\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(f+g x)^3} \, dx\) [73]

Optimal result

Integrand size = 32, antiderivative size = 389 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^3} \, dx=\frac {B (b c-a d) g n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b f-a g)^2 (d f-c g) (f+g x)}+\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 g (b f-a g)^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 g (f+g x)^2}+\frac {B^2 (b c-a d)^2 g n^2 \log \left (\frac {f+g x}{c+d x}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac {B (b c-a d) (2 b d f-b c g-a d g) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) n^2 \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g)^2 (d f-c g)^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.40 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.58 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^3} \, dx=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {B n (f+g x) \left (2 (b c-a d) g (b f-a g) (d f-c g) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 b^2 (d f-c g)^2 (f+g x) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 d^2 (b f-a g)^2 (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+2 (b c-a d) g (-2 b d f+b c g+a d g) (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)-2 B (b c-a d) g n (f+g x) (b (d f-c g) \log (a+b x)+(-b d f+a d g) \log (c+d x)+(b c-a d) g \log (f+g x))+b^2 B (d f-c g)^2 n (f+g x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-B d^2 (b f-a g)^2 n (f+g x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )-2 B (b c-a d) g (-2 b d f+b c g+a d g) n (f+g x) \left (\left (\log \left (\frac {g (a+b x)}{-b f+a g}\right )-\log \left (\frac {g (c+d x)}{-d f+c g}\right )\right ) \log (f+g x)+\operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )-\operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )\right )\right )}{(b f-a g)^2 (d f-c g)^2}}{2 g (f+g x)^2} \] Input:

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(f + g*x)^3,x]
 

Output:

-1/2*((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + (B*n*(f + g*x)*(2*(b*c - 
a*d)*g*(b*f - a*g)*(d*f - c*g)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 2* 
b^2*(d*f - c*g)^2*(f + g*x)*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x) 
)^n]) + 2*d^2*(b*f - a*g)^2*(f + g*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n 
])*Log[c + d*x] + 2*(b*c - a*d)*g*(-2*b*d*f + b*c*g + a*d*g)*(f + g*x)*(A 
+ B*Log[e*((a + b*x)/(c + d*x))^n])*Log[f + g*x] - 2*B*(b*c - a*d)*g*n*(f 
+ g*x)*(b*(d*f - c*g)*Log[a + b*x] + (-(b*d*f) + a*d*g)*Log[c + d*x] + (b* 
c - a*d)*g*Log[f + g*x]) + b^2*B*(d*f - c*g)^2*n*(f + g*x)*(Log[a + b*x]*( 
Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b* 
x))/(-(b*c) + a*d)]) - B*d^2*(b*f - a*g)^2*n*(f + g*x)*((2*Log[(d*(a + b*x 
))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x 
))/(b*c - a*d)]) - 2*B*(b*c - a*d)*g*(-2*b*d*f + b*c*g + a*d*g)*n*(f + g*x 
)*((Log[(g*(a + b*x))/(-(b*f) + a*g)] - Log[(g*(c + d*x))/(-(d*f) + c*g)]) 
*Log[f + g*x] + PolyLog[2, (b*(f + g*x))/(b*f - a*g)] - PolyLog[2, (d*(f + 
 g*x))/(d*f - c*g)])))/((b*f - a*g)^2*(d*f - c*g)^2))/(g*(f + g*x)^2)
 

Rubi [A] (verified)

Time = 1.07 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.33, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2953, 2798, 2804, 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.

