\(\int \frac {(f+g x)^3}{(d+e (F^{c (a+b x)})^n)^2} \, dx\) [39]

Optimal result

Integrand size = 25, antiderivative size = 388 \[ \int \frac {(f+g x)^3}{\left (d+e \left (F^{c (a+b x)}\right )^n\right )^2} \, dx=\frac {(f+g x)^4}{4 d^2 g}-\frac {(f+g x)^3}{b c d^2 n \log (F)}+\frac {(f+g x)^3}{b c d \left (d+e \left (F^{c (a+b x)}\right )^n\right ) n \log (F)}+\frac {3 g (f+g x)^2 \log \left (1+\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 d^2 n^2 \log ^2(F)}-\frac {(f+g x)^3 \log \left (1+\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c d^2 n \log (F)}+\frac {6 g^2 (f+g x) \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 d^2 n^3 \log ^3(F)}-\frac {3 g (f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 d^2 n^2 \log ^2(F)}-\frac {6 g^3 \operatorname {PolyLog}\left (3,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 d^2 n^4 \log ^4(F)}+\frac {6 g^2 (f+g x) \operatorname {PolyLog}\left (3,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 d^2 n^3 \log ^3(F)}-\frac {6 g^3 \operatorname {PolyLog}\left (4,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^4 c^4 d^2 n^4 \log ^4(F)} \] Output:

1/4*(g*x+f)^4/d^2/g-(g*x+f)^3/b/c/d^2/n/ln(F)+(g*x+f)^3/b/c/d/(d+e*(F^(c*( 
b*x+a)))^n)/n/ln(F)+3*g*(g*x+f)^2*ln(1+e*(F^(c*(b*x+a)))^n/d)/b^2/c^2/d^2/ 
n^2/ln(F)^2-(g*x+f)^3*ln(1+e*(F^(c*(b*x+a)))^n/d)/b/c/d^2/n/ln(F)+6*g^2*(g 
*x+f)*polylog(2,-e*(F^(c*(b*x+a)))^n/d)/b^3/c^3/d^2/n^3/ln(F)^3-3*g*(g*x+f 
)^2*polylog(2,-e*(F^(c*(b*x+a)))^n/d)/b^2/c^2/d^2/n^2/ln(F)^2-6*g^3*polylo 
g(3,-e*(F^(c*(b*x+a)))^n/d)/b^4/c^4/d^2/n^4/ln(F)^4+6*g^2*(g*x+f)*polylog( 
3,-e*(F^(c*(b*x+a)))^n/d)/b^3/c^3/d^2/n^3/ln(F)^3-6*g^3*polylog(4,-e*(F^(c 
*(b*x+a)))^n/d)/b^4/c^4/d^2/n^4/ln(F)^4
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 4.28 (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.

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

\(\Big \downarrow \) 2616

\(\displaystyle \frac {\int \frac {(f+g x)^3}{e \left (F^{c (a+b x)}\right )^n+d}dx}{d}-\frac {e \int \frac {\left (F^{c (a+b x)}\right )^n (f+g x)^3}{\left (e \left (F^{c (a+b x)}\right )^n+d\right )^2}dx}{d}\)

\(\Big \downarrow \) 2615

\(\displaystyle \frac {\frac {(f+g x)^4}{4 d g}-\frac {e \int \frac {\left (F^{c (a+b x)}\right )^n (f+g x)^3}{e \left (F^{c (a+b x)}\right )^n+d}dx}{d}}{d}-\frac {e \int \frac {\left (F^{c (a+b x)}\right )^n (f+g x)^3}{\left (e \left (F^{c (a+b x)}\right )^n+d\right )^2}dx}{d}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\frac {(f+g x)^4}{4 d g}-\frac {e \left (\frac {(f+g x)^3 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {3 g \int (f+g x)^2 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )dx}{b c e n \log (F)}\right )}{d}}{d}-\frac {e \int \frac {\left (F^{c (a+b x)}\right )^n (f+g x)^3}{\left (e \left (F^{c (a+b x)}\right )^n+d\right )^2}dx}{d}\)

