Integrand size = 25, antiderivative size = 439 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\frac {(c+d x)^3}{3 a^3 d}+\frac {d^2 x}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {d (c+d x)}{a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac {3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac {(c+d x)^2}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {(c+d x)^2}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {d^2 \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac {3 d (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}+\frac {3 d^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac {2 d^2 \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)} \]
1/3*(d*x+c)^3/a^3/d+d^2*x/a^3/f^2/g^2/n^2/ln(F)^2-d*(d*x+c)/a^2/f^2/(a+b*( F^(g*(f*x+e)))^n)/g^2/n^2/ln(F)^2-3/2*(d*x+c)^2/a^3/f/g/n/ln(F)+1/2*(d*x+c )^2/a/f/(a+b*(F^(g*(f*x+e)))^n)^2/g/n/ln(F)+(d*x+c)^2/a^2/f/(a+b*(F^(g*(f* x+e)))^n)/g/n/ln(F)-d^2*ln(a+b*(F^(g*(f*x+e)))^n)/a^3/f^3/g^3/n^3/ln(F)^3+ 3*d*(d*x+c)*ln(1+b*(F^(g*(f*x+e)))^n/a)/a^3/f^2/g^2/n^2/ln(F)^2-(d*x+c)^2* ln(1+b*(F^(g*(f*x+e)))^n/a)/a^3/f/g/n/ln(F)+3*d^2*polylog(2,-b*(F^(g*(f*x+ e)))^n/a)/a^3/f^3/g^3/n^3/ln(F)^3-2*d*(d*x+c)*polylog(2,-b*(F^(g*(f*x+e))) ^n/a)/a^3/f^2/g^2/n^2/ln(F)^2+2*d^2*polylog(3,-b*(F^(g*(f*x+e)))^n/a)/a^3/ f^3/g^3/n^3/ln(F)^3
\[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx \]
Time = 4.03 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.42, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {2616, 2616, 2615, 2620, 2621, 2615, 2616, 2615, 2620, 2621, 2715, 2720, 798, 47, 14, 16, 2838, 3011, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx\) |
\(\Big \downarrow \) 2616 |
\(\displaystyle \frac {\int \frac {(c+d x)^2}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^3}dx}{a}\) |
\(\Big \downarrow \) 2616 |
\(\displaystyle \frac {\frac {\int \frac {(c+d x)^2}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^3}dx}{a}\) |
\(\Big \downarrow \) 2615 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^3}dx}{a}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^3}dx}{a}\) |
\(\Big \downarrow \) 2621 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \int \frac {c+d x}{b \left (F^{g (e+f x)}\right )^n+a}dx}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \int \frac {c+d x}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 2615 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \int \frac {c+d x}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 2616 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\int \frac {c+d x}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 2615 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 2621 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {d \int \frac {1}{b \left (F^{g (e+f x)}\right )^n+a}dx}{b f g n \log (F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \int \frac {1}{b \left (F^{g (e+f x)}\right )^n+a}dx}{b f g n \log (F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \int \frac {F^{-g (e+f x)}}{b \left (F^{g (e+f x)}\right )^n+a}dF^{g (e+f x)}}{b f^2 g^2 n \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \int \frac {F^{-g (e+f x)}}{b \left (F^{g (e+f x)}\right )^n+a}d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 47 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\int F^{-g (e+f x)}d\left (F^{g (e+f x)}\right )^n}{a}-\frac {b \int \frac {1}{b \left (F^{g (e+f x)}\right )^n+a}d\left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a}-\frac {b \int \frac {1}{b \left (F^{g (e+f x)}\right )^n+a}d\left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}+\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}+\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \left (\frac {d \int \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )dx}{f g n \log (F)}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}+\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}+\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \left (\frac {d \int F^{-g (e+f x)} \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )dF^{g (e+f x)}}{f^2 g^2 n \log ^2(F)}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}+\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}+\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}+\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}+\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
