\(\int \frac {(c+d x)^2}{(a+b (F^{g (e+f x)})^n)^3} \, dx\) [59]

Optimal result

Integrand size = 25, antiderivative size = 439 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\frac {(c+d x)^3}{3 a^3 d}+\frac {d^2 x}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {d (c+d x)}{a^2 f^2 \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g^2 n^2 \log ^2(F)}-\frac {3 (c+d x)^2}{2 a^3 f g n \log (F)}+\frac {(c+d x)^2}{2 a f \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2 g n \log (F)}+\frac {(c+d x)^2}{a^2 f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}-\frac {d^2 \log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}+\frac {3 d (c+d x) \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f g n \log (F)}+\frac {3 d^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)}-\frac {2 d (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^2 g^2 n^2 \log ^2(F)}+\frac {2 d^2 \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^3 f^3 g^3 n^3 \log ^3(F)} \] Output:

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
 

Mathematica [F]

\[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx \] Input:

Integrate[(c + d*x)^2/(a + b*(F^(g*(e + f*x)))^n)^3,x]
 

Output:

Integrate[(c + d*x)^2/(a + b*(F^(g*(e + f*x)))^n)^3, x]
 

Rubi [A] (verified)

Time = 6.82 (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

\(\displaystyle \frac {\frac {\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {2 d \left (\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}+\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\frac {(c+d x)^2}{2 a d}-\frac {b \left (\frac {(c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}+\frac {d \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}\right )}{a}}{a}-\frac {b \left (\frac {d \left (\frac {\log \left (\left (F^{g (e+f x)}\right )^n\right )}{a}-\frac {\log \left (a+b \left (F^{g (e+f x)}\right )^n\right )}{a}\right )}{b f^2 g^2 n^2 \log ^2(F)}-\frac {c+d x}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^2}{2 b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )^2}\right )}{a}\)

Input:

Int[(c + d*x)^2/(a + b*(F^(g*(e + f*x)))^n)^3,x]
 

Output:

-((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
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 47

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]
 

rule 798

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]]
 

rule 2615

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]
 

rule 2616

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]
 

rule 2620

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]
 

rule 2621

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]
 

rule 2715

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]
 

rule 2720

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]]]
 

rule 2838

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]
 

rule 3011

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]
 

rule 7143

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1886\) vs. \(2(433)=866\).

Time = 0.33 (sec) , antiderivative size = 1887, normalized size of antiderivative = 4.30

method	result	size
risch	\(\text {Expression too large to display}\)	\(1887\)

Input:

int((d*x+c)^2/(a+b*(F^(g*(f*x+e)))^n)^3,x,method=_RETURNVERBOSE)
 

Output:

1/a^3/ln(F)^2/f^2/g^2*d^2*ln(F^(g*(f*x+e)))^2*x-3/2/a^3/ln(F)^3/f^3/g^3/n* 
d^2*ln(F^(g*(f*x+e)))^2+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)/f/g/n*c^2*ln(F^(n*g*f*x 
)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)+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)))^n*F^(-n*g*f*x)*F^(n*g*f*x)*b+a)*ln(F^(g*(f*x 
+e)))+1/a^3/ln(F)^3/f^3/g^3/n*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)))^2-1/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^3/ 
n*d^2*ln(F^(g*(f*x+e)))^2*ln(1+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^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)-3/a^3/ln(F)^2/f^2/g^2/n^2*c*d*ln(F^(n*g*f* 
x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)+3/a^3/ln(F)^2/f^2/g^2/n^2*c*d*ln((F^(g* 
(f*x+e)))^n*F^(-n*g*f*x)*F^(n*g*f*x)*b+a)+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)^2-2/3/a^3/ln(F)^ 
3/f^3/g^3*d^2*ln(F^(g*(f*x+e)))^3-3/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)^2/f^2/g^2/n^2*d^2*ln...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1518 vs. \(2 (431) = 862\).

Time = 0.11 (sec) , antiderivative size = 1518, normalized size of antiderivative = 3.46 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^2/(a+b*(F^(g*(f*x+e)))^n)^3,x, algorithm="fricas")
 

Output:

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...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**2/(a+b*(F**(g*(f*x+e)))**n)**3,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.58 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^2/(a+b*(F^(g*(f*x+e)))^n)^3,x, algorithm="maxima")
 

Output:

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)
 

Giac [F]

\[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{3}} \,d x } \] Input:

integrate((d*x+c)^2/(a+b*(F^(g*(f*x+e)))^n)^3,x, algorithm="giac")
 

Output:

integrate((d*x + c)^2/((F^((f*x + e)*g))^n*b + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^3} \,d x \] Input:

int((c + d*x)^2/(a + b*(F^(g*(e + f*x)))^n)^3,x)
 

Output:

int((c + d*x)^2/(a + b*(F^(g*(e + f*x)))^n)^3, x)
 