\(\Big \downarrow \) 2621

\(\displaystyle \frac {\frac {(f+g x)^4}{4 d g}-\frac {e \left (\frac {(f+g x)^3 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {3 g \int (f+g x)^2 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )dx}{b c e n \log (F)}\right )}{d}}{d}-\frac {e \left (\frac {3 g \int \frac {(f+g x)^2}{e \left (F^{c (a+b x)}\right )^n+d}dx}{b c e n \log (F)}-\frac {(f+g x)^3}{b c e n \log (F) \left (e \left (F^{c (a+b x)}\right )^n+d\right )}\right )}{d}\)

\(\Big \downarrow \) 2615

\(\displaystyle \frac {\frac {(f+g x)^4}{4 d g}-\frac {e \left (\frac {(f+g x)^3 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {3 g \int (f+g x)^2 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )dx}{b c e n \log (F)}\right )}{d}}{d}-\frac {e \left (\frac {3 g \left (\frac {(f+g x)^3}{3 d g}-\frac {e \int \frac {\left (F^{c (a+b x)}\right )^n (f+g x)^2}{e \left (F^{c (a+b x)}\right )^n+d}dx}{d}\right )}{b c e n \log (F)}-\frac {(f+g x)^3}{b c e n \log (F) \left (e \left (F^{c (a+b x)}\right )^n+d\right )}\right )}{d}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\frac {(f+g x)^4}{4 d g}-\frac {e \left (\frac {(f+g x)^3 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {3 g \int (f+g x)^2 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )dx}{b c e n \log (F)}\right )}{d}}{d}-\frac {e \left (\frac {3 g \left (\frac {(f+g x)^3}{3 d g}-\frac {e \left (\frac {(f+g x)^2 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {2 g \int (f+g x) \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )dx}{b c e n \log (F)}\right )}{d}\right )}{b c e n \log (F)}-\frac {(f+g x)^3}{b c e n \log (F) \left (e \left (F^{c (a+b x)}\right )^n+d\right )}\right )}{d}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {\frac {(f+g x)^4}{4 d g}-\frac {e \left (\frac {(f+g x)^3 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {3 g \left (\frac {2 g \int (f+g x) \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )dx}{b c n \log (F)}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}\right )}{b c e n \log (F)}\right )}{d}}{d}-\frac {e \left (\frac {3 g \left (\frac {(f+g x)^3}{3 d g}-\frac {e \left (\frac {(f+g x)^2 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {2 g \left (\frac {g \int \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )dx}{b c n \log (F)}-\frac {(f+g x) \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}\right )}{b c e n \log (F)}\right )}{d}\right )}{b c e n \log (F)}-\frac {(f+g x)^3}{b c e n \log (F) \left (e \left (F^{c (a+b x)}\right )^n+d\right )}\right )}{d}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {(f+g x)^4}{4 d g}-\frac {e \left (\frac {(f+g x)^3 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {3 g \left (\frac {2 g \int (f+g x) \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )dx}{b c n \log (F)}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}\right )}{b c e n \log (F)}\right )}{d}}{d}-\frac {e \left (\frac {3 g \left (\frac {(f+g x)^3}{3 d g}-\frac {e \left (\frac {(f+g x)^2 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {2 g \left (\frac {g \int F^{-c (a+b x)} \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )dF^{c (a+b x)}}{b^2 c^2 n \log ^2(F)}-\frac {(f+g x) \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}\right )}{b c e n \log (F)}\right )}{d}\right )}{b c e n \log (F)}-\frac {(f+g x)^3}{b c e n \log (F) \left (e \left (F^{c (a+b x)}\right )^n+d\right )}\right )}{d}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\frac {(f+g x)^4}{4 d g}-\frac {e \left (\frac {(f+g x)^3 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {3 g \left (\frac {2 g \int (f+g x) \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )dx}{b c n \log (F)}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}\right )}{b c