-((b*(-1/2*(c + d*x)^2/(b*f*(a + b*(F^(g*(e + f*x)))^n)^2*g*n*Log[F]) + (d *(-((b*(-((c + d*x)/(b*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F])) + (d*(Lo g[(F^(g*(e + f*x)))^n]/a - Log[a + b*(F^(g*(e + f*x)))^n]/a))/(b*f^2*g^2*n ^2*Log[F]^2)))/a) + ((c + d*x)^2/(2*a*d) - (b*(((c + d*x)*Log[1 + (b*(F^(g *(e + f*x)))^n)/a])/(b*f*g*n*Log[F]) + (d*PolyLog[2, -((b*(F^(g*(e + f*x)) )^n)/a)])/(b*f^2*g^2*n^2*Log[F]^2)))/a)/a))/(b*f*g*n*Log[F])))/a) + (-((b* (-((c + d*x)^2/(b*f*(a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F])) + (2*d*((c + d*x)^2/(2*a*d) - (b*(((c + d*x)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g *n*Log[F]) + (d*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(b*f^2*g^2*n^2*L og[F]^2)))/a))/(b*f*g*n*Log[F])))/a) + ((c + d*x)^3/(3*a*d) - (b*(((c + d* x)^2*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]) - (2*d*(-(((c + d*x)*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(f*g*n*Log[F])) + (d*PolyLo g[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(f^2*g^2*n^2*Log[F]^2)))/(b*f*g*n*Log[ F])))/a)/a)/a
3.1.59.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x _))))^(n_.)), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(a*d*(m + 1)), x] - Simp[ b/a Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n)), x] , x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[1/a Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Simp[b/a Int[(c + d*x)^m*(F^(g*(e + f*x)))^ n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n }, x] && ILtQ[p, 0] && IGtQ[m, 0]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((a_.) + (b_.)*((F_)^((g_.)*( (e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*((a + b*(F^(g*(e + f*x)))^n)^(p + 1)/(b*f*g*n*(p + 1)*Log [F])), x] - Simp[d*(m/(b*f*g*n*(p + 1)*Log[F])) Int[(c + d*x)^(m - 1)*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, m, n, p}, x] && NeQ[p, -1]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Leaf count of result is larger than twice the leaf count of optimal. \(1886\) vs. \(2(433)=866\).
Time = 0.43 (sec) , antiderivative size = 1887, normalized size of antiderivative = 4.30
-2/3/a^3/ln(F)^3/f^3/g^3*d^2*ln(F^(g*(f*x+e)))^3-1/a^3/ln(F)/f/g/n*c^2*ln( (F^(g*(f*x+e)))^n*F^(-n*g*f*x)*F^(n*g*f*x)*b+a)+1/a^3/ln(F)/f/g/n*c^2*ln(F ^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)+2/a^3/ln(F)^3/f^3/g^3/n^3*d^2*p olylog(3,-b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)+3/a^3/ln(F)^3/f^ 3/g^3/n^3*d^2*polylog(2,-b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)-1 /a^3/ln(F)^3/f^3/g^3/n^3*d^2*ln((F^(g*(f*x+e)))^n*F^(-n*g*f*x)*F^(n*g*f*x) *b+a)+1/a^3/ln(F)^3/f^3/g^3/n^3*d^2*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x +e)))^n)-3/2/a^3/ln(F)^3/f^3/g^3/n*d^2*ln(F^(g*(f*x+e)))^2+1/a^3/ln(F)^2/f ^2/g^2*c*d*ln(F^(g*(f*x+e)))^2+1/a^3/ln(F)^2/f^2/g^2*d^2*ln(F^(g*(f*x+e))) ^2*x+1/2*(2*ln(F)*b*d^2*f*g*n*x^2*(F^(g*(f*x+e)))^n+3*ln(F)*a*d^2*f*g*n*x^ 2+4*ln(F)*b*c*d*f*g*n*x*(F^(g*(f*x+e)))^n+6*ln(F)*a*c*d*f*g*n*x+2*ln(F)*b* c^2*f*g*n*(F^(g*(f*x+e)))^n+3*ln(F)*a*c^2*f*g*n-2*b*d^2*x*(F^(g*(f*x+e)))^ n-2*a*d^2*x-2*b*c*d*(F^(g*(f*x+e)))^n-2*a*c*d)/n^2/g^2/f^2/ln(F)^2/a^2/(a+ b*(F^(g*(f*x+e)))^n)^2-3/a^3/ln(F)^2/f^2/g^2/n^2*d^2*ln(F^(n*g*f*x)*F^(-n* g*f*x)*(F^(g*(f*x+e)))^n)*x-3/a^3/ln(F)^3/f^3/g^3/n^2*d^2*ln((F^(g*(f*x+e) ))^n*F^(-n*g*f*x)*F^(n*g*f*x)*b+a)*ln(F^(g*(f*x+e)))+3/a^3/ln(F)^3/f^3/g^3 /n^2*d^2*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))+ 3/a^3/ln(F)^3/f^3/g^3/n^2*d^2*ln(F^(g*(f*x+e)))*ln(1+b*F^(n*g*f*x)*F^(-n*g *f*x)*(F^(g*(f*x+e)))^n/a)-1/a^3/ln(F)^3/f^3/g^3/n*d^2*ln((F^(g*(f*x+e)))^ n*F^(-n*g*f*x)*F^(n*g*f*x)*b+a)*ln(F^(g*(f*x+e)))^2+1/a^3/ln(F)^3/f^3/g...
