\(\int \frac {(c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(a g+b g x)^3} \, dx\) [123]

Optimal result

Integrand size = 43, antiderivative size = 242 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^3} \, dx=-\frac {B d i^2 n (c+d x)}{b^2 g^3 (a+b x)}-\frac {B i^2 n (c+d x)^2}{4 b g^3 (a+b x)^2}-\frac {d i^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^3 (a+b x)}-\frac {i^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b g^3 (a+b x)^2}-\frac {d^2 i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3}+\frac {B d^2 i^2 n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.07 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^3} \, dx=\frac {i^2 \left (-\frac {B (b c-a d)^2 n}{(a+b x)^2}+\frac {6 B d (-b c+a d) n}{a+b x}-6 B d^2 n \log (a+b x)-\frac {2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2}+\frac {8 d (-b c+a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}+4 d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+6 B d^2 n \log (c+d x)-2 B d^2 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{4 b^3 g^3} \] Input:

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

Output:

(i^2*(-((B*(b*c - a*d)^2*n)/(a + b*x)^2) + (6*B*d*(-(b*c) + a*d)*n)/(a + b 
*x) - 6*B*d^2*n*Log[a + b*x] - (2*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c 
 + d*x))^n]))/(a + b*x)^2 + (8*d*(-(b*c) + a*d)*(A + B*Log[e*((a + b*x)/(c 
 + d*x))^n]))/(a + b*x) + 4*d^2*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + 
d*x))^n]) + 6*B*d^2*n*Log[c + d*x] - 2*B*d^2*n*(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)])))/(4*b^3*g^3)
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.89, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {2961, 2780, 2741, 2780, 2741, 2779, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2780

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

\(\Big \downarrow \) 2741

\(\displaystyle \frac {i^2 \left (\frac {d \int \frac {(c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}+\frac {-\frac {(c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}-\frac {B n (c+d x)^2}{4 (a+b x)^2}}{b}\right )}{g^3}\)

\(\Big \downarrow \) 2780

\(\displaystyle \frac {i^2 \left (\frac {d \left (\frac {\int \frac {(c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2}d\frac {a+b x}{c+d x}}{b}+\frac {d \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}\right )}{b}+\frac {-\frac {(c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}-\frac {B n (c+d x)^2}{4 (a+b x)^2}}{b}\right )}{g^3}\)

\(\Big \downarrow \) 2741

\(\displaystyle \frac {i^2 \left (\frac {d \left (\frac {d \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}+\frac {-\frac {(c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {B n (c+d x)}{a+b x}}{b}\right )}{b}+\frac {-\frac {(c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}-\frac {B n (c+d x)^2}{4 (a+b x)^2}}{b}\right )}{g^3}\)

\(\Big \downarrow \) 2779

\(\displaystyle \frac {i^2 \left (\frac {d \left (\frac {d \left (\frac {B n \int \frac {(c+d x) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{b}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b}\right )}{b}+\frac {-\frac {(c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {B n (c+d x)}{a+b x}}{b}\right )}{b}+\frac {-\frac {(c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}-\frac {B n (c+d x)^2}{4 (a+b x)^2}}{b}\right )}{g^3}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {i^2 \left (\frac {d \left (\frac {d \left (\frac {B n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b}-\frac {\log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b}\right )}{b}+\frac {-\frac {(c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {B n (c+d x)}{a+b x}}{b}\right )}{b}+\frac {-\frac {(c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}-\frac {B n (c+d x)^2}{4 (a+b x)^2}}{b}\right )}{g^3}\)

Input:

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

Output:

(i^2*((-1/4*(B*n*(c + d*x)^2)/(a + b*x)^2 - ((c + d*x)^2*(A + B*Log[e*((a 
+ b*x)/(c + d*x))^n]))/(2*(a + b*x)^2))/b + (d*((-((B*n*(c + d*x))/(a + b* 
x)) - ((c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x))/b + (d 
*(-(((A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + 
b*x))])/b) + (B*n*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/b))/b))/b))/g^3
 

Defintions of rubi rules used

rule 2741

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> 
Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( 
m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
 

rule 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2780

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.))/((d_) + (e_.)* 
(x_)^(r_.)), x_Symbol] :> Simp[1/d   Int[x^m*(a + b*Log[c*x^n])^p, x], x] - 
 Simp[e/d   Int[(x^(m + r)*(a + b*Log[c*x^n])^p)/(d + e*x^r), x], x] /; Fre 
eQ[{a, b, c, d, e, m, n, r}, x] && IGtQ[p, 0] && IGtQ[r, 0] && ILtQ[m, -1]
 

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 2961

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + B*L 
og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

i**2*(Integral(A*c**2/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3), x) 
+ Integral(A*d**2*x**2/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3), x) 
 + Integral(B*c**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(a**3 + 3*a**2* 
b*x + 3*a*b**2*x**2 + b**3*x**3), x) + Integral(2*A*c*d*x/(a**3 + 3*a**2*b 
*x + 3*a*b**2*x**2 + b**3*x**3), x) + Integral(B*d**2*x**2*log(e*(a/(c + d 
*x) + b*x/(c + d*x))**n)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3), 
x) + Integral(2*B*c*d*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(a**3 + 3* 
a**2*b*x + 3*a*b**2*x**2 + b**3*x**3), x))/g**3
 