e n \log (F)}\right )}{d}}{d}-\frac {e \left (\frac {3 g \left (\frac {(f+g x)^3}{3 d g}-\frac {e \left (\frac {(f+g x)^2 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {2 g \left (\frac {g \operatorname {PolyLog}\left (3,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(F)}-\frac {(f+g x) \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}\right )}{b c e n \log (F)}\right )}{d}\right )}{b c e n \log (F)}-\frac {(f+g x)^3}{b c e n \log (F) \left (e \left (F^{c (a+b x)}\right )^n+d\right )}\right )}{d}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {\frac {(f+g x)^4}{4 d g}-\frac {e \left (\frac {(f+g x)^3 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {3 g \left (\frac {2 g \left (\frac {(f+g x) \operatorname {PolyLog}\left (3,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}-\frac {g \int \operatorname {PolyLog}\left (3,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )dx}{b c n \log (F)}\right )}{b c n \log (F)}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}\right )}{b c e n \log (F)}\right )}{d}}{d}-\frac {e \left (\frac {3 g \left (\frac {(f+g x)^3}{3 d g}-\frac {e \left (\frac {(f+g x)^2 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {2 g \left (\frac {g \operatorname {PolyLog}\left (3,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(F)}-\frac {(f+g x) \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}\right )}{b c e n \log (F)}\right )}{d}\right )}{b c e n \log (F)}-\frac {(f+g x)^3}{b c e n \log (F) \left (e \left (F^{c (a+b x)}\right )^n+d\right )}\right )}{d}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {(f+g x)^4}{4 d g}-\frac {e \left (\frac {(f+g x)^3 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {3 g \left (\frac {2 g \left (\frac {(f+g x) \operatorname {PolyLog}\left (3,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}-\frac {g \int F^{-c (a+b x)} \operatorname {PolyLog}\left (3,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )dF^{c (a+b x)}}{b^2 c^2 n \log ^2(F)}\right )}{b c n \log (F)}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}\right )}{b c e n \log (F)}\right )}{d}}{d}-\frac {e \left (\frac {3 g \left (\frac {(f+g x)^3}{3 d g}-\frac {e \left (\frac {(f+g x)^2 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {2 g \left (\frac {g \operatorname {PolyLog}\left (3,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(F)}-\frac {(f+g x) \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}\right )}{b c e n \log (F)}\right )}{d}\right )}{b c e n \log (F)}-\frac {(f+g x)^3}{b c e n \log (F) \left (e \left (F^{c (a+b x)}\right )^n+d\right )}\right )}{d}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {\frac {(f+g x)^4}{4 d g}-\frac {e \left (\frac {(f+g x)^3 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {3 g \left (\frac {2 g \left (\frac {(f+g x) \operatorname {PolyLog}\left (3,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}-\frac {g \operatorname {PolyLog}\left (4,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(F)}\right )}{b c n \log (F)}-\frac {(f+g x)^2 \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}\right )}{b c e n \log (F)}\right )}{d}}{d}-\frac {e \left (\frac {3 g \left (\frac {(f+g x)^3}{3 d g}-\frac {e \left (\frac {(f+g x)^2 \log \left (\frac {e \left (F^{c (a+b x)}\right )^n}{d}+1\right )}{b c e n \log (F)}-\frac {2 g \left (\frac {g \operatorname {PolyLog}\left (3,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 n^2 \log ^2(F)}-\frac {(f+g x) \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c n \log (F)}\right )}{b c e n \log (F)}\right )}{d}\right )}{b c e n \log (F)}-\frac {(f+g x)^3}{b c e n \log (F) \left (e \left (F^{c (a+b x)}\right )^n+d\right )}\right )}{d}\)

Input:

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

Output:

-((e*(-((f + g*x)^3/(b*c*e*(d + e*(F^(c*(a + b*x)))^n)*n*Log[F])) + (3*g*( 
(f + g*x)^3/(3*d*g) - (e*(((f + g*x)^2*Log[1 + (e*(F^(c*(a + b*x)))^n)/d]) 
/(b*c*e*n*Log[F]) - (2*g*(-(((f + g*x)*PolyLog[2, -((e*(F^(c*(a + b*x)))^n 
)/d)])/(b*c*n*Log[F])) + (g*PolyLog[3, -((e*(F^(c*(a + b*x)))^n)/d)])/(b^2 
*c^2*n^2*Log[F]^2)))/(b*c*e*n*Log[F])))/d))/(b*c*e*n*Log[F])))/d) + ((f + 
g*x)^4/(4*d*g) - (e*(((f + g*x)^3*Log[1 + (e*(F^(c*(a + b*x)))^n)/d])/(b*c 
*e*n*Log[F]) - (3*g*(-(((f + g*x)^2*PolyLog[2, -((e*(F^(c*(a + b*x)))^n)/d 
)])/(b*c*n*Log[F])) + (2*g*(((f + g*x)*PolyLog[3, -((e*(F^(c*(a + b*x)))^n 
)/d)])/(b*c*n*Log[F]) - (g*PolyLog[4, -((e*(F^(c*(a + b*x)))^n)/d)])/(b^2* 
c^2*n^2*Log[F]^2)))/(b*c*n*Log[F])))/(b*c*e*n*Log[F])))/d)/d
 

Defintions of rubi rules used

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 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 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]
 

rule 7163

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3318\) vs. \(2(386)=772\).

Time = 0.34 (sec) , antiderivative size = 3319, normalized size of antiderivative = 8.55

method	result	size
risch	\(\text {Expression too large to display}\)	\(3319\)

Input:

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

Output:

1/n/c/b/ln(F)/d*(g^3*x^3+3*f*g^2*x^2+3*f^2*g*x+f^3)/(d+e*(F^(c*(b*x+a)))^n 
)+3/n^2/c^2/b^2/ln(F)^2/d^2*f^2*g*ln((F^(c*(b*x+a)))^n*F^(-n*c*b*x)*F^(n*c 
*b*x)*e+d)-3/n^2/c^2/b^2/ln(F)^2/d^2*g^3*polylog(2,-e*F^(n*c*b*x)*F^(-n*c* 
b*x)*(F^(c*(b*x+a)))^n/d)*x^2+6/n^3/c^3/b^3/ln(F)^3/d^2*g^3*polylog(3,-e*F 
^(n*c*b*x)*F^(-n*c*b*x)*(F^(c*(b*x+a)))^n/d)*x+3/n^2/c^2/b^2/ln(F)^2/d^2*g 
^3*ln((F^(c*(b*x+a)))^n*F^(-n*c*b*x)*F^(n*c*b*x)*e+d)*x^2+6/n^3/c^3/b^3/ln 
(F)^3/d^2*g^3*polylog(2,-e*F^(n*c*b*x)*F^(-n*c*b*x)*(F^(c*(b*x+a)))^n/d)*x 
+6/n^3/c^3/b^3/ln(F)^3/d^2*f*g^2*polylog(2,-e*F^(n*c*b*x)*F^(-n*c*b*x)*(F^ 
(c*(b*x+a)))^n/d)+3/n^2/c^4/b^4/ln(F)^4/d^2*g^3*ln((F^(c*(b*x+a)))^n*F^(-n 
*c*b*x)*F^(n*c*b*x)*e+d)*ln(F^(c*(b*x+a)))^2-3/n^2/c^2/b^2/ln(F)^2/d^2*g^3 
*ln(F^(n*c*b*x)*F^(-n*c*b*x)*(F^(c*(b*x+a)))^n)*x^2-3/n^2/c^4/b^4/ln(F)^4/ 
d^2*g^3*ln(F^(n*c*b*x)*F^(-n*c*b*x)*(F^(c*(b*x+a)))^n)*ln(F^(c*(b*x+a)))^2 
-3/n/c^3/b^3/ln(F)^3/d^2*f*g^2*ln(F^(c*(b*x+a)))^2-1/n/c/b/ln(F)/d^2*g^3*l 
n((F^(c*(b*x+a)))^n*F^(-n*c*b*x)*F^(n*c*b*x)*e+d)*x^3+1/n/c^4/b^4/ln(F)^4/ 
d^2*g^3*ln((F^(c*(b*x+a)))^n*F^(-n*c*b*x)*F^(n*c*b*x)*e+d)*ln(F^(c*(b*x+a) 
))^3+1/n/c/b/ln(F)/d^2*g^3*ln(F^(n*c*b*x)*F^(-n*c*b*x)*(F^(c*(b*x+a)))^n)* 
x^3-1/n/c^4/b^4/ln(F)^4/d^2*g^3*ln(F^(n*c*b*x)*F^(-n*c*b*x)*(F^(c*(b*x+a)) 
)^n)*ln(F^(c*(b*x+a)))^3+3/c^2/b^2/ln(F)^2/d^2*f*g^2*ln(F^(c*(b*x+a)))^2*x 
-1/n/c^4/b^4/ln(F)^4/d^2*g^3*ln(1+e*F^(n*c*b*x)*F^(-n*c*b*x)*(F^(c*(b*x+a) 
))^n/d)*ln(F^(c*(b*x+a)))^3-3/n/c^3/b^3/ln(F)^3/d^2*g^3*ln(F^(c*(b*x+a)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1469 vs. \(2 (384) = 768\).

