Integrand size = 25, antiderivative size = 594 \[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\frac {(c+d x)^4}{4 a^3 d}+\frac {3 d (c+d x)^2}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {3 d (c+d x)^2}{2 a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac {3 (c+d x)^3}{2 a^3 f g n \log (F)}+\frac {(c+d x)^3}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {(c+d x)^3}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {3 d^2 (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac {9 d (c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{2 a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}-\frac {3 d^3 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}+\frac {9 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {9 d^3 \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {6 d^3 \operatorname {PolyLog}\left (4,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^4 g^4 n^4 \log ^4(F)} \]
1/4*(d*x+c)^4/a^3/d+3/2*d*(d*x+c)^2/a^3/f^2/g^2/n^2/ln(F)^2-3/2*d*(d*x+c)^ 2/a^2/f^2/(a+b*(F^(g*(f*x+e)))^n)/g^2/n^2/ln(F)^2-3/2*(d*x+c)^3/a^3/f/g/n/ ln(F)+1/2*(d*x+c)^3/a/f/(a+b*(F^(g*(f*x+e)))^n)^2/g/n/ln(F)+(d*x+c)^3/a^2/ f/(a+b*(F^(g*(f*x+e)))^n)/g/n/ln(F)-3*d^2*(d*x+c)*ln(1+b*(F^(g*(f*x+e)))^n /a)/a^3/f^3/g^3/n^3/ln(F)^3+9/2*d*(d*x+c)^2*ln(1+b*(F^(g*(f*x+e)))^n/a)/a^ 3/f^2/g^2/n^2/ln(F)^2-(d*x+c)^3*ln(1+b*(F^(g*(f*x+e)))^n/a)/a^3/f/g/n/ln(F )-3*d^3*polylog(2,-b*(F^(g*(f*x+e)))^n/a)/a^3/f^4/g^4/n^4/ln(F)^4+9*d^2*(d *x+c)*polylog(2,-b*(F^(g*(f*x+e)))^n/a)/a^3/f^3/g^3/n^3/ln(F)^3-3*d*(d*x+c )^2*polylog(2,-b*(F^(g*(f*x+e)))^n/a)/a^3/f^2/g^2/n^2/ln(F)^2-9*d^3*polylo g(3,-b*(F^(g*(f*x+e)))^n/a)/a^3/f^4/g^4/n^4/ln(F)^4+6*d^2*(d*x+c)*polylog( 3,-b*(F^(g*(f*x+e)))^n/a)/a^3/f^3/g^3/n^3/ln(F)^3-6*d^3*polylog(4,-b*(F^(g *(f*x+e)))^n/a)/a^3/f^4/g^4/n^4/ln(F)^4
\[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx \]
Time = 6.08 (sec) , antiderivative size = 867, normalized size of antiderivative = 1.46, 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, 2615, 2620, 2715, 2838, 3011, 2720, 7143, 7163, 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.
