Integrand size = 25, antiderivative size = 388 \[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx=\frac {(c+d x)^4}{4 a^2 d}-\frac {(c+d x)^3}{a^2 f g n \log (F)}+\frac {(c+d x)^3}{a f \left (a+b \left (F^{g (e+f x)}\right )^n\right ) g n \log (F)}+\frac {3 d (c+d x)^2 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x)^3 \log \left (1+\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f g n \log (F)}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^2 g^2 n^2 \log ^2(F)}-\frac {6 d^3 \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)}+\frac {6 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^3 g^3 n^3 \log ^3(F)}-\frac {6 d^3 \operatorname {PolyLog}\left (4,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{a^2 f^4 g^4 n^4 \log ^4(F)} \]
1/4*(d*x+c)^4/a^2/d-(d*x+c)^3/a^2/f/g/n/ln(F)+(d*x+c)^3/a/f/(a+b*(F^(g*(f* x+e)))^n)/g/n/ln(F)+3*d*(d*x+c)^2*ln(1+b*(F^(g*(f*x+e)))^n/a)/a^2/f^2/g^2/ n^2/ln(F)^2-(d*x+c)^3*ln(1+b*(F^(g*(f*x+e)))^n/a)/a^2/f/g/n/ln(F)+6*d^2*(d *x+c)*polylog(2,-b*(F^(g*(f*x+e)))^n/a)/a^2/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^2/f^2/g^2/n^2/ln(F)^2-6*d^3*polylo g(3,-b*(F^(g*(f*x+e)))^n/a)/a^2/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^2/f^3/g^3/n^3/ln(F)^3-6*d^3*polylog(4,-b*(F^(g *(f*x+e)))^n/a)/a^2/f^4/g^4/n^4/ln(F)^4
\[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx=\int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx \]
Time = 2.48 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.17, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2616, 2615, 2620, 2621, 2615, 2620, 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.
| \(\displaystyle \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx\) |
| \(\Big \downarrow \) 2616 |
| \(\displaystyle \frac {\int \frac {(c+d x)^3}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}\) |
| \(\Big \downarrow \) 2615 |
| \(\displaystyle \frac {\frac {(c+d x)^4}{4 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^3}{\left (b \left (F^{g (e+f x)}\right )^n+a\right )^2}dx}{a}\) |
| \(\Big \downarrow \) 2621 |
| \(\displaystyle \frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \int \frac {(c+d x)^2}{b \left (F^{g (e+f x)}\right )^n+a}dx}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\) |
| \(\Big \downarrow \) 2615 |
| \(\displaystyle \frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \int \frac {\left (F^{g (e+f x)}\right )^n (c+d x)^2}{b \left (F^{g (e+f x)}\right )^n+a}dx}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \int (c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \int (c+d x) \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )dx}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )dx}{f g n \log (F)}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \left (\frac {d \int \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )dx}{f g n \log (F)}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )dx}{f g n \log (F)}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \left (\frac {d \int F^{-g (e+f x)} \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )dF^{g (e+f x)}}{f^2 g^2 n \log ^2(F)}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \left (\frac {2 d \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )dx}{f g n \log (F)}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}-\frac {d \int \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )dx}{f g n \log (F)}\right )}{f g n \log (F)}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}-\frac {d \int F^{-g (e+f x)} \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )dF^{g (e+f x)}}{f^2 g^2 n \log ^2(F)}\right )}{f g n \log (F)}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\frac {(c+d x)^4}{4 a d}-\frac {b \left (\frac {(c+d x)^3 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {3 d \left (\frac {2 d \left (\frac {(c+d x) \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}-\frac {d \operatorname {PolyLog}\left (4,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f^2 g^2 n^2 \log ^2(F)}\right )}{f g n \log (F)}-\frac {(c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}}{a}-\frac {b \left (\frac {3 d \left (\frac {(c+d x)^3}{3 a d}-\frac {b \left (\frac {(c+d x)^2 \log \left (\frac {b \left (F^{g (e+f x)}\right )^n}{a}+1\right )}{b f g n \log (F)}-\frac {2 d \left (\frac {d \operatorname {PolyLog}\left (3,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f^2 g^2 n^2 \log ^2(F)}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-\frac {b \left (F^{g (e+f x)}\right )^n}{a}\right )}{f g n \log (F)}\right )}{b f g n \log (F)}\right )}{a}\right )}{b f g n \log (F)}-\frac {(c+d x)^3}{b f g n \log (F) \left (a+b \left (F^{g (e+f x)}\right )^n\right )}\right )}{a}\) |
-((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
3.1.52.3.1 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[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[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. \(3318\) vs. \(2(386)=772\).
