\(\int \frac {x^2 (a+b \log (c (d+e x)^n))^2}{(f+g x^2)^2} \, dx\) [326]

Optimal result

Integrand size = 29, antiderivative size = 815 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{3/2}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b e n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{3/2}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b^2 e n^2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{3/2}}-\frac {b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} g^{3/2}}+\frac {b^2 e n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{3/2}}+\frac {b n \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} g^{3/2}}+\frac {b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} g^{3/2}}-\frac {b^2 n^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 \sqrt {-f} g^{3/2}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 4.95 (sec) , antiderivative size = 1132, normalized size of antiderivative = 1.39 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\left (f+g x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

((-2*Sqrt[g]*x*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2 
) + (2*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*(a - b*n*Log[d + e*x] + b*Log[c*(d + e* 
x)^n])^2)/Sqrt[f] + 2*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(( 
-(Sqrt[g]*(d + e*x)*Log[d + e*x]) + e*((-I)*Sqrt[f] + Sqrt[g]*x)*Log[I*Sqr 
t[f] - Sqrt[g]*x])/((e*Sqrt[f] - I*d*Sqrt[g])*(Sqrt[f] + I*Sqrt[g]*x)) + ( 
-(Sqrt[g]*(d + e*x)*Log[d + e*x]) + e*(I*Sqrt[f] + Sqrt[g]*x)*Log[I*Sqrt[f 
] + Sqrt[g]*x])/((e*Sqrt[f] + I*d*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)) - (I*( 
Log[d + e*x]*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/(e*Sqrt[f] - I*d*Sqrt[g])] + 
PolyLog[2, ((-I)*Sqrt[g]*(d + e*x))/(e*Sqrt[f] - I*d*Sqrt[g])]))/Sqrt[f] + 
 (I*(Log[d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sqrt[f] + I*d*Sqrt[g] 
)] + PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*Sqrt[f] + I*d*Sqrt[g])]))/Sqrt[f] 
) + b^2*n^2*((-(Sqrt[g]*(d + e*x)*Log[d + e*x]^2) + 2*e*(I*Sqrt[f] + Sqrt[ 
g]*x)*Log[d + e*x]*Log[(e*(Sqrt[f] - I*Sqrt[g]*x))/(e*Sqrt[f] + I*d*Sqrt[g 
])] + 2*e*(I*Sqrt[f] + Sqrt[g]*x)*PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*Sqrt 
[f] + I*d*Sqrt[g])])/((e*Sqrt[f] + I*d*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)) - 
 (Log[d + e*x]*(Sqrt[g]*(d + e*x)*Log[d + e*x] + (2*I)*e*(Sqrt[f] + I*Sqrt 
[g]*x)*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/(e*Sqrt[f] - I*d*Sqrt[g])]) + (2*I) 
*e*(Sqrt[f] + I*Sqrt[g]*x)*PolyLog[2, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d 
*Sqrt[g])])/((e*Sqrt[f] - I*d*Sqrt[g])*(Sqrt[f] + I*Sqrt[g]*x)) + (I*(Log[ 
d + e*x]^2*Log[1 - (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] + ...
 

Rubi [A] (verified)

Time = 3.08 (sec) , antiderivative size = 815, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2863, 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.

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*(e*Sqrt[-f] + d*Sqrt[g])*g*(Sq 
rt[-f] - Sqrt[g]*x)) + ((d + e*x)*(a + b*Log[c*(d + e*x)^n])^2)/(4*(e*Sqrt 
[-f] - d*Sqrt[g])*g*(Sqrt[-f] + Sqrt[g]*x)) + (b*e*n*(a + b*Log[c*(d + e*x 
)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*(e*Sqrt 
[-f] + d*Sqrt[g])*g^(3/2)) + ((a + b*Log[c*(d + e*x)^n])^2*Log[(e*(Sqrt[-f 
] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*Sqrt[-f]*g^(3/2)) - (b*e*n*( 
a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*S 
qrt[g])])/(2*(e*Sqrt[-f] - d*Sqrt[g])*g^(3/2)) - ((a + b*Log[c*(d + e*x)^n 
])^2*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(4*Sqrt[-f] 
*g^(3/2)) - (b^2*e*n^2*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sq 
rt[g]))])/(2*(e*Sqrt[-f] - d*Sqrt[g])*g^(3/2)) - (b*n*(a + b*Log[c*(d + e* 
x)^n])*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*Sqr 
t[-f]*g^(3/2)) + (b^2*e*n^2*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d 
*Sqrt[g])])/(2*(e*Sqrt[-f] + d*Sqrt[g])*g^(3/2)) + (b*n*(a + b*Log[c*(d + 
e*x)^n])*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*Sqrt 
[-f]*g^(3/2)) + (b^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d* 
Sqrt[g]))])/(2*Sqrt[-f]*g^(3/2)) - (b^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x)) 
/(e*Sqrt[-f] + d*Sqrt[g])])/(2*Sqrt[-f]*g^(3/2))
 