Time = 0.12 (sec) , antiderivative size = 1469, normalized size of antiderivative = 3.79 \[ \int \frac {(f+g x)^3}{\left (d+e \left (F^{c (a+b x)}\right )^n\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(4*(b^3*c^3*d*f^3 - 3*a*b^2*c^3*d*f^2*g + 3*a^2*b*c^3*d*f*g^2 - a^3*c^ 
3*d*g^3)*n^3*log(F)^3 + (b^4*c^4*d*g^3*n^4*x^4 + 4*b^4*c^4*d*f*g^2*n^4*x^3 
 + 6*b^4*c^4*d*f^2*g*n^4*x^2 + 4*b^4*c^4*d*f^3*n^4*x + (4*a*b^3*c^4*d*f^3 
- 6*a^2*b^2*c^4*d*f^2*g + 4*a^3*b*c^4*d*f*g^2 - a^4*c^4*d*g^3)*n^4)*log(F) 
^4 + ((b^4*c^4*e*g^3*n^4*x^4 + 4*b^4*c^4*e*f*g^2*n^4*x^3 + 6*b^4*c^4*e*f^2 
*g*n^4*x^2 + 4*b^4*c^4*e*f^3*n^4*x + (4*a*b^3*c^4*e*f^3 - 6*a^2*b^2*c^4*e* 
f^2*g + 4*a^3*b*c^4*e*f*g^2 - a^4*c^4*e*g^3)*n^4)*log(F)^4 - 4*(b^3*c^3*e* 
g^3*n^3*x^3 + 3*b^3*c^3*e*f*g^2*n^3*x^2 + 3*b^3*c^3*e*f^2*g*n^3*x + (3*a*b 
^2*c^3*e*f^2*g - 3*a^2*b*c^3*e*f*g^2 + a^3*c^3*e*g^3)*n^3)*log(F)^3)*F^(b* 
c*n*x + a*c*n) - 12*((b^2*c^2*d*g^3*n^2*x^2 + 2*b^2*c^2*d*f*g^2*n^2*x + b^ 
2*c^2*d*f^2*g*n^2)*log(F)^2 + ((b^2*c^2*e*g^3*n^2*x^2 + 2*b^2*c^2*e*f*g^2* 
n^2*x + b^2*c^2*e*f^2*g*n^2)*log(F)^2 - 2*(b*c*e*g^3*n*x + b*c*e*f*g^2*n)* 
log(F))*F^(b*c*n*x + a*c*n) - 2*(b*c*d*g^3*n*x + b*c*d*f*g^2*n)*log(F))*di 
log(-(F^(b*c*n*x + a*c*n)*e + d)/d + 1) - 4*((b^3*c^3*d*f^3 - 3*a*b^2*c^3* 
d*f^2*g + 3*a^2*b*c^3*d*f*g^2 - a^3*c^3*d*g^3)*n^3*log(F)^3 - 3*(b^2*c^2*d 
*f^2*g - 2*a*b*c^2*d*f*g^2 + a^2*c^2*d*g^3)*n^2*log(F)^2 + ((b^3*c^3*e*f^3 
 - 3*a*b^2*c^3*e*f^2*g + 3*a^2*b*c^3*e*f*g^2 - a^3*c^3*e*g^3)*n^3*log(F)^3 
 - 3*(b^2*c^2*e*f^2*g - 2*a*b*c^2*e*f*g^2 + a^2*c^2*e*g^3)*n^2*log(F)^2)*F 
^(b*c*n*x + a*c*n))*log(F^(b*c*n*x + a*c*n)*e + d) - 4*((b^3*c^3*d*g^3*n^3 
*x^3 + 3*b^3*c^3*d*f*g^2*n^3*x^2 + 3*b^3*c^3*d*f^2*g*n^3*x + (3*a*b^2*c...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 699, normalized size of antiderivative = 1.80 \[ \int \frac {(f+g x)^3}{\left (d+e \left (F^{c (a+b x)}\right )^n\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