Time = 0.46 (sec) , antiderivative size = 3319, normalized size of antiderivative = 8.55
1/n/g/f/ln(F)/a*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/(a+b*(F^(g*(f*x+e)))^n )+3/4/g^4/f^4/ln(F)^4/a^2*d^3*ln(F^(g*(f*x+e)))^4+3/2/g^2/f^2/ln(F)^2/a^2* c^2*d*ln(F^(g*(f*x+e)))^2-1/n/g/f/ln(F)/a^2*c^3*ln((F^(g*(f*x+e)))^n*F^(-n *g*f*x)*F^(n*g*f*x)*b+a)+1/n/g/f/ln(F)/a^2*c^3*ln(F^(n*g*f*x)*F^(-n*g*f*x) *(F^(g*(f*x+e)))^n)+2/n/g^4/f^4/ln(F)^4/a^2*d^3*ln(F^(g*(f*x+e)))^3-6/n^4/ g^4/f^4/ln(F)^4/a^2*d^3*polylog(3,-b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e )))^n/a)-6/n^4/g^4/f^4/ln(F)^4/a^2*d^3*polylog(4,-b*F^(n*g*f*x)*F^(-n*g*f* x)*(F^(g*(f*x+e)))^n/a)+3/2/g^2/f^2/ln(F)^2/a^2*d^3*ln(F^(g*(f*x+e)))^2*x^ 2-2/g^3/f^3/ln(F)^3/a^2*d^3*ln(F^(g*(f*x+e)))^3*x-2/g^3/f^3/ln(F)^3/a^2*c* d^2*ln(F^(g*(f*x+e)))^3+3/n/g^3/f^3/ln(F)^3/a^2*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)))^2+3/n/g/f/ln(F)/a^2*c^2 *d*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)*x-3/n/g^2/f^2/ln(F)^2/a^ 2*c^2*d*ln(F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g*(f*x+e)))^n)*ln(F^(g*(f*x+e)))-6 /n^2/g^2/f^2/ln(F)^2/a^2*c*d^2*polylog(2,-b*F^(n*g*f*x)*F^(-n*g*f*x)*(F^(g *(f*x+e)))^n/a)*x-6/n^2/g^3/f^3/ln(F)^3/a^2*d^3*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)))*x+6/n^2/g^3/f^3/ln(F)^3/a^2*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)))*x+6/n^2/ g^3/f^3/ln(F)^3/a^2*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-3/n/g/f/ln(F)/a^2*c*d^2*ln((F^(g*(f*x+e)))^n*F^(-n* g*f*x)*F^(n*g*f*x)*b+a)*x^2-3/n/g^3/f^3/ln(F)^3/a^2*c*d^2*ln((F^(g*(f*x...
Leaf count of result is larger than twice the leaf count of optimal. 1390 vs. \(2 (384) = 768\).
Time = 0.31 (sec) , antiderivative size = 1390, normalized size of antiderivative = 3.58 \[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx=\text {Too large to display} \]
-1/4*(4*(a*d^3*e^3 - 3*a*c*d^2*e^2*f + 3*a*c^2*d*e*f^2 - a*c^3*f^3)*g^3*n^ 3*log(F)^3 - (a*d^3*f^4*g^4*n^4*x^4 + 4*a*c*d^2*f^4*g^4*n^4*x^3 + 6*a*c^2* d*f^4*g^4*n^4*x^2 + 4*a*c^3*f^4*g^4*n^4*x - (a*d^3*e^4 - 4*a*c*d^2*e^3*f + 6*a*c^2*d*e^2*f^2 - 4*a*c^3*e*f^3)*g^4*n^4)*log(F)^4 - ((b*d^3*f^4*g^4*n^ 4*x^4 + 4*b*c*d^2*f^4*g^4*n^4*x^3 + 6*b*c^2*d*f^4*g^4*n^4*x^2 + 4*b*c^3*f^ 4*g^4*n^4*x - (b*d^3*e^4 - 4*b*c*d^2*e^3*f + 6*b*c^2*d*e^2*f^2 - 4*b*c^3*e *f^3)*g^4*n^4)*log(F)^4 - 4*(b*d^3*f^3*g^3*n^3*x^3 + 3*b*c*d^2*f^3*g^3*n^3 *x^2 + 3*b*c^2*d*f^3*g^3*n^3*x + (b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d* e*f^2)*g^3*n^3)*log(F)^3)*F^(f*g*n*x + e*g*n) + 12*((a*d^3*f^2*g^2*n^2*x^2 + 2*a*c*d^2*f^2*g^2*n^2*x + a*c^2*d*f^2*g^2*n^2)*log(F)^2 + ((b*d^3*f^2*g ^2*n^2*x^2 + 2*b*c*d^2*f^2*g^2*n^2*x + b*c^2*d*f^2*g^2*n^2)*log(F)^2 - 2*( b*d^3*f*g*n*x + b*c*d^2*f*g*n)*log(F))*F^(f*g*n*x + e*g*n) - 2*(a*d^3*f*g* n*x + a*c*d^2*f*g*n)*log(F))*dilog(-(F^(f*g*n*x + e*g*n)*b + a)/a + 1) - 4 *((a*d^3*e^3 - 3*a*c*d^2*e^2*f + 3*a*c^2*d*e*f^2 - a*c^3*f^3)*g^3*n^3*log( F)^3 + 3*(a*d^3*e^2 - 2*a*c*d^2*e*f + a*c^2*d*f^2)*g^2*n^2*log(F)^2 + ((b* d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*g^3*n^3*log(F)^3 + 3*(b*d^3*e^2 - 2*b*c*d^2*e*f + b*c^2*d*f^2)*g^2*n^2*log(F)^2)*F^(f*g*n*x + e*g*n))*log(F^(f*g*n*x + e*g*n)*b + a) + 4*((a*d^3*f^3*g^3*n^3*x^3 + 3* a*c*d^2*f^3*g^3*n^3*x^2 + 3*a*c^2*d*f^3*g^3*n^3*x + (a*d^3*e^3 - 3*a*c*d^2 *e^2*f + 3*a*c^2*d*e*f^2)*g^3*n^3)*log(F)^3 - 3*(a*d^3*f^2*g^2*n^2*x^2 ...