| \(\displaystyle \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx\) |
| \(\Big \downarrow \) 2616 |
| \(\displaystyle \frac {\int \frac {(c+d x)^3}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^3}dx}{a}\) |
| \(\Big \downarrow \) 2616 |
| \(\displaystyle \frac {\frac {\int \frac {(c+d x)^3}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^3}dx}{a}\) |
| \(\Big \downarrow \) 2615 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^3}dx}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^3}dx}{a}\) |
| \(\Big \downarrow \) 2621 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \int \frac {(c+d x)^2}{b \left (F^{g (e+f x)}\right )^n+a}dx}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {3 d \int \frac {(c+d x)^2}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
| \(\Big \downarrow \) 2615 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {3 d \int \frac {(c+d x)^2}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
| \(\Big \downarrow \) 2616 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {\int \frac {(c+d x)^2}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
| \(\Big \downarrow \) 2615 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
| \(\Big \downarrow \) 2621 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \int \frac {c+d x}{b \left (F^{g (e+f x)}\right )^n+a}dx}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
| \(\Big \downarrow \) 2615 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {d \int \left (F^{g (e+f x)}\right )^{-n} \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )d\left (F^{g (e+f x)}\right )^n}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}+\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \left (\frac {2 d \int (c+d x) \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)^2 \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 {3 d \left (\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}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f \left (b \left (F^{g (e+f x)}\right )^n+a\right ) g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\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 \left (b \left (F^{g (e+f x)}\right )^n+a\right ) g n \log (F)}\right )}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f \left (b \left (F^{g (e+f x)}\right )^n+a\right )^2 g n \log (F)}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \left (\frac {2 d \int (c+d x) \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)^2 \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 {3 d \left (\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}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f \left (b \left (F^{g (e+f x)}\right )^n+a\right ) g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\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 \left (b \left (F^{g (e+f x)}\right )^n+a\right ) g n \log (F)}\right )}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f \left (b \left (F^{g (e+f x)}\right )^n+a\right )^2 g n \log (F)}\right )}{a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \left (\frac {2 d \int (c+d x) \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)^2 \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 {3 d \left (\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}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f \left (b \left (F^{g (e+f x)}\right )^n+a\right ) g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\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 \left (b \left (F^{g (e+f x)}\right )^n+a\right ) g n \log (F)}\right )}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f \left (b \left (F^{g (e+f x)}\right )^n+a\right )^2 g n \log (F)}\right )}{a}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}-\frac {d \int \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )dx}{f g n \log (F)}\right )}{f g n \log (F)}-\frac {(c+d x)^2 \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 {3 d \left (\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}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f \left (b \left (F^{g (e+f x)}\right )^n+a\right ) g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\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 \left (b \left (F^{g (e+f x)}\right )^n+a\right ) g n \log (F)}\right )}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f \left (b \left (F^{g (e+f x)}\right )^n+a\right )^2 g n \log (F)}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}-\frac {d \int F^{-g (e+f x)} \operatorname {PolyLog}\left (3,-\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)}\right )}{f g n \log (F)}-\frac {(c+d x)^2 \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 {3 d \left (\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}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f \left (b \left (F^{g (e+f x)}\right )^n+a\right ) g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\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 \left (b \left (F^{g (e+f x)}\right )^n+a\right ) g n \log (F)}\right )}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f \left (b \left (F^{g (e+f x)}\right )^n+a\right )^2 g n \log (F)}\right )}{a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}-\frac {d \operatorname {PolyLog}\left (4,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f^2 g^2 n^2 \log ^2(F)}\right )}{f g n \log (F)}-\frac {(c+d x)^2 \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 {3 d \left (\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}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f \left (b \left (F^{g (e+f x)}\right )^n+a\right ) g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\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 \left (b \left (F^{g (e+f x)}\right )^n+a\right ) g n \log (F)}\right )}{a}\right )}{2 b f g n \log (F)}-\frac {(c+d x)^3}{2 b f \left (b \left (F^{g (e+f x)}\right )^n+a\right )^2 g n \log (F)}\right )}{a}\) |
-((b*(-1/2*(c + d*x)^3/(b*f*(a + b*(F^(g*(e + f*x)))^n)^2*g*n*Log[F]) + (3 *d*(-((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*Log[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*PolyLog[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))/(2*b*f*g*n*Log[F])))/a) + (-((b*(-((c + d*x)^3/(b*f* (a + b*(F^(g*(e + f*x)))^n)*g*n*Log[F])) + (3*d*((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*PolyLog[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))/(b*f*g*n*Log[F])))/a) + ((c + d*x)^4/(4*a*d) - (b*(((c + d*x)^3*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]) - (3*d*(-((( c + d*x)^2*PolyLog[2, -((b*(F^(g*(e + f*x)))^n)/a)])/(f*g*n*Log[F])) + (2* d*(((c + d*x)*PolyLog[3, -((b*(F^(g*(e + f*x)))^n)/a)])/(f*g*n*Log[F]) - ( d*PolyLog[4, -((b*(F^(g*(e + f*x)))^n)/a)])/(f^2*g^2*n^2*Log[F]^2)))/(f*g* n*Log[F])))/(b*f*g*n*Log[F])))/a)/a)/a
3.1.58.3.1 Defintions of rubi rules used
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]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(4004\) vs. \(2(582)=1164\).