Leaf count of result is larger than twice the leaf count of optimal. 1518 vs. \(2 (431) = 862\).
Time = 0.30 (sec) , antiderivative size = 1518, normalized size of antiderivative = 3.46 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\text {Too large to display} \]
1/6*(9*(a^2*d^2*e^2 - 2*a^2*c*d*e*f + a^2*c^2*f^2)*g^2*n^2*log(F)^2 + 6*(a ^2*d^2*e - a^2*c*d*f)*g*n*log(F) + 2*(a^2*d^2*f^3*g^3*n^3*x^3 + 3*a^2*c*d* f^3*g^3*n^3*x^2 + 3*a^2*c^2*f^3*g^3*n^3*x + (a^2*d^2*e^3 - 3*a^2*c*d*e^2*f + 3*a^2*c^2*e*f^2)*g^3*n^3)*log(F)^3 + (2*(b^2*d^2*f^3*g^3*n^3*x^3 + 3*b^ 2*c*d*f^3*g^3*n^3*x^2 + 3*b^2*c^2*f^3*g^3*n^3*x + (b^2*d^2*e^3 - 3*b^2*c*d *e^2*f + 3*b^2*c^2*e*f^2)*g^3*n^3)*log(F)^3 - 9*(b^2*d^2*f^2*g^2*n^2*x^2 + 2*b^2*c*d*f^2*g^2*n^2*x - (b^2*d^2*e^2 - 2*b^2*c*d*e*f)*g^2*n^2)*log(F)^2 + 6*(b^2*d^2*f*g*n*x + b^2*d^2*e*g*n)*log(F))*F^(2*f*g*n*x + 2*e*g*n) + 2 *(2*(a*b*d^2*f^3*g^3*n^3*x^3 + 3*a*b*c*d*f^3*g^3*n^3*x^2 + 3*a*b*c^2*f^3*g ^3*n^3*x + (a*b*d^2*e^3 - 3*a*b*c*d*e^2*f + 3*a*b*c^2*e*f^2)*g^3*n^3)*log( F)^3 - 3*(2*a*b*d^2*f^2*g^2*n^2*x^2 + 4*a*b*c*d*f^2*g^2*n^2*x - (3*a*b*d^2 *e^2 - 6*a*b*c*d*e*f + a*b*c^2*f^2)*g^2*n^2)*log(F)^2 + 3*(a*b*d^2*f*g*n*x + (2*a*b*d^2*e - a*b*c*d*f)*g*n)*log(F))*F^(f*g*n*x + e*g*n) + 6*(3*a^2*d ^2 + (3*b^2*d^2 - 2*(b^2*d^2*f*g*n*x + b^2*c*d*f*g*n)*log(F))*F^(2*f*g*n*x + 2*e*g*n) + 2*(3*a*b*d^2 - 2*(a*b*d^2*f*g*n*x + a*b*c*d*f*g*n)*log(F))*F ^(f*g*n*x + e*g*n) - 2*(a^2*d^2*f*g*n*x + a^2*c*d*f*g*n)*log(F))*dilog(-(F ^(f*g*n*x + e*g*n)*b + a)/a + 1) - 6*((a^2*d^2*e^2 - 2*a^2*c*d*e*f + a^2*c ^2*f^2)*g^2*n^2*log(F)^2 + a^2*d^2 + 3*(a^2*d^2*e - a^2*c*d*f)*g*n*log(F) + ((b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*g^2*n^2*log(F)^2 + b^2*d^2 + 3*(b^2*d^2*e - b^2*c*d*f)*g*n*log(F))*F^(2*f*g*n*x + 2*e*g*n) + 2*((a...
