{e^3 \sqrt {x}}+\frac {12 b^3 d^3 n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^4}-\frac {9 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^4}+\frac {4 b^2 d n^2 \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 e^4}-\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{16 e^4}-\frac {6 b d^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^4}+\frac {9 b d^2 n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^4}-\frac {2 b d n \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^4}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{8 e^4}+\frac {2 d^3 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}-\frac {3 d^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 e^4} \]
[Out]
Time = 0.43 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.00, number of steps used = 20, 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 \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=-\frac {9 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^4}-\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{16 e^4}+\frac {4 b^2 d n^2 \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 e^4}+\frac {12 a b^2 d^3 n^2}{e^3 \sqrt {x}}+\frac {2 d^3 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}-\frac {6 b d^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^4}-\frac {3 d^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}+\frac {9 b d^2 n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^4}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 e^4}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{8 e^4}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}-\frac {2 b d n \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^4}+\frac {12 b^3 d^3 n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^4}-\frac {12 b^3 d^3 n^3}{e^3 \sqrt {x}}+\frac {9 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^4}+\frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{64 e^4}-\frac {4 b^3 d n^3 \left (d+\frac {e}{\sqrt {x}}\right )^3}{9 e^4} \]
[In]
[Out]
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2436
Rule 2437
Rule 2448
Rule 2504
Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -\left (2 \text {Subst}\left (\int \left (-\frac {d^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {3 d^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}-\frac {3 d (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {(d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -\frac {2 \text {Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^3}+\frac {(6 d) \text {Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^3}-\frac {\left (6 d^2\right ) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^3}+\frac {\left (2 d^3\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt {x}}\right )}{e^3} \\ & = -\frac {2 \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^4}+\frac {(6 d) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^4}-\frac {\left (6 d^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^4}+\frac {\left (2 d^3\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^4} \\ & = \frac {2 d^3 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}-\frac {3 d^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 e^4}+\frac {(3 b n) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{2 e^4}-\frac {(6 b d n) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^4}+\frac {\left (9 b d^2 n\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^4}-\frac {\left (6 b d^3 n\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^4} \\ & = -\frac {6 b d^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^4}+\frac {9 b d^2 n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^4}-\frac {2 b d n \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^4}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{8 e^4}+\frac {2 d^3 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}-\frac {3 d^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 e^4}-\frac {\left (3 b^2 n^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{4 e^4}+\frac {\left (4 b^2 d n^2\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^4}-\frac {\left (9 b^2 d^2 n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^4}+\frac {\left (12 b^2 d^3 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^4} \\ & = \frac {9 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^4}-\frac {4 b^3 d n^3 \left (d+\frac {e}{\sqrt {x}}\right )^3}{9 e^4}+\frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{64 e^4}+\frac {12 a b^2 d^3 n^2}{e^3 \sqrt {x}}-\frac {9 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^4}+\frac {4 b^2 d n^2 \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 e^4}-\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{16 e^4}-\frac {6 b d^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^4}+\frac {9 b d^2 n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^4}-\frac {2 b d n \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^4}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{8 e^4}+\frac {2 d^3 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}-\frac {3 d^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 e^4}+\frac {\left (12 b^3 d^3 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^4} \\ & = \frac {9 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^4}-\frac {4 b^3 d n^3 \left (d+\frac {e}{\sqrt {x}}\right )^3}{9 e^4}+\frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^4}{64 e^4}+\frac {12 a b^2 d^3 n^2}{e^3 \sqrt {x}}-\frac {12 b^3 d^3 n^3}{e^3 \sqrt {x}}+\frac {12 b^3 d^3 n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^4}-\frac {9 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^4}+\frac {4 b^2 d n^2 \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{3 e^4}-\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{16 e^4}-\frac {6 b d^3 n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^4}+\frac {9 b d^2 n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^4}-\frac {2 b d n \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^4}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{8 e^4}+\frac {2 d^3 \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}-\frac {3 d^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^4}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{2 e^4} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 766, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=\frac {-288 a^3 e^4+216 a^2 b e^4 n-108 a b^2 e^4 n^2+27 b^3 e^4 n^3-288 a^2 b d e^3 n \sqrt {x}+336 a b^2 d e^3 n^2 \sqrt {x}-148 b^3 d e^3 n^3 \sqrt {x}+432 a^2 b d^2 e^2 n x-936 a b^2 d^2 e^2 n^2 x+690 b^3 d^2 e^2 n^3 x-864 a^2 b d^3 e n x^{3/2}+3600 a b^2 d^3 e n^2 x^{3/2}-4980 b^3 d^3 e n^3 x^{3/2}-576 b^3 d^4 n^3 x^2 \log ^3\left (d+\frac {e}{\sqrt {x}}\right )-288 b^3 e^4 \log ^3\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+864 a^2 b d^4 n x^2 \log \left (e+d \sqrt {x}\right )-3600 a b^2 d^4 n^2 x^2 \log \left (e+d \sqrt {x}\right )+4980 b^3 d^4 n^3 x^2 \log \left (e+d \sqrt {x}\right )+72 b^2 d^4 n^2 x^2 \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (-12 a+25 b n-12 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (2 \log \left (e+d \sqrt {x}\right )-\log (x)\right )-432 a^2 b d^4 n x^2 \log (x)+1800 a b^2 d^4 n^2 x^2 \log (x)-2490 b^3 d^4 n^3 x^2 \log (x)+72 b^2 d^4 n^2 x^2 \log ^2\left (d+\frac {e}{\sqrt {x}}\right ) \left (12 a-25 b n+12 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+12 b n \log \left (e+d \sqrt {x}\right )-6 b n \log (x)\right )+72 b^2 \log ^2\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (e \left (-12 a e^3+3 b e^3 n-4 b d e^2 n \sqrt {x}+6 b d^2 e n x-12 b d^3 n x^{3/2}\right )+12 b d^4 n x^2 \log \left (e+d \sqrt {x}\right )-6 b d^4 n x^2 \log (x)\right )-12 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (72 a^2 e^4+b^2 e n^2 \left (9 e^3-28 d e^2 \sqrt {x}+78 d^2 e x-300 d^3 x^{3/2}\right )-12 a b e n \left (3 e^3-4 d e^2 \sqrt {x}+6 d^2 e x-12 d^3 x^{3/2}\right )+12 b d^4 n (-12 a+25 b n) x^2 \log \left (e+d \sqrt {x}\right )+6 b d^4 n (12 a-25 b n) x^2 \log (x)\right )}{576 e^4 x^2} \]
[In]
[Out]
\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{3}}{x^{3}}d x\]
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 869, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=\frac {27 \, b^{3} e^{4} n^{3} - 288 \, b^{3} e^{4} \log \left (c\right )^{3} - 108 \, a b^{2} e^{4} n^{2} + 216 \, a^{2} b e^{4} n - 288 \, a^{3} e^{4} + 288 \, {\left (b^{3} d^{4} n^{3} x^{2} - b^{3} e^{4} n^{3}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right )^{3} + 216 \, {\left (2 \, b^{3} d^{2} e^{2} n x + b^{3} e^{4} n - 4 \, a b^{2} e^{4}\right )} \log \left (c\right )^{2} + 72 \, {\left (6 \, b^{3} d^{2} e^{2} n^{3} x + 3 \, b^{3} e^{4} n^{3} - 12 \, a b^{2} e^{4} n^{2} - {\left (25 \, b^{3} d^{4} n^{3} - 12 \, a b^{2} d^{4} n^{2}\right )} x^{2} + 12 \, {\left (b^{3} d^{4} n^{2} x^{2} - b^{3} e^{4} n^{2}\right )} \log \left (c\right ) - 4 \, {\left (3 \, b^{3} d^{3} e n^{3} x + b^{3} d e^{3} n^{3}\right )} \sqrt {x}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right )^{2} + 6 \, {\left (115 \, b^{3} d^{2} e^{2} n^{3} - 156 \, a b^{2} d^{2} e^{2} n^{2} + 72 \, a^{2} b d^{2} e^{2} n\right )} x - 36 \, {\left (3 \, b^{3} e^{4} n^{2} - 