\(\int \frac {(c+d x)^2}{(a+b (F^{g (e+f x)})^n)^2} \, dx\) [53]

Optimal result

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

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 3.03 (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.

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

\(\Big \downarrow \) 2616

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

\(\Big \downarrow \) 2615

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2621

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

\(\Big \downarrow \) 2615

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 = 0.15 (sec) , antiderivative size = 1650, normalized size of antiderivative = 5.61

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 834 vs. \(2 (290) = 580\).

Time = 0.09 (sec) , antiderivative size = 834, normalized size of antiderivative = 2.84 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.59 \[ \int \frac {(c+d x)^2}{\left (a+b \left (F^{g (e+f x)}\right )^n\right )^2} \, dx=c^{2} {\left (\frac {f g n x + e g n}{a^{2} f g n} + \frac {1}{{\left (F^{f g n x + e g n} a b + a^{2}\right )} f g n \log \left (F\right )} - \frac {\log \left (F^{f g n x + e g n} b + a\right )}{a^{2} f g n \log \left (F\right )}\right )} + \frac {d^{2} x^{2} + 2 \, c d x}{F^{f g n x} F^{e g n} a b f g n \log \left (F\right ) + a^{2} f g n \log \left (F\right )} - \frac {2 \, c d x}{a^{2} f g n \log \left (F\right )} + \frac {2 \, c d \log \left (F^{f g n x} F^{e g n} b + a\right )}{a^{2} f^{2} g^{2} n^{2} \log \left (F\right )^{2}} - \frac {{\left (f^{2} g^{2} n^{2} x^{2} \log \left (\frac {F^{f g n x} F^{e g n} b}{a} + 1\right ) \log \left (F\right )^{2} + 2 \, f g n x {\rm Li}_2\left (-\frac {F^{f g n x} F^{e g n} b}{a}\right ) \log \left (F\right ) - 2 \, {\rm Li}_{3}(-\frac {F^{f g n x} F^{e g n} b}{a})\right )} d^{2}}{a^{2} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} - \frac {2 \, {\left (c d f g n \log \left (F\right ) - d^{2}\right )} {\left (f g n x \log \left (\frac {F^{f g n x} F^{e g n} b}{a} + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{f g n x} F^{e g n} b}{a}\right )\right )}}{a^{2} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} + \frac {d^{2} f^{3} g^{3} n^{3} x^{3} \log \left (F\right )^{3} + 3 \, {\left (c d f g n \log \left (F\right ) - d^{2}\right )} f^{2} g^{2} n^{2} x^{2} \log \left (F\right )^{2}}{3 \, a^{2} f^{3} g^{3} n^{3} \log \left (F\right )^{3}} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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