\[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {\left (c + d x\right )^{2}}{\left (a + b \left (F^{e g + f g x}\right )^{n}\right )^{3}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.58 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\frac {1}{2} \, c^{2} {\left (\frac {2 \, F^{f g n x + e g n} b + 3 \, a}{{\left (2 \, F^{f g n x + e g n} a^{3} b + F^{2 \, f g n x + 2 \, e g n} a^{2} b^{2} + a^{4}\right )} f g n \log \left (F\right )} + \frac {2 \, {\left (f g n x + e g n\right )}}{a^{3} f g n} - \frac {2 \, \log \left (F^{f g n x + e g n} b + a\right )}{a^{3} f g n \log \left (F\right )}\right )} + \frac {3 \, a d^{2} f g n x^{2} \log \left (F\right ) - 2 \, a c d + 2 \, {\left (F^{e g n} b d^{2} f g n x^{2} \log \left (F\right ) - F^{e g n} b c d + {\left (2 \, F^{e g n} b c d f g n \log \left (F\right ) - F^{e g n} b d^{2}\right )} x\right )} F^{f g n x} + 2 \, {\left (3 \, a c d f g n \log \left (F\right ) - a d^{2}\right )} x}{2 \, {\left (2 \, F^{f g n x} F^{e g n} a^{3} b f^{2} g^{2} n^{2} \log \left (F\right )^{2} + F^{2 \, f g n x} F^{2 \, e g n} a^{2} b^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2} + a^{4} f^{2} g^{2} n^{2} \log \left (F\right )^{2}\right )}} - \frac {{\left (3 \, c d f g n \log \left (F\right ) - d^{2}\right )} x}{a^{3} f^{2} g^{2} n^{2} \log \left (F\right )^{2}} - \frac {{\left (f^{2} g^{2} n^{2} x^{2} \log \left (\frac {F^{f g n x} F^{e g n} b}{a} + 1\right ) \log \left (F\right )^{2} + 2 \, f g n x {\rm Li}_2\left (-\frac {F^{f g n x} F^{e g n} b}{a}\right ) \log \left (F\right ) - 2 \, {\rm Li}_{3}(-\frac {F^{f g n x} F^{e g n} b}{a})\right )} d^{2}}{a^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} - \frac {{\left (2 \, c d f g n \log \left (F\right ) - 3 \, d^{2}\right )} {\left (f g n x \log \left (\frac {F^{f g n x} F^{e g n} b}{a} + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{f g n x} F^{e g n} b}{a}\right )\right )}}{a^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {{\left (3 \, c d f g n \log \left (F\right ) - d^{2}\right )} \log \left (F^{f g n x} F^{e g n} b + a\right )}{a^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {2 \, d^{2} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} + 3 \, {\left (2 \, c d f g n \log \left (F\right ) - 3 \, d^{2}\right )} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2}}{6 \, a^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \]
1/2*c^2*((2*F^(f*g*n*x + e*g*n)*b + 3*a)/((2*F^(f*g*n*x + e*g*n)*a^3*b + F ^(2*f*g*n*x + 2*e*g*n)*a^2*b^2 + a^4)*f*g*n*log(F)) + 2*(f*g*n*x + e*g*n)/ (a^3*f*g*n) - 2*log(F^(f*g*n*x + e*g*n)*b + a)/(a^3*f*g*n*log(F))) + 1/2*( 3*a*d^2*f*g*n*x^2*log(F) - 2*a*c*d + 2*(F^(e*g*n)*b*d^2*f*g*n*x^2*log(F) - F^(e*g*n)*b*c*d + (2*F^(e*g*n)*b*c*d*f*g*n*log(F) - F^(e*g*n)*b*d^2)*x)*F ^(f*g*n*x) + 2*(3*a*c*d*f*g*n*log(F) - a*d^2)*x)/(2*F^(f*g*n*x)*F^(e*g*n)* a^3*b*f^2*g^2*n^2*log(F)^2 + F^(2*f*g*n*x)*F^(2*e*g*n)*a^2*b^2*f^2*g^2*n^2 *log(F)^2 + a^4*f^2*g^2*n^2*log(F)^2) - (3*c*d*f*g*n*log(F) - d^2)*x/(a^3* f^2*g^2*n^2*log(F)^2) - (f^2*g^2*n^2*x^2*log(F^(f*g*n*x)*F^(e*g*n)*b/a + 1 )*log(F)^2 + 2*f*g*n*x*dilog(-F^(f*g*n*x)*F^(e*g*n)*b/a)*log(F) - 2*polylo g(3, -F^(f*g*n*x)*F^(e*g*n)*b/a))*d^2/(a^3*f^3*g^3*n^3*log(F)^3) - (2*c*d* f*g*n*log(F) - 3*d^2)*(f*g*n*x*log(F^(f*g*n*x)*F^(e*g*n)*b/a + 1)*log(F) + dilog(-F^(f*g*n*x)*F^(e*g*n)*b/a))/(a^3*f^3*g^3*n^3*log(F)^3) + (3*c*d*f* g*n*log(F) - d^2)*log(F^(f*g*n*x)*F^(e*g*n)*b + a)/(a^3*f^3*g^3*n^3*log(F) ^3) + 1/6*(2*d^2*f^3*g^3*n^3*x^3*log(F)^3 + 3*(2*c*d*f*g*n*log(F) - 3*d^2) *f^2*g^2*n^2*x^2*log(F)^2)/(a^3*f^3*g^3*n^3*log(F)^3)
\[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^3} \,d x \]