Maxima [F]

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

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

Output:

-1/2*B*c*d*i^2*n*((3*a*b*c - a^2*d + 2*(2*b^2*c - a*b*d)*x)/((b^5*c - a*b^ 
4*d)*g^3*x^2 + 2*(a*b^4*c - a^2*b^3*d)*g^3*x + (a^2*b^3*c - a^3*b^2*d)*g^3 
) + 2*(2*b*c*d - a*d^2)*log(b*x + a)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 
)*g^3) - 2*(2*b*c*d - a*d^2)*log(d*x + c)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^ 
2*d^2)*g^3)) + 1/4*B*c^2*i^2*n*((2*b*d*x - b*c + 3*a*d)/((b^4*c - a*b^3*d) 
*g^3*x^2 + 2*(a*b^3*c - a^2*b^2*d)*g^3*x + (a^2*b^2*c - a^3*b*d)*g^3) + 2* 
d^2*log(b*x + a)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3) - 2*d^2*log(d*x 
 + c)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3)) + 1/2*A*d^2*i^2*((4*a*b*x 
 + 3*a^2)/(b^5*g^3*x^2 + 2*a*b^4*g^3*x + a^2*b^3*g^3) + 2*log(b*x + a)/(b^ 
3*g^3)) + 1/2*B*d^2*i^2*(((4*a*b*x + 3*a^2 + 2*(b^2*x^2 + 2*a*b*x + a^2)*l 
og(b*x + a))*log((b*x + a)^n) - (4*a*b*x + 3*a^2 + 2*(b^2*x^2 + 2*a*b*x + 
a^2)*log(b*x + a))*log((d*x + c)^n))/(b^5*g^3*x^2 + 2*a*b^4*g^3*x + a^2*b^ 
3*g^3) + 2*integrate(1/2*(2*b^3*d*x^3*log(e) + 2*b^3*c*x^2*log(e) - 3*a^2* 
b*c*n + 3*a^3*d*n - 4*(a*b^2*c*n - a^2*b*d*n)*x - 2*(a^2*b*c*n - a^3*d*n + 
 (b^3*c*n - a*b^2*d*n)*x^2 + 2*(a*b^2*c*n - a^2*b*d*n)*x)*log(b*x + a))/(b 
^6*d*g^3*x^4 + a^3*b^3*c*g^3 + (b^6*c*g^3 + 3*a*b^5*d*g^3)*x^3 + 3*(a*b^5* 
c*g^3 + a^2*b^4*d*g^3)*x^2 + (3*a^2*b^4*c*g^3 + a^3*b^3*d*g^3)*x), x)) - ( 
2*b*x + a)*B*c*d*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^4*g^3*x^2 + 
 2*a*b^3*g^3*x + a^2*b^2*g^3) - (2*b*x + a)*A*c*d*i^2/(b^4*g^3*x^2 + 2*a*b 
^3*g^3*x + a^2*b^2*g^3) - 1/2*B*c^2*i^2*log(e*(b*x/(d*x + c) + a/(d*x +...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 4*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2)/(a**3 + 3*a**2*b*x + 3 
*a*b**2*x**2 + b**3*x**3),x)*a**6*b**4*d**4 + 8*int((log(((a + b*x)**n*e)/ 
(c + d*x)**n)*x**2)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*a** 
5*b**5*c*d**3 - 8*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2)/(a**3 + 3* 
a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*a**5*b**5*d**4*x - 4*int((log(((a 
 + b*x)**n*e)/(c + d*x)**n)*x**2)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b** 
3*x**3),x)*a**4*b**6*c**2*d**2 + 16*int((log(((a + b*x)**n*e)/(c + d*x)**n 
)*x**2)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*a**4*b**6*c*d** 
3*x - 4*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2)/(a**3 + 3*a**2*b*x + 
 3*a*b**2*x**2 + b**3*x**3),x)*a**4*b**6*d**4*x**2 - 8*int((log(((a + b*x) 
**n*e)/(c + d*x)**n)*x**2)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3) 
,x)*a**3*b**7*c**2*d**2*x + 8*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2 
)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*a**3*b**7*c*d**3*x**2 
 - 4*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2)/(a**3 + 3*a**2*b*x + 3* 
a*b**2*x**2 + b**3*x**3),x)*a**2*b**8*c**2*d**2*x**2 - 4*log(a + b*x)*a**7 
*d**4 + 8*log(a + b*x)*a**6*b*c*d**3 - 8*log(a + b*x)*a**6*b*d**4*x - 4*lo 
g(a + b*x)*a**5*b**2*c**2*d**2 + 16*log(a + b*x)*a**5*b**2*c*d**3*x - 4*lo 
g(a + b*x)*a**5*b**2*d**4*x**2 - 8*log(a + b*x)*a**4*b**3*c**2*d**2*x + 8* 
log(a + b*x)*a**4*b**3*c*d**3*x**2 - 4*log(a + b*x)*a**3*b**4*c**2*d**2*x* 
*2 + 2*log(a + b*x)*a**2*b**5*c**4*n + 4*log(a + b*x)*a*b**6*c**4*n*x +...