12 \, a b^{2} e^{4} n + 24 \, a^{2} b e^{4} + 2 \, {\left (13 \, b^{3} d^{2} e^{2} n^{2} - 12 \, a b^{2} d^{2} e^{2} n\right )} x\right )} \log \left (c\right ) - 12 \, {\left (9 \, b^{3} e^{4} n^{3} - 36 \, a b^{2} e^{4} n^{2} + 72 \, a^{2} b e^{4} n - {\left (415 \, b^{3} d^{4} n^{3} - 300 \, a b^{2} d^{4} n^{2} + 72 \, a^{2} b d^{4} n\right )} x^{2} - 72 \, {\left (b^{3} d^{4} n x^{2} - b^{3} e^{4} n\right )} \log \left (c\right )^{2} + 6 \, {\left (13 \, b^{3} d^{2} e^{2} n^{3} - 12 \, a b^{2} d^{2} e^{2} n^{2}\right )} x - 12 \, {\left (6 \, b^{3} d^{2} e^{2} n^{2} x + 3 \, b^{3} e^{4} n^{2} - 12 \, a b^{2} e^{4} n - {\left (25 \, b^{3} d^{4} n^{2} - 12 \, a b^{2} d^{4} n\right )} x^{2}\right )} \log \left (c\right ) - 4 \, {\left (7 \, b^{3} d e^{3} n^{3} - 12 \, a b^{2} d e^{3} n^{2} + 3 \, {\left (25 \, b^{3} d^{3} e n^{3} - 12 \, a b^{2} d^{3} e n^{2}\right )} x - 12 \, {\left (3 \, b^{3} d^{3} e n^{2} x + b^{3} d e^{3} n^{2}\right )} \log \left (c\right )\right )} \sqrt {x}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) - 4 \, {\left (37 \, b^{3} d e^{3} n^{3} - 84 \, a b^{2} d e^{3} n^{2} + 72 \, a^{2} b d e^{3} n + 72 \, {\left (3 \, b^{3} d^{3} e n x + b^{3} d e^{3} n\right )} \log \left (c\right )^{2} + 3 \, {\left (415 \, b^{3} d^{3} e n^{3} - 300 \, a b^{2} d^{3} e n^{2} + 72 \, a^{2} b d^{3} e n\right )} x - 12 \, {\left (7 \, b^{3} d e^{3} n^{2} - 12 \, a b^{2} d e^{3} n + 3 \, {\left (25 \, b^{3} d^{3} e n^{2} - 12 \, a b^{2} d^{3} e n\right )} x\right )} \log \left (c\right )\right )} \sqrt {x}}{576 \, e^{4} x^{2}} \]
[In]
[Out]
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{3}}{x^{3}}\, dx \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 732, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=\frac {1}{8} \, a^{2} b e n {\left (\frac {12 \, d^{4} \log \left (d \sqrt {x} + e\right )}{e^{5}} - \frac {6 \, d^{4} \log \left (x\right )}{e^{5}} - \frac {12 \, d^{3} x^{\frac {3}{2}} - 6 \, d^{2} e x + 4 \, d e^{2} \sqrt {x} - 3 \, e^{3}}{e^{4} x^{2}}\right )} + \frac {1}{48} \, {\left (12 \, e n {\left (\frac {12 \, d^{4} \log \left (d \sqrt {x} + e\right )}{e^{5}} - \frac {6 \, d^{4} \log \left (x\right )}{e^{5}} - \frac {12 \, d^{3} x^{\frac {3}{2}} - 6 \, d^{2} e x + 4 \, d e^{2} \sqrt {x} - 3 \, e^{3}}{e^{4} x^{2}}\right )} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) - \frac {{\left (72 \, d^{4} x^{2} \log \left (d \sqrt {x} + e\right )^{2} + 18 \, d^{4} x^{2} \log \left (x\right )^{2} - 150 \, d^{4} x^{2} \log \left (x\right ) - 300 \, d^{3} e x^{\frac {3}{2}} + 78 \, d^{2} e^{2} x - 28 \, d e^{3} \sqrt {x} + 9 \, e^{4} - 12 \, {\left (6 \, d^{4} x^{2} \log \left (x\right ) - 25 \, d^{4} x^{2}\right )} \log \left (d \sqrt {x} + e\right )\right )} n^{2}}{e^{4} x^{2}}\right )} a b^{2} + \frac {1}{576} \, {\left (72 \, e n {\left (\frac {12 \, d^{4} \log \left (d \sqrt {x} + e\right )}{e^{5}} - \frac {6 \, d^{4} \log \left (x\right )}{e^{5}} - \frac {12 \, d^{3} x^{\frac {3}{2}} - 6 \, d^{2} e x + 4 \, d e^{2} \sqrt {x} - 3 \, e^{3}}{e^{4} x^{2}}\right )} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{2} + e n {\left (\frac {{\left (288 \, d^{4} x^{2} \log \left (d \sqrt {x} + e\right )^{3} - 36 \, d^{4} x^{2} \log \left (x\right )^{3} + 450 \, d^{4} x^{2} \log \left (x\right )^{2} - 2490 \, d^{4} x^{2} \log \left (x\right ) - 4980 \, d^{3} e x^{\frac {3}{2}} + 690 \, d^{2} e^{2} x - 148 \, d e^{3} \sqrt {x} + 27 \, e^{4} - 72 \, {\left (6 \, d^{4} x^{2} \log \left (x\right ) - 25 \, d^{4} x^{2}\right )} \log \left (d \sqrt {x} + e\right )^{2} + 12 \, {\left (18 \, d^{4} x^{2} \log \left (x\right )^{2} - 150 \, d^{4} x^{2} \log \left (x\right ) + 415 \, d^{4} x^{2}\right )} \log \left (d \sqrt {x} + e\right )\right )} n^{2}}{e^{5} x^{2}} - \frac {12 \, {\left (72 \, d^{4} x^{2} \log \left (d \sqrt {x} + e\right )^{2} + 18 \, d^{4} x^{2} \log \left (x\right )^{2} - 150 \, d^{4} x^{2} \log \left (x\right ) - 300 \, d^{3} e x^{\frac {3}{2}} + 78 \, d^{2} e^{2} x - 28 \, d e^{3} \sqrt {x} + 9 \, e^{4} - 12 \, {\left (6 \, d^{4} x^{2} \log \left (x\right ) - 25 \, d^{4} x^{2}\right )} \log \left (d \sqrt {x} + e\right )\right )} n \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{e^{5} x^{2}}\right )}\right )} b^{3} - \frac {b^{3} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{3}}{2 \, x^{2}} - \frac {3 \, a b^{2} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{2}}{2 \, x^{2}} - \frac {3 \, a^{2} b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{2 \, x^{2}} - \frac {a^{3}}{2 \, x^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1147 vs. \(2 (519) = 1038\).