Reduce [F]

\[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^3} \, dx=\frac {2 f^{2 f g n x +2 e g n} \left (\int \frac {x^{2}}{f^{3 f g n x +3 e g n} b^{3}+3 f^{2 f g n x +2 e g n} a \,b^{2}+3 f^{f g n x +e g n} a^{2} b +a^{3}}d x \right ) \mathrm {log}\left (f \right ) a^{3} b^{2} d^{2} f g n +4 f^{2 f g n x +2 e g n} \left (\int \frac {x}{f^{3 f g n x +3 e g n} b^{3}+3 f^{2 f g n x +2 e g n} a \,b^{2}+3 f^{f g n x +e g n} a^{2} b +a^{3}}d x \right ) \mathrm {log}\left (f \right ) a^{3} b^{2} c d f g n -2 f^{2 f g n x +2 e g n} \mathrm {log}\left (f^{f g n x +e g n} b +a \right ) b^{2} c^{2}+2 f^{2 f g n x +2 e g n} \mathrm {log}\left (f \right ) b^{2} c^{2} f g n x -f^{2 f g n x +2 e g n} b^{2} c^{2}+4 f^{f g n x +e g n} \left (\int \frac {x^{2}}{f^{3 f g n x +3 e g n} b^{3}+3 f^{2 f g n x +2 e g n} a \,b^{2}+3 f^{f g n x +e g n} a^{2} b +a^{3}}d x \right ) \mathrm {log}\left (f \right ) a^{4} b \,d^{2} f g n +8 f^{f g n x +e g n} \left (\int \frac {x}{f^{3 f g n x +3 e g n} b^{3}+3 f^{2 f g n x +2 e g n} a \,b^{2}+3 f^{f g n x +e g n} a^{2} b +a^{3}}d x \right ) \mathrm {log}\left (f \right ) a^{4} b c d f g n -4 f^{f g n x +e g n} \mathrm {log}\left (f^{f g n x +e g n} b +a \right ) a b \,c^{2}+4 f^{f g n x +e g n} \mathrm {log}\left (f \right ) a b \,c^{2} f g n x +2 \left (\int \frac {x^{2}}{f^{3 f g n x +3 e g n} b^{3}+3 f^{2 f g n x +2 e g n} a \,b^{2}+3 f^{f g n x +e g n} a^{2} b +a^{3}}d x \right ) \mathrm {log}\left (f \right ) a^{5} d^{2} f g n +4 \left (\int \frac {x}{f^{3 f g n x +3 e g n} b^{3}+3 f^{2 f g n x +2 e g n} a \,b^{2}+3 f^{f g n x +e g n} a^{2} b +a^{3}}d x \right ) \mathrm {log}\left (f \right ) a^{5} c d f g n -2 \,\mathrm {log}\left (f^{f g n x +e g n} b +a \right ) a^{2} c^{2}+2 \,\mathrm {log}\left (f \right ) a^{2} c^{2} f g n x +2 a^{2} c^{2}}{2 \,\mathrm {log}\left (f \right ) a^{3} f g n \left (f^{2 f g n x +2 e g n} b^{2}+2 f^{f g n x +e g n} a b +a^{2}\right )} \] Input:

int((d*x+c)^2/(a+b*(F^(g*(f*x+e)))^n)^3,x)
 

Output:

(2*f**(2*e*g*n + 2*f*g*n*x)*int(x**2/(f**(3*e*g*n + 3*f*g*n*x)*b**3 + 3*f* 
*(2*e*g*n + 2*f*g*n*x)*a*b**2 + 3*f**(e*g*n + f*g*n*x)*a**2*b + a**3),x)*l 
og(f)*a**3*b**2*d**2*f*g*n + 4*f**(2*e*g*n + 2*f*g*n*x)*int(x/(f**(3*e*g*n 
 + 3*f*g*n*x)*b**3 + 3*f**(2*e*g*n + 2*f*g*n*x)*a*b**2 + 3*f**(e*g*n + f*g 
*n*x)*a**2*b + a**3),x)*log(f)*a**3*b**2*c*d*f*g*n - 2*f**(2*e*g*n + 2*f*g 
*n*x)*log(f**(e*g*n + f*g*n*x)*b + a)*b**2*c**2 + 2*f**(2*e*g*n + 2*f*g*n* 
x)*log(f)*b**2*c**2*f*g*n*x - f**(2*e*g*n + 2*f*g*n*x)*b**2*c**2 + 4*f**(e 
*g*n + f*g*n*x)*int(x**2/(f**(3*e*g*n + 3*f*g*n*x)*b**3 + 3*f**(2*e*g*n + 
2*f*g*n*x)*a*b**2 + 3*f**(e*g*n + f*g*n*x)*a**2*b + a**3),x)*log(f)*a**4*b 
*d**2*f*g*n + 8*f**(e*g*n + f*g*n*x)*int(x/(f**(3*e*g*n + 3*f*g*n*x)*b**3 
+ 3*f**(2*e*g*n + 2*f*g*n*x)*a*b**2 + 3*f**(e*g*n + f*g*n*x)*a**2*b + a**3 
),x)*log(f)*a**4*b*c*d*f*g*n - 4*f**(e*g*n + f*g*n*x)*log(f**(e*g*n + f*g* 
n*x)*b + a)*a*b*c**2 + 4*f**(e*g*n + f*g*n*x)*log(f)*a*b*c**2*f*g*n*x + 2* 
int(x**2/(f**(3*e*g*n + 3*f*g*n*x)*b**3 + 3*f**(2*e*g*n + 2*f*g*n*x)*a*b** 
2 + 3*f**(e*g*n + f*g*n*x)*a**2*b + a**3),x)*log(f)*a**5*d**2*f*g*n + 4*in 
t(x/(f**(3*e*g*n + 3*f*g*n*x)*b**3 + 3*f**(2*e*g*n + 2*f*g*n*x)*a*b**2 + 3 
*f**(e*g*n + f*g*n*x)*a**2*b + a**3),x)*log(f)*a**5*c*d*f*g*n - 2*log(f**( 
e*g*n + f*g*n*x)*b + a)*a**2*c**2 + 2*log(f)*a**2*c**2*f*g*n*x + 2*a**2*c* 
*2)/(2*log(f)*a**3*f*g*n*(f**(2*e*g*n + 2*f*g*n*x)*b**2 + 2*f**(e*g*n + f* 
g*n*x)*a*b + a**2))