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

\(\Big \downarrow \) 2953

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

\(\Big \downarrow \) 2798

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

\(\Big \downarrow \) 2804

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(f + g*x)^3,x]
 

Output:

(b*c - a*d)*(-1/2*((b - (d*(a + b*x))/(c + d*x))^2*(A + B*Log[e*((a + b*x) 
/(c + d*x))^n])^2)/((b*c - a*d)*g*(b*f - a*g - ((d*f - c*g)*(a + b*x))/(c 
+ d*x))^2) + (B*n*(((b*c - a*d)^2*g^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c 
 + d*x))^n]))/((b*f - a*g)^2*(d*f - c*g)*(c + d*x)*(b*f - a*g - ((d*f - c* 
g)*(a + b*x))/(c + d*x))) + (b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2) 
/(2*B*(b*f - a*g)^2*n) + (B*(b*c - a*d)^2*g^2*n*Log[b*f - a*g - ((d*f - c* 
g)*(a + b*x))/(c + d*x)])/((b*f - a*g)^2*(d*f - c*g)^2) + ((b*c - a*d)*g*( 
2*b*d*f - b*c*g - a*d*g)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (( 
d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)^2*(d*f - c*g) 
^2) + (B*(b*c - a*d)*g*(2*b*d*f - b*c*g - a*d*g)*n*PolyLog[2, ((d*f - c*g) 
*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*f - a*g)^2*(d*f - c*g)^2)))/((b* 
c - a*d)*g))
 

Defintions of rubi rules used

rule 2009

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

rule 2798

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_)*(( 
f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*(d + e*x)^(q + 
 1)*((a + b*Log[c*x^n])^p/((q + 1)*(e*f - d*g))), x] - Simp[b*n*(p/((q + 1) 
*(e*f - d*g)))   Int[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] 
)^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[e*f 
 - d*g, 0] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 2804

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ 
u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / 
; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
 

rule 2953

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)   Sub 
st[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2 
)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n} 
, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
 

Maple [F]

\[\int \frac {{\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}}{\left (g x +f \right )^{3}}d x\]

Input:

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^3,x)
 

Output:

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^3,x)
 

Fricas [F]

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

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^3,x, algorithm="frica 
s")
 

Output:

integral((B^2*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*A*B*log(e*((b*x + a)/(d 
*x + c))^n) + A^2)/(g^3*x^3 + 3*f*g^2*x^2 + 3*f^2*g*x + f^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^3} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(g*x+f)**3,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^3,x, algorithm="maxim 
a")
 

Output:

(b^2*log(b*x + a)/(b^2*f^2*g - 2*a*b*f*g^2 + a^2*g^3) - d^2*log(d*x + c)/( 
d^2*f^2*g - 2*c*d*f*g^2 + c^2*g^3) + (2*(b^2*c*d - a*b*d^2)*f - (b^2*c^2 - 
 a^2*d^2)*g)*log(g*x + f)/(b^2*d^2*f^4 + a^2*c^2*g^4 - 2*(b^2*c*d + a*b*d^ 
2)*f^3*g + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^2*g^2 - 2*(a*b*c^2 + a^2*c*d) 
*f*g^3) - (b*c - a*d)/(b*d*f^3 + a*c*f*g^2 - (b*c + a*d)*f^2*g + (b*d*f^2* 
g + a*c*g^3 - (b*c + a*d)*f*g^2)*x))*A*B*n - 1/2*B^2*(log((d*x + c)^n)^2/( 
g^3*x^2 + 2*f*g^2*x + f^2*g) + 2*integrate(-(d*g*x*log(e)^2 + c*g*log(e)^2 
 + (d*g*x + c*g)*log((b*x + a)^n)^2 + 2*(d*g*x*log(e) + c*g*log(e))*log((b 
*x + a)^n) + (d*f*n + (g*n - 2*g*log(e))*d*x - 2*c*g*log(e) - 2*(d*g*x + c 
*g)*log((b*x + a)^n))*log((d*x + c)^n))/(d*g^4*x^4 + c*f^3*g + (3*d*f*g^3 
+ c*g^4)*x^3 + 3*(d*f^2*g^2 + c*f*g^3)*x^2 + (d*f^3*g + 3*c*f^2*g^2)*x), x 
)) - A*B*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(g^3*x^2 + 2*f*g^2*x + f^2 
*g) - 1/2*A^2/(g^3*x^2 + 2*f*g^2*x + f^2*g)
 

