\(\int \frac {(f+g x)^2}{(d+e (F^{c (a+b x)})^n)^2} \, dx\) [40]

Optimal result

Integrand size = 25, antiderivative size = 294 \[ \int \frac {(f+g x)^2}{\left (d+e \left (F^{c (a+b x)}\right )^n\right )^2} \, dx=\frac {(f+g x)^3}{3 d^2 g}-\frac {(f+g x)^2}{b c d^2 n \log (F)}+\frac {(f+g x)^2}{b c d \left (d+e \left (F^{c (a+b x)}\right )^n\right ) n \log (F)}+\frac {2 g (f+g x) \log \left (1+\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 d^2 n^2 \log ^2(F)}-\frac {(f+g x)^2 \log \left (1+\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c d^2 n \log (F)}+\frac {2 g^2 \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 d^2 n^3 \log ^3(F)}-\frac {2 g (f+g x) \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 d^2 n^2 \log ^2(F)}+\frac {2 g^2 \operatorname {PolyLog}\left (3,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^3 c^3 d^2 n^3 \log ^3(F)} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 3.02 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2616, 2615, 2620, 2621, 2615, 2620, 2715, 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.

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

\(\Big \downarrow \) 2616

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

\(\Big \downarrow \) 2615

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2621

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

\(\Big \downarrow \) 2615

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-((e*(-((f + g*x)^2/(b*c*e*(d + e*(F^(c*(a + b*x)))^n)*n*Log[F])) + (2*g*( 
(f + g*x)^2/(2*d*g) - (e*(((f + g*x)*Log[1 + (e*(F^(c*(a + b*x)))^n)/d])/( 
b*c*e*n*Log[F]) + (g*PolyLog[2, -((e*(F^(c*(a + b*x)))^n)/d)])/(b^2*c^2*e* 
n^2*Log[F]^2)))/d))/(b*c*e*n*Log[F])))/d) + ((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*P 
olyLog[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)/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 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. \(1649\) vs. \(2(292)=584\).

Time = 0.14 (sec) , antiderivative size = 1650, normalized size of antiderivative = 5.61

method	result	size
risch	\(\text {Expression too large to display}\)	\(1650\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 875 vs. \(2 (290) = 580\).

Time = 0.09 (sec) , antiderivative size = 875, normalized size of antiderivative = 2.98 \[ \int \frac {(f+g x)^2}{\left (d+e \left (F^{c (a+b x)}\right )^n\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.59 \[ \int \frac {(f+g x)^2}{\left (d+e \left (F^{c (a+b x)}\right )^n\right )^2} \, dx=f^{2} {\left (\frac {b c n x + a c n}{b c d^{2} n} + \frac {1}{{\left (F^{b c n x + a c n} d e + d^{2}\right )} b c n \log \left (F\right )} - \frac {\log \left (F^{b c n x + a c n} e + d\right )}{b c d^{2} n \log \left (F\right )}\right )} + \frac {g^{2} x^{2} + 2 \, f g x}{F^{b c n x} F^{a c n} b c d e n \log \left (F\right ) + b c d^{2} n \log \left (F\right )} - \frac {2 \, f g x}{b c d^{2} n \log \left (F\right )} + \frac {2 \, f g \log \left (F^{b c n x} F^{a c n} e + d\right )}{b^{2} c^{2} d^{2} n^{2} \log \left (F\right )^{2}} - \frac {{\left (b^{2} c^{2} n^{2} x^{2} \log \left (\frac {F^{b c n x} F^{a c n} e}{d} + 1\right ) \log \left (F\right )^{2} + 2 \, b c n x {\rm Li}_2\left (-\frac {F^{b c n x} F^{a c n} e}{d}\right ) \log \left (F\right ) - 2 \, {\rm Li}_{3}(-\frac {F^{b c n x} F^{a c n} e}{d})\right )} g^{2}}{b^{3} c^{3} d^{2} n^{3} \log \left (F\right )^{3}} - \frac {2 \, {\left (b c f g n \log \left (F\right ) - g^{2}\right )} {\left (b c n x \log \left (\frac {F^{b c n x} F^{a c n} e}{d} + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{b c n x} F^{a c n} e}{d}\right )\right )}}{b^{3} c^{3} d^{2} n^{3} \log \left (F\right )^{3}} + \frac {b^{3} c^{3} g^{2} n^{3} x^{3} \log \left (F\right )^{3} + 3 \, {\left (b c f g n \log \left (F\right ) - g^{2}\right )} b^{2} c^{2} n^{2} x^{2} \log \left (F\right )^{2}}{3 \, b^{3} c^{3} d^{2} n^{3} \log \left (F\right )^{3}} \] Input:

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

Output:

f^2*((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^2*x^ 
2 + 2*f*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)) - 
 2*f*g*x/(b*c*d^2*n*log(F)) + 2*f*g*log(F^(b*c*n*x)*F^(a*c*n)*e + d)/(b^2* 
c^2*d^2*n^2*log(F)^2) - (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*polylo 
g(3, -F^(b*c*n*x)*F^(a*c*n)*e/d))*g^2/(b^3*c^3*d^2*n^3*log(F)^3) - 2*(b*c* 
f*g*n*log(F) - g^2)*(b*c*n*x*log(F^(b*c*n*x)*F^(a*c*n)*e/d + 1)*log(F) + d 
ilog(-F^(b*c*n*x)*F^(a*c*n)*e/d))/(b^3*c^3*d^2*n^3*log(F)^3) + 1/3*(b^3*c^ 
3*g^2*n^3*x^3*log(F)^3 + 3*(b*c*f*g*n*log(F) - g^2)*b^2*c^2*n^2*x^2*log(F) 
^2)/(b^3*c^3*d^2*n^3*log(F)^3)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(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*g**2*n + 2*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*g*n - f**(a*c*n + b*c*n*x)*log(f**(a*c*n + b*c* 
n*x)*e + d)*e*f**2 + f**(a*c*n + b*c*n*x)*log(f)*b*c*e*f**2*n*x - f**(a*c* 
n + b*c*n*x)*e*f**2 + 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*g**2*n + 2*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*g*n - log(f**(a*c*n + b*c*n*x)*e + d)*d*f**2 + log(f)*b*c*d*f**2*n*x)/(l 
og(f)*b*c*d**2*n*(f**(a*c*n + b*c*n*x)*e + d))