Time = 0.44 (sec) , antiderivative size = 4005, normalized size of antiderivative = 6.74
-9/n^2/g^3/f^3/ln(F)^3/a^3*c*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)))-9/n^2/g^2/f^2/ln(F)^2/a^3*c*d^2*ln(F^(n*g*f*x) *F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)*x+9/n^2/g^3/f^3/ln(F)^3/a^3*c*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)))+9/n^2/g^3/f^3/ln (F)^3/a^3*c*d^2*ln(1+b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n/a)*ln(F^ (g*(f*x+e)))+9/n^2/g^3/f^3/ln(F)^3/a^3*d^3*ln(1+b*F^(n*g*f*x)*F^(-n*g*f*x) *(F^(g*(f*x+e)))^n/a)*ln(F^(g*(f*x+e)))*x+1/2*(2*ln(F)*b*d^3*f*g*n*x^3*(F^ (g*(f*x+e)))^n+3*ln(F)*a*d^3*f*g*n*x^3+6*ln(F)*b*c*d^2*f*g*n*x^2*(F^(g*(f* x+e)))^n+9*ln(F)*a*c*d^2*f*g*n*x^2+6*ln(F)*b*c^2*d*f*g*n*x*(F^(g*(f*x+e))) ^n+9*ln(F)*a*c^2*d*f*g*n*x+2*ln(F)*b*c^3*f*g*n*(F^(g*(f*x+e)))^n+3*ln(F)*a *c^3*f*g*n-3*b*d^3*x^2*(F^(g*(f*x+e)))^n-3*a*d^3*x^2-6*b*c*d^2*x*(F^(g*(f* x+e)))^n-6*a*c*d^2*x-3*b*c^2*d*(F^(g*(f*x+e)))^n-3*a*d*c^2)/n^2/g^2/f^2/ln (F)^2/a^2/(a+b*(F^(g*(f*x+e)))^n)^2+3/g^2/f^2/ln(F)^2/a^3*c*d^2*ln(F^(g*(f *x+e)))^2*x+6/n^3/g^3/f^3/ln(F)^3/a^3*c*d^2*polylog(3,-b*F^(n*g*f*x)*F^(-n *g*f*x)*(F^(g*(f*x+e)))^n/a)-9/2/n/g^3/f^3/ln(F)^3/a^3*d^3*ln(F^(g*(f*x+e) ))^2*x-9/2/n^2/g^4/f^4/ln(F)^4/a^3*d^3*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*( f*x+e)))^n)*ln(F^(g*(f*x+e)))^2-9/2/n/g^3/f^3/ln(F)^3/a^3*c*d^2*ln(F^(g*(f *x+e)))^2-9/2/n^2/g^4/f^4/ln(F)^4/a^3*d^3*ln(1+b*F^(n*g*f*x)*F^(-n*g*f*x)* (F^(g*(f*x+e)))^n/a)*ln(F^(g*(f*x+e)))^2+9/2/n^2/g^2/f^2/ln(F)^2/a^3*d^3*l n((F^(g*(f*x+e)))^n*F^(-n*g*f*x)*F^(n*g*f*x)*b+a)*x^2+9/2/n^2/g^4/f^4/l...
Leaf count of result is larger than twice the leaf count of optimal. 2704 vs. \(2 (579) = 1158\).