Giac [F]

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

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^3,x, algorithm="giac" 
)
 

Output:

integrate((B*log(e*((b*x + a)/(d*x + c))^n) + A)^2/(g*x + f)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^3} \, dx=\int \frac {{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (f+g\,x\right )}^3} \,d x \] Input:

int((A + B*log(e*((a + b*x)/(c + d*x))^n))^2/(f + g*x)^3,x)
 

Output:

int((A + B*log(e*((a + b*x)/(c + d*x))^n))^2/(f + g*x)^3, x)
 

Reduce [F]

\[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^3} \, dx=\text {too large to display} \] Input:

int((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^3,x)
 

Output:

( - 4*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a**3*c**2*d*f**3*g**3 + 
3*a**3*c**2*d*f**2*g**4*x + 3*a**3*c**2*d*f*g**5*x**2 + a**3*c**2*d*g**6*x 
**3 + a**3*c*d**2*f**3*g**3*x + 3*a**3*c*d**2*f**2*g**4*x**2 + 3*a**3*c*d* 
*2*f*g**5*x**3 + a**3*c*d**2*g**6*x**4 + a**2*b*c**3*f**3*g**3 + 3*a**2*b* 
c**3*f**2*g**4*x + 3*a**2*b*c**3*f*g**5*x**2 + a**2*b*c**3*g**6*x**3 - 3*a 
**2*b*c**2*d*f**4*g**2 - 7*a**2*b*c**2*d*f**3*g**3*x - 3*a**2*b*c**2*d*f** 
2*g**4*x**2 + 3*a**2*b*c**2*d*f*g**5*x**3 + 2*a**2*b*c**2*d*g**6*x**4 - 3* 
a**2*b*c*d**2*f**4*g**2*x - 8*a**2*b*c*d**2*f**3*g**3*x**2 - 6*a**2*b*c*d* 
*2*f**2*g**4*x**3 + a**2*b*c*d**2*g**6*x**5 + a*b**2*c**3*f**3*g**3*x + 3* 
a*b**2*c**3*f**2*g**4*x**2 + 3*a*b**2*c**3*f*g**5*x**3 + a*b**2*c**3*g**6* 
x**4 - 3*a*b**2*c**2*d*f**4*g**2*x - 8*a*b**2*c**2*d*f**3*g**3*x**2 - 6*a* 
b**2*c**2*d*f**2*g**4*x**3 + a*b**2*c**2*d*g**6*x**5 + a*b**2*c*d**2*f**6 
+ 3*a*b**2*c*d**2*f**5*g*x - 8*a*b**2*c*d**2*f**3*g**3*x**3 - 9*a*b**2*c*d 
**2*f**2*g**4*x**4 - 3*a*b**2*c*d**2*f*g**5*x**5 + a*b**2*d**3*f**6*x + 3* 
a*b**2*d**3*f**5*g*x**2 + 3*a*b**2*d**3*f**4*g**2*x**3 + a*b**2*d**3*f**3* 
g**3*x**4 + b**3*c*d**2*f**6*x + 3*b**3*c*d**2*f**5*g*x**2 + 3*b**3*c*d**2 
*f**4*g**2*x**3 + b**3*c*d**2*f**3*g**3*x**4 + b**3*d**3*f**6*x**2 + 3*b** 
3*d**3*f**5*g*x**3 + 3*b**3*d**3*f**4*g**2*x**4 + b**3*d**3*f**3*g**3*x**5 
),x)*a**9*b**2*c**4*d**5*f**4*g**13*n - 8*int((log(((a + b*x)**n*e)/(c + d 
*x)**n)*x)/(a**3*c**2*d*f**3*g**3 + 3*a**3*c**2*d*f**2*g**4*x + 3*a**3*...