Time = 0.40 (sec) , antiderivative size = 1147, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 2.03 (sec) , antiderivative size = 846, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^3} \, dx=\frac {\frac {d\,\left (2\,a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{4}-\frac {3\,b^3\,n^3}{16}\right )}{3\,e}-\frac {d\,\left (24\,a^3-12\,a\,b^2\,n^2+7\,b^3\,n^3\right )}{36\,e}}{x^{3/2}}-{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^3\,\left (\frac {b^3}{2\,x^2}-\frac {b^3\,d^4}{2\,e^4}\right )+\frac {\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (2\,a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{4}-\frac {3\,b^3\,n^3}{16}\right )}{e}-\frac {d\,\left (24\,a^3-12\,a\,b^2\,n^2+7\,b^3\,n^3\right )}{12\,e}\right )}{e}+\frac {b^2\,d^2\,n^2\,\left (12\,a-13\,b\,n\right )}{8\,e^2}\right )}{e}+\frac {b^2\,d^3\,n^2\,\left (12\,a-25\,b\,n\right )}{4\,e^3}}{\sqrt {x}}+{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^2\,\left (\frac {\frac {b^2\,d\,\left (4\,a-b\,n\right )}{e}-\frac {4\,a\,b^2\,d}{e}}{2\,x^{3/2}}-\frac {3\,b^2\,\left (4\,a-b\,n\right )}{8\,x^2}+\frac {d\,\left (12\,a\,b^2\,d^3-25\,b^3\,d^3\,n\right )}{8\,e^4}-\frac {d\,\left (\frac {6\,b^2\,d\,\left (4\,a-b\,n\right )}{e}-\frac {24\,a\,b^2\,d}{e}\right )}{8\,e\,x}+\frac {d^2\,\left (\frac {6\,b^2\,d\,\left (4\,a-b\,n\right )}{e}-\frac {24\,a\,b^2\,d}{e}\right )}{4\,e^2\,\sqrt {x}}\right )-\frac {\frac {d\,\left (\frac {d\,\left (2\,a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{4}-\frac {3\,b^3\,n^3}{16}\right )}{e}-\frac {d\,\left (24\,a^3-12\,a\,b^2\,n^2+7\,b^3\,n^3\right )}{12\,e}\right )}{2\,e}+\frac {b^2\,d^2\,n^2\,\left (12\,a-13\,b\,n\right )}{16\,e^2}}{x}-\frac {\frac {a^3}{2}-\frac {3\,a^2\,b\,n}{8}+\frac {3\,a\,b^2\,n^2}{16}-\frac {3\,b^3\,n^3}{64}}{x^2}-\frac {\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\,\left (\frac {16\,b\,d\,e^3\,\left (6\,a^2-b^2\,n^2\right )-12\,b\,d\,e^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{12\,e^2\,x^{3/2}}+\frac {\frac {d\,\left (\frac {d\,\left (16\,b\,d\,e^3\,\left (6\,a^2-b^2\,n^2\right )-12\,b\,d\,e^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )\right )}{e}-24\,b^3\,d^2\,e^2\,n^2\right )}{e}-48\,b^3\,d^3\,e\,n^2}{4\,e^2\,\sqrt {x}}-\frac {\frac {d\,\left (16\,b\,d\,e^3\,\left (6\,a^2-b^2\,n^2\right )-12\,b\,d\,e^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )\right )}{e}-24\,b^3\,d^2\,e^2\,n^2}{8\,e^2\,x}+\frac {3\,b\,e^2\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{4\,x^2}\right )}{4\,e^2}+\frac {\ln \left (d+\frac {e}{\sqrt {x}}\right )\,\left (72\,a^2\,b\,d^4\,n-300\,a\,b^2\,d^4\,n^2+415\,b^3\,d^4\,n^3\right )}{48\,e^4} \]
[In]
[Out]