Time = 0.32 (sec) , antiderivative size = 2704, normalized size of antiderivative = 4.55 \[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\text {Too large to display} \]
-1/4*(6*(a^2*d^3*e^3 - 3*a^2*c*d^2*e^2*f + 3*a^2*c^2*d*e*f^2 - a^2*c^3*f^3 )*g^3*n^3*log(F)^3 + 6*(a^2*d^3*e^2 - 2*a^2*c*d^2*e*f + a^2*c^2*d*f^2)*g^2 *n^2*log(F)^2 - (a^2*d^3*f^4*g^4*n^4*x^4 + 4*a^2*c*d^2*f^4*g^4*n^4*x^3 + 6 *a^2*c^2*d*f^4*g^4*n^4*x^2 + 4*a^2*c^3*f^4*g^4*n^4*x - (a^2*d^3*e^4 - 4*a^ 2*c*d^2*e^3*f + 6*a^2*c^2*d*e^2*f^2 - 4*a^2*c^3*e*f^3)*g^4*n^4)*log(F)^4 - ((b^2*d^3*f^4*g^4*n^4*x^4 + 4*b^2*c*d^2*f^4*g^4*n^4*x^3 + 6*b^2*c^2*d*f^4 *g^4*n^4*x^2 + 4*b^2*c^3*f^4*g^4*n^4*x - (b^2*d^3*e^4 - 4*b^2*c*d^2*e^3*f + 6*b^2*c^2*d*e^2*f^2 - 4*b^2*c^3*e*f^3)*g^4*n^4)*log(F)^4 - 6*(b^2*d^3*f^ 3*g^3*n^3*x^3 + 3*b^2*c*d^2*f^3*g^3*n^3*x^2 + 3*b^2*c^2*d*f^3*g^3*n^3*x + (b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2)*g^3*n^3)*log(F)^3 + 6*(b^2*d^3*f^2*g^2*n^2*x^2 + 2*b^2*c*d^2*f^2*g^2*n^2*x - (b^2*d^3*e^2 - 2* b^2*c*d^2*e*f)*g^2*n^2)*log(F)^2)*F^(2*f*g*n*x + 2*e*g*n) - 2*((a*b*d^3*f^ 4*g^4*n^4*x^4 + 4*a*b*c*d^2*f^4*g^4*n^4*x^3 + 6*a*b*c^2*d*f^4*g^4*n^4*x^2 + 4*a*b*c^3*f^4*g^4*n^4*x - (a*b*d^3*e^4 - 4*a*b*c*d^2*e^3*f + 6*a*b*c^2*d *e^2*f^2 - 4*a*b*c^3*e*f^3)*g^4*n^4)*log(F)^4 - 2*(2*a*b*d^3*f^3*g^3*n^3*x ^3 + 6*a*b*c*d^2*f^3*g^3*n^3*x^2 + 6*a*b*c^2*d*f^3*g^3*n^3*x + (3*a*b*d^3* e^3 - 9*a*b*c*d^2*e^2*f + 9*a*b*c^2*d*e*f^2 - a*b*c^3*f^3)*g^3*n^3)*log(F) ^3 + 3*(a*b*d^3*f^2*g^2*n^2*x^2 + 2*a*b*c*d^2*f^2*g^2*n^2*x - (2*a*b*d^3*e ^2 - 4*a*b*c*d^2*e*f + a*b*c^2*d*f^2)*g^2*n^2)*log(F)^2)*F^(f*g*n*x + e*g* n) + 12*(a^2*d^3 + (a^2*d^3*f^2*g^2*n^2*x^2 + 2*a^2*c*d^2*f^2*g^2*n^2*x...