Defintions of rubi rules used

rule 2009

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

rule 2863

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) 
^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a 
 + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((b^2*x^2*log((e*x + d)^n*c)^2 + 2*a*b*x^2*log((e*x + d)^n*c) + a^ 
2*x^2)/(g^2*x^4 + 2*f*g*x^2 + f^2), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

-1/2*a^2*(x/(g^2*x^2 + f*g) - arctan(g*x/sqrt(f*g))/(sqrt(f*g)*g)) + integ 
rate((b^2*x^2*log((e*x + d)^n)^2 + 2*(b^2*log(c) + a*b)*x^2*log((e*x + d)^ 
n) + (b^2*log(c)^2 + 2*a*b*log(c))*x^2)/(g^2*x^4 + 2*f*g*x^2 + f^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a**2*d**2*e*f*g + sqrt(g)*s 
qrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a**2*d**2*e*g**2*x**2 + sqrt(g)*sqrt( 
f)*atan((g*x)/(sqrt(g)*sqrt(f)))*a**2*e**3*f**2 + sqrt(g)*sqrt(f)*atan((g* 
x)/(sqrt(g)*sqrt(f)))*a**2*e**3*f*g*x**2 + 4*sqrt(g)*sqrt(f)*atan((g*x)/(s 
qrt(g)*sqrt(f)))*a*b*e**3*f**2*n + 4*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*s 
qrt(f)))*a*b*e**3*f*g*n*x**2 - 2*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt( 
f)))*b**2*d**2*e*f*g*n**2 - 2*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f)) 
)*b**2*d**2*e*g**2*n**2*x**2 + 4*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt( 
f)))*b**2*e**3*f**2*n**2 + 4*sqrt(g)*sqrt(f)*atan((g*x)/(sqrt(g)*sqrt(f))) 
*b**2*e**3*f*g*n**2*x**2 + 2*int(log((d + e*x)**n*c)**2/(d*f**2 + 2*d*f*g* 
x**2 + d*g**2*x**4 + e*f**2*x + 2*e*f*g*x**3 + e*g**2*x**5),x)*b**2*d**3*e 
*f**3*g**2 + 2*int(log((d + e*x)**n*c)**2/(d*f**2 + 2*d*f*g*x**2 + d*g**2* 
x**4 + e*f**2*x + 2*e*f*g*x**3 + e*g**2*x**5),x)*b**2*d**3*e*f**2*g**3*x** 
2 + 2*int(log((d + e*x)**n*c)**2/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e* 
f**2*x + 2*e*f*g*x**3 + e*g**2*x**5),x)*b**2*d*e**3*f**4*g + 2*int(log((d 
+ e*x)**n*c)**2/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e*f**2*x + 2*e*f*g* 
x**3 + e*g**2*x**5),x)*b**2*d*e**3*f**3*g**2*x**2 + 4*int(log((d + e*x)**n 
*c)/(d*f**2 + 2*d*f*g*x**2 + d*g**2*x**4 + e*f**2*x + 2*e*f*g*x**3 + e*g** 
2*x**5),x)*a*b*d**3*e*f**3*g**2 + 4*int(log((d + e*x)**n*c)/(d*f**2 + 2*d* 
f*g*x**2 + d*g**2*x**4 + e*f**2*x + 2*e*f*g*x**3 + e*g**2*x**5),x)*a*b*...