\(\int \frac {f+g x}{(a+b \log (c (d+e x)^n))^2} \, dx\) [96]

Optimal result

Integrand size = 22, antiderivative size = 177 \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {e^{-\frac {a}{b n}} (e f-d g) (d+e x) \left (c (d+e x)^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b^2 e^2 n^2}+\frac {2 e^{-\frac {2 a}{b n}} g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b^2 e^2 n^2}-\frac {(d+e x) (f+g x)}{b e n \left (a+b \log \left (c (d+e x)^n\right )\right )} \] Output:

(-d*g+e*f)*(e*x+d)*Ei((a+b*ln(c*(e*x+d)^n))/b/n)/b^2/e^2/exp(a/b/n)/n^2/(( 
c*(e*x+d)^n)^(1/n))+2*g*(e*x+d)^2*Ei(2*(a+b*ln(c*(e*x+d)^n))/b/n)/b^2/e^2/ 
exp(2*a/b/n)/n^2/((c*(e*x+d)^n)^(2/n))-(e*x+d)*(g*x+f)/b/e/n/(a+b*ln(c*(e* 
x+d)^n))
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.18 \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=-\frac {e^{-\frac {2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (b e e^{\frac {2 a}{b n}} n \left (c (d+e x)^n\right )^{2/n} (f+g x)-e^{\frac {a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac {1}{n}} \operatorname {ExpIntegralEi}\left (\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-2 g (d+e x) \operatorname {ExpIntegralEi}\left (\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )\right )}{b^2 e^2 n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \] Input:

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

Output:

-(((d + e*x)*(b*e*E^((2*a)/(b*n))*n*(c*(d + e*x)^n)^(2/n)*(f + g*x) - E^(a 
/(b*n))*(e*f - d*g)*(c*(d + e*x)^n)^n^(-1)*ExpIntegralEi[(a + b*Log[c*(d + 
 e*x)^n])/(b*n)]*(a + b*Log[c*(d + e*x)^n]) - 2*g*(d + e*x)*ExpIntegralEi[ 
(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)]*(a + b*Log[c*(d + e*x)^n])))/(b^2*e^ 
2*E^((2*a)/(b*n))*n^2*(c*(d + e*x)^n)^(2/n)*(a + b*Log[c*(d + e*x)^n])))
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.45, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2847, 2836, 2737, 2609, 2846, 2009}

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.

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

\(\Big \downarrow \) 2847

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

\(\Big \downarrow \) 2836

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

\(\Big \downarrow \) 2737

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

\(\Big \downarrow \) 2609

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

\(\Big \downarrow \) 2846

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-(((e*f - d*g)*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/ 
(b^2*e^2*E^(a/(b*n))*n^2*(c*(d + e*x)^n)^n^(-1))) + (2*(((e*f - d*g)*(d + 
e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b*e^2*E^(a/(b*n))*n 
*(c*(d + e*x)^n)^n^(-1)) + (g*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d 
 + e*x)^n]))/(b*n)])/(b*e^2*E^((2*a)/(b*n))*n*(c*(d + e*x)^n)^(2/n))))/(b* 
n) - ((d + e*x)*(f + g*x))/(b*e*n*(a + b*Log[c*(d + e*x)^n]))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2609

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si 
mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F 
reeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2737

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x/(n*(c*x 
^n)^(1/n))   Subst[Int[E^(x/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ 
[{a, b, c, n, p}, x]
 

rule 2836

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

rule 2846

Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.) 
]*(b_.)), x_Symbol] :> Int[ExpandIntegrand[(f + g*x)^q/(a + b*Log[c*(d + e* 
x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] & 
& IGtQ[q, 0]
 

rule 2847

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. 
)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)*(f + g*x)^q*((a + b*Log[c*(d + e 
*x)^n])^(p + 1)/(b*e*n*(p + 1))), x] + (-Simp[(q + 1)/(b*n*(p + 1))   Int[( 
f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Simp[q*((e*f - d*g) 
/(b*e*n*(p + 1)))   Int[(f + g*x)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1 
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && Lt 
Q[p, -1] && GtQ[q, 0]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.14 (sec) , antiderivative size = 2300, normalized size of antiderivative = 12.99

method	result	size
risch	\(\text {Expression too large to display}\)	\(2300\)

Input:

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

Output:

-2*(e*x+d)*(g*x+f)/b/e/n/(I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I 
*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*b*Pi*csgn(I*c*(e*x 
+d)^n)^3+I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+2*ln((e*x+d)^n)*b+2*b*ln(c 
)+2*a)-1/b^2/n^2*f*c^(-1/n)*((e*x+d)^n)^(-1/n)*exp(-1/2*(I*b*Pi*csgn(I*(e* 
x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n) 
*csgn(I*c)-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn( 
I*c)+2*a)/n/b)*Ei(1,-ln(e*x+d)-1/2*(I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x 
+d)^n)^2-I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*b*Pi*csg 
n(I*c*(e*x+d)^n)^3+I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+2*b*ln(c)+2*b*(l 
n((e*x+d)^n)-n*ln(e*x+d))+2*a)/n/b)*x-1/b^2/e/n^2*f*c^(-1/n)*((e*x+d)^n)^( 
-1/n)*exp(-1/2*(I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn 
(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+I 
*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+2*a)/n/b)*Ei(1,-ln(e*x+d)-1/2*(I*b*P 
i*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I* 
c*(e*x+d)^n)*csgn(I*c)-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+I*b*Pi*csgn(I*c*(e*x+d 
)^n)^2*csgn(I*c)+2*b*ln(c)+2*b*(ln((e*x+d)^n)-n*ln(e*x+d))+2*a)/n/b)*d-2/b 
^2/n^2*g*c^(-2/n)*((e*x+d)^n)^(-2/n)*exp(-(I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I 
*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*csgn(I*c)-I*b 
*Pi*csgn(I*c*(e*x+d)^n)^3+I*b*Pi*csgn(I*c*(e*x+d)^n)^2*csgn(I*c)+2*a)/n/b) 
*Ei(1,-2*ln(e*x+d)-(I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*...
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.35 \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\frac {{\left ({\left (a e f - a d g + {\left (b e f - b d g\right )} n \log \left (e x + d\right ) + {\left (b e f - b d g\right )} \log \left (c\right )\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )} \operatorname {log\_integral}\left ({\left (e x + d\right )} e^{\left (\frac {b \log \left (c\right ) + a}{b n}\right )}\right ) - {\left (b e^{2} g n x^{2} + b d e f n + {\left (b e^{2} f + b d e g\right )} n x\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )} + 2 \, {\left (b g n \log \left (e x + d\right ) + b g \log \left (c\right ) + a g\right )} \operatorname {log\_integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} e^{\left (\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right )\right )} e^{\left (-\frac {2 \, {\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{b^{3} e^{2} n^{3} \log \left (e x + d\right ) + b^{3} e^{2} n^{2} \log \left (c\right ) + a b^{2} e^{2} n^{2}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 973 vs. \(2 (177) = 354\).

Time = 0.15 (sec) , antiderivative size = 973, normalized size of antiderivative = 5.50 \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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