f^3*((b*c*n*x + a*c*n)/(b*c*d^2*n) + 1/((F^(b*c*n*x + a*c*n)*d*e + d^2)*b* 
c*n*log(F)) - log(F^(b*c*n*x + a*c*n)*e + d)/(b*c*d^2*n*log(F))) + (g^3*x^ 
3 + 3*f*g^2*x^2 + 3*f^2*g*x)/(F^(b*c*n*x)*F^(a*c*n)*b*c*d*e*n*log(F) + b*c 
*d^2*n*log(F)) - 3*f^2*g*x/(b*c*d^2*n*log(F)) + 3*f^2*g*log(F^(b*c*n*x)*F^ 
(a*c*n)*e + d)/(b^2*c^2*d^2*n^2*log(F)^2) - 3*(b*c*f^2*g*n*log(F) - 2*f*g^ 
2)*(b*c*n*x*log(F^(b*c*n*x)*F^(a*c*n)*e/d + 1)*log(F) + dilog(-F^(b*c*n*x) 
*F^(a*c*n)*e/d))/(b^3*c^3*d^2*n^3*log(F)^3) - (b^3*c^3*n^3*x^3*log(F^(b*c* 
n*x)*F^(a*c*n)*e/d + 1)*log(F)^3 + 3*b^2*c^2*n^2*x^2*dilog(-F^(b*c*n*x)*F^ 
(a*c*n)*e/d)*log(F)^2 - 6*b*c*n*x*log(F)*polylog(3, -F^(b*c*n*x)*F^(a*c*n) 
*e/d) + 6*polylog(4, -F^(b*c*n*x)*F^(a*c*n)*e/d))*g^3/(b^4*c^4*d^2*n^4*log 
(F)^4) - 3*(b^2*c^2*n^2*x^2*log(F^(b*c*n*x)*F^(a*c*n)*e/d + 1)*log(F)^2 + 
2*b*c*n*x*dilog(-F^(b*c*n*x)*F^(a*c*n)*e/d)*log(F) - 2*polylog(3, -F^(b*c* 
n*x)*F^(a*c*n)*e/d))*(b*c*f*g^2*n*log(F) - g^3)/(b^4*c^4*d^2*n^4*log(F)^4) 
 + 1/4*(b^4*c^4*g^3*n^4*x^4*log(F)^4 + 4*(b*c*f*g^2*n*log(F) - g^3)*b^3*c^ 
3*n^3*x^3*log(F)^3 + 6*(b^2*c^2*f^2*g*n^2*log(F)^2 - 2*b*c*f*g^2*n*log(F)) 
*b^2*c^2*n^2*x^2*log(F)^2)/(b^4*c^4*d^2*n^4*log(F)^4)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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