\[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {\left (c + d x\right )^{3}}{\left (a + b \left (F^{e g + f g x}\right )^{n}\right )^{3}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 1005, normalized size of antiderivative = 1.69 \[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\frac {1}{2} \, c^{3} {\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^{3} f g n x^{3} \log \left (F\right ) - 3 \, a c^{2} d + 3 \, {\left (3 \, a c d^{2} f g n \log \left (F\right ) - a d^{3}\right )} x^{2} + {\left (2 \, F^{e g n} b d^{3} f g n x^{3} \log \left (F\right ) - 3 \, F^{e g n} b c^{2} d + 3 \, {\left (2 \, F^{e g n} b c d^{2} f g n \log \left (F\right ) - F^{e g n} b d^{3}\right )} x^{2} + 6 \, {\left (F^{e g n} b c^{2} d f g n \log \left (F\right ) - F^{e g n} b c d^{2}\right )} x\right )} F^{f g n x} + 3 \, {\left (3 \, a c^{2} d f g n \log \left (F\right ) - 2 \, a c 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 {3 \, {\left (3 \, c^{2} d f g n \log \left (F\right ) - 2 \, c d^{2}\right )} x}{2 \, a^{3} f^{2} g^{2} n^{2} \log \left (F\right )^{2}} + \frac {3 \, {\left (3 \, c^{2} d f g n \log \left (F\right ) - 2 \, c d^{2}\right )} \log \left (F^{f g n x} F^{e g n} b + a\right )}{2 \, a^{3} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} - \frac {{\left (f^{3} g^{3} n^{3} x^{3} \log \left (\frac {F^{f g n x} F^{e g n} b}{a} + 1\right ) \log \left (F\right )^{3} + 3 \, f^{2} g^{2} n^{2} x^{2} {\rm Li}_2\left (-\frac {F^{f g n x} F^{e g n} b}{a}\right ) \log \left (F\right )^{2} - 6 \, f g n x \log \left (F\right ) {\rm Li}_{3}(-\frac {F^{f g n x} F^{e g n} b}{a}) + 6 \, {\rm Li}_{4}(-\frac {F^{f g n x} F^{e g n} b}{a})\right )} d^{3}}{a^{3} f^{4} g^{4} n^{4} \log \left (F\right )^{4}} - \frac {3 \, {\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 )} {\left (2 \, c d^{2} f g n \log \left (F\right ) - 3 \, d^{3}\right )}}{2 \, a^{3} f^{4} g^{4} n^{4} \log \left (F\right )^{4}} - \frac {3 \, {\left (c^{2} d f^{2} g^{2} n^{2} \log \left (F\right )^{2} - 3 \, c d^{2} f g n \log \left (F\right ) + d^{3}\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^{4} g^{4} n^{4} \log \left (F\right )^{4}} + \frac {d^{3} f^{4} g^{4} n^{4} x^{4} \log \left (F\right )^{4} + 2 \, {\left (2 \, c d^{2} f g n \log \left (F\right ) - 3 \, d^{3}\right )} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} + 6 \, {\left (c^{2} d f^{2} g^{2} n^{2} \log \left (F\right )^{2} - 3 \, c d^{2} f g n \log \left (F\right ) + d^{3}\right )} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2}}{4 \, a^{3} f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \]
1/2*c^3*((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^3*f*g*n*x^3*log(F) - 3*a*c^2*d + 3*(3*a*c*d^2*f*g*n*log(F) - a*d^3)* x^2 + (2*F^(e*g*n)*b*d^3*f*g*n*x^3*log(F) - 3*F^(e*g*n)*b*c^2*d + 3*(2*F^( e*g*n)*b*c*d^2*f*g*n*log(F) - F^(e*g*n)*b*d^3)*x^2 + 6*(F^(e*g*n)*b*c^2*d* f*g*n*log(F) - F^(e*g*n)*b*c*d^2)*x)*F^(f*g*n*x) + 3*(3*a*c^2*d*f*g*n*log( F) - 2*a*c*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*lo g(F)^2) - 3/2*(3*c^2*d*f*g*n*log(F) - 2*c*d^2)*x/(a^3*f^2*g^2*n^2*log(F)^2 ) + 3/2*(3*c^2*d*f*g*n*log(F) - 2*c*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) - (f^3*g^3*n^3*x^3*log(F^(f*g*n*x)*F^(e*g*n)*b/ a + 1)*log(F)^3 + 3*f^2*g^2*n^2*x^2*dilog(-F^(f*g*n*x)*F^(e*g*n)*b/a)*log( F)^2 - 6*f*g*n*x*log(F)*polylog(3, -F^(f*g*n*x)*F^(e*g*n)*b/a) + 6*polylog (4, -F^(f*g*n*x)*F^(e*g*n)*b/a))*d^3/(a^3*f^4*g^4*n^4*log(F)^4) - 3/2*(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*polylog(3, -F^(f*g*n*x)*F^(e*g*n)* b/a))*(2*c*d^2*f*g*n*log(F) - 3*d^3)/(a^3*f^4*g^4*n^4*log(F)^4) - 3*(c^2*d *f^2*g^2*n^2*log(F)^2 - 3*c*d^2*f*g*n*log(F) + d^3)*(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*...
\[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^3} \,d x \]