Timed out. \[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 699, normalized size of antiderivative = 1.80 \[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx=c^{3} {\left (\frac {f g n x + e g n}{a^{2} f g n} + \frac {1}{{\left (F^{f g n x + e g n} a b + a^{2}\right )} f g n \log \left (F\right )} - \frac {\log \left (F^{f g n x + e g n} b + a\right )}{a^{2} f g n \log \left (F\right )}\right )} + \frac {d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x}{F^{f g n x} F^{e g n} a b f g n \log \left (F\right ) + a^{2} f g n \log \left (F\right )} - \frac {3 \, c^{2} d x}{a^{2} f g n \log \left (F\right )} + \frac {3 \, c^{2} d \log \left (F^{f g n x} F^{e g n} b + a\right )}{a^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2}} - \frac {3 \, {\left (c^{2} d f g n \log \left (F\right ) - 2 \, c 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^{2} 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^{2} 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 (c d^{2} f g n \log \left (F\right ) - d^{3}\right )}}{a^{2} 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} + 4 \, {\left (c d^{2} f g n \log \left (F\right ) - 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} - 2 \, c d^{2} f g n \log \left (F\right )\right )} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2}}{4 \, a^{2} f^{4} g^{4} n^{4} \log \left (F\right )^{4}} \]
c^3*((f*g*n*x + e*g*n)/(a^2*f*g*n) + 1/((F^(f*g*n*x + e*g*n)*a*b + a^2)*f* g*n*log(F)) - log(F^(f*g*n*x + e*g*n)*b + a)/(a^2*f*g*n*log(F))) + (d^3*x^ 3 + 3*c*d^2*x^2 + 3*c^2*d*x)/(F^(f*g*n*x)*F^(e*g*n)*a*b*f*g*n*log(F) + a^2 *f*g*n*log(F)) - 3*c^2*d*x/(a^2*f*g*n*log(F)) + 3*c^2*d*log(F^(f*g*n*x)*F^ (e*g*n)*b + a)/(a^2*f^2*g^2*n^2*log(F)^2) - 3*(c^2*d*f*g*n*log(F) - 2*c*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^2*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^2*f^4*g^4*n^4*log (F)^4) - 3*(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))*(c*d^2*f*g*n*log(F) - d^3)/(a^2*f^4*g^4*n^4*log(F)^4) + 1/4*(d^3*f^4*g^4*n^4*x^4*log(F)^4 + 4*(c*d^2*f*g*n*log(F) - d^3)*f^3*g^ 3*n^3*x^3*log(F)^3 + 6*(c^2*d*f^2*g^2*n^2*log(F)^2 - 2*c*d^2*f*g*n*log(F)) *f^2*g^2*n^2*x^2*log(F)^2)/(a^2*f^4*g^4*n^4*log(F)^4)
\[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left ({\left (F^{{\left (f x + e\right )} g}\right )}^{n} b + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^3}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{{\left (a+b\,{\left (F^{g\,\left (e+f\,x\right )}\right )}^n\right )}^2} \,d x \]