\(\int \frac {f+g x}{(d+e (F^{c (a+b x)})^n)^2} \, dx\) [41]

Optimal result

Integrand size = 23, antiderivative size = 191 \[ \int \frac {f+g x}{\left (d+e \left (F^{c (a+b x)}\right )^n\right )^2} \, dx=\frac {(f+g x)^2}{2 d^2 g}-\frac {g x}{b c d^2 n \log (F)}+\frac {f+g x}{b c d \left (d+e \left (F^{c (a+b x)}\right )^n\right ) n \log (F)}+\frac {g \log \left (d+e \left (F^{c (a+b x)}\right )^n\right )}{b^2 c^2 d^2 n^2 \log ^2(F)}-\frac {(f+g x) \log \left (1+\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b c d^2 n \log (F)}-\frac {g \operatorname {PolyLog}\left (2,-\frac {e \left (F^{c (a+b x)}\right )^n}{d}\right )}{b^2 c^2 d^2 n^2 \log ^2(F)} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2616, 2615, 2620, 2621, 2715, 2720, 798, 47, 14, 16, 2838}

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.

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

\(\Big \downarrow \) 2616

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

\(\Big \downarrow \) 2615

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2621

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 798

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 14

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

rule 16

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

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2615

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x 
_))))^(n_.)), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(a*d*(m + 1)), x] - Simp[ 
b/a   Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n)), x] 
, x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2616

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + 
(d_.)*(x_))^(m_.), x_Symbol] :> Simp[1/a   Int[(c + d*x)^m*(a + b*(F^(g*(e 
+ f*x)))^n)^(p + 1), x], x] - Simp[b/a   Int[(c + d*x)^m*(F^(g*(e + f*x)))^ 
n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n 
}, x] && ILtQ[p, 0] && IGtQ[m, 0]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2621

Int[((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((a_.) + (b_.)*((F_)^((g_.)*( 
(e_.) + (f_.)*(x_))))^(n_.))^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> 
 Simp[(c + d*x)^m*((a + b*(F^(g*(e + f*x)))^n)^(p + 1)/(b*f*g*n*(p + 1)*Log 
[F])), x] - Simp[d*(m/(b*f*g*n*(p + 1)*Log[F]))   Int[(c + d*x)^(m - 1)*(a 
+ b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, 
m, n, p}, x] && NeQ[p, -1]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(590\) vs. \(2(189)=378\).

Time = 0.16 (sec) , antiderivative size = 591, normalized size of antiderivative = 3.09

method	result	size
risch	\(\frac {g x +f}{b c d \left (d +e \left (F^{c \left (b x +a \right )}\right )^{n}\right ) n \ln \left (F \right )}+\frac {g \ln \left (F^{c \left (b x +a \right )}\right )^{2}}{2 d^{2} c^{2} b^{2} \ln \left (F \right )^{2}}-\frac {g \ln \left (F^{c \left (b x +a \right )}\right ) \ln \left (1+\frac {e \,F^{n c b x} F^{-n c b x} \left (F^{c \left (b x +a \right )}\right )^{n}}{d}\right )}{d^{2} n \,c^{2} b^{2} \ln \left (F \right )^{2}}-\frac {g \operatorname {polylog}\left (2, -\frac {e \,F^{n c b x} F^{-n c b x} \left (F^{c \left (b x +a \right )}\right )^{n}}{d}\right )}{d^{2} n^{2} c^{2} b^{2} \ln \left (F \right )^{2}}-\frac {f \ln \left (\left (F^{c \left (b x +a \right )}\right )^{n} F^{-n c b x} F^{n c b x} e +d \right )}{d^{2} n c b \ln \left (F \right )}+\frac {f \ln \left (F^{n c b x} F^{-n c b x} \left (F^{c \left (b x +a \right )}\right )^{n}\right )}{d^{2} n c b \ln \left (F \right )}+\frac {g \ln \left (\left (F^{c \left (b x +a \right )}\right )^{n} F^{-n c b x} F^{n c b x} e +d \right )}{d^{2} n^{2} c^{2} b^{2} \ln \left (F \right )^{2}}-\frac {g \ln \left (F^{n c b x} F^{-n c b x} \left (F^{c \left (b x +a \right )}\right )^{n}\right )}{d^{2} n^{2} c^{2} b^{2} \ln \left (F \right )^{2}}-\frac {g \ln \left (\left (F^{c \left (b x +a \right )}\right )^{n} F^{-n c b x} F^{n c b x} e +d \right ) x}{d^{2} n c b \ln \left (F \right )}+\frac {g \ln \left (\left (F^{c \left (b x +a \right )}\right )^{n} F^{-n c b x} F^{n c b x} e +d \right ) \ln \left (F^{c \left (b x +a \right )}\right )}{d^{2} n \,c^{2} b^{2} \ln \left (F \right )^{2}}+\frac {g \ln \left (F^{n c b x} F^{-n c b x} \left (F^{c \left (b x +a \right )}\right )^{n}\right ) x}{d^{2} n c b \ln \left (F \right )}-\frac {g \ln \left (F^{n c b x} F^{-n c b x} \left (F^{c \left (b x +a \right )}\right )^{n}\right ) \ln \left (F^{c \left (b x +a \right )}\right )}{d^{2} n \,c^{2} b^{2} \ln \left (F \right )^{2}}\)	\(591\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 409 vs. \(2 (188) = 376\).

Time = 0.08 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.14 \[ \int \frac {f+g x}{\left (d+e \left (F^{c (a+b x)}\right )^n\right )^2} \, dx=\frac {2 \, {\left (b c d f - a c d g\right )} n \log \left (F\right ) + {\left (b^{2} c^{2} d g n^{2} x^{2} + 2 \, b^{2} c^{2} d f n^{2} x + {\left (2 \, a b c^{2} d f - a^{2} c^{2} d g\right )} n^{2}\right )} \log \left (F\right )^{2} + {\left ({\left (b^{2} c^{2} e g n^{2} x^{2} + 2 \, b^{2} c^{2} e f n^{2} x + {\left (2 \, a b c^{2} e f - a^{2} c^{2} e g\right )} n^{2}\right )} \log \left (F\right )^{2} - 2 \, {\left (b c e g n x + a c e g n\right )} \log \left (F\right )\right )} F^{b c n x + a c n} - 2 \, {\left (F^{b c n x + a c n} e g + d g\right )} {\rm Li}_2\left (-\frac {F^{b c n x + a c n} e + d}{d} + 1\right ) - 2 \, {\left ({\left (b c d f - a c d g\right )} n \log \left (F\right ) + {\left ({\left (b c e f - a c e g\right )} n \log \left (F\right ) - e g\right )} F^{b c n x + a c n} - d g\right )} \log \left (F^{b c n x + a c n} e + d\right ) - 2 \, {\left ({\left (b c e g n x + a c e g n\right )} F^{b c n x + a c n} \log \left (F\right ) + {\left (b c d g n x + a c d g n\right )} \log \left (F\right )\right )} \log \left (\frac {F^{b c n x + a c n} e + d}{d}\right )}{2 \, {\left (F^{b c n x + a c n} b^{2} c^{2} d^{2} e n^{2} \log \left (F\right )^{2} + b^{2} c^{2} d^{3} n^{2} \log \left (F\right )^{2}\right )}} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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)) + integra 
te((b*c*n*x*log(F) - 1)/(F^(b*c*n*x)*F^(a*c*n)*b*c*d*e*n*log(F) + b*c*d^2* 
n*log(F)), x)) + f*((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)))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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