\(\int \frac {1}{(a g+b g x)^3 (A+B \log (\frac {e (c+d x)^2}{(a+b x)^2}))^2} \, dx\) [228]

Optimal result

Integrand size = 34, antiderivative size = 206 \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\frac {d e^{-\frac {A}{2 B}} (c+d x) \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{2 B}\right )}{4 B^2 (b c-a d)^2 g^3 (a+b x) \sqrt {\frac {e (c+d x)^2}{(a+b x)^2}}}-\frac {b e^{-\frac {A}{B}} \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{B}\right )}{2 B^2 (b c-a d)^2 e g^3}+\frac {c+d x}{2 B (b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )} \] Output:

1/4*d*(d*x+c)*Ei(1/2*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/B)/B^2/(-a*d+b*c)^2/e 
xp(1/2*A/B)/g^3/(b*x+a)/(e*(d*x+c)^2/(b*x+a)^2)^(1/2)-1/2*b*Ei((A+B*ln(e*( 
d*x+c)^2/(b*x+a)^2))/B)/B^2/(-a*d+b*c)^2/e/exp(A/B)/g^3+1/2*(d*x+c)/B/(-a* 
d+b*c)/g^3/(b*x+a)^2/(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx \] Input:

Integrate[1/((a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2),x 
]
 

Output:

Integrate[1/((a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2), 
x]
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.37, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2952, 2757, 2737, 2609, 2767, 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.

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

\(\Big \downarrow \) 2952

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

\(\Big \downarrow \) 2757

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

\(\Big \downarrow \) 2737

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

\(\Big \downarrow \) 2609

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

\(\Big \downarrow \) 2767

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

\(\Big \downarrow \) 2009

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

Input:

Int[1/((a*g + b*g*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2),x]
 

Output:

(-1/4*(d*(c + d*x)*ExpIntegralEi[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/ 
(2*B)])/(B^2*E^(A/(2*B))*(a + b*x)*Sqrt[(e*(c + d*x)^2)/(a + b*x)^2]) + (( 
d*(c + d*x)*ExpIntegralEi[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/(2*B)]) 
/(2*B*E^(A/(2*B))*(a + b*x)*Sqrt[(e*(c + d*x)^2)/(a + b*x)^2]) - (b*ExpInt 
egralEi[(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])/B])/(2*B*e*E^(A/B)))/B - 
((c + d*x)*(d - (b*(c + d*x))/(a + b*x)))/(2*B*(a + b*x)*(A + B*Log[(e*(c 
+ d*x)^2)/(a + b*x)^2])))/((b*c - a*d)^2*g^3)
 

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 2757

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x 
_Symbol] :> Simp[x*(d + e*x)^q*((a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1))), 
x] + (-Simp[(q + 1)/(b*n*(p + 1))   Int[(d + e*x)^q*(a + b*Log[c*x^n])^(p + 
 1), x], x] + Simp[d*(q/(b*n*(p + 1)))   Int[(d + e*x)^(q - 1)*(a + b*Log[c 
*x^n])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, n}, x] && LtQ[p, -1] && Gt 
Q[q, 0]
 

rule 2767

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^( 
q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x 
^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] 
&& IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))
 

rule 2952

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( 
m + 1)*(g/d)^m   Subst[Int[(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] && EqQ[ 
n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f 
 - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

Maple [F]

\[\int \frac {1}{\left (b g x +a g \right )^{3} {\left (A +B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )\right )}^{2}}d x\]

Input:

int(1/(b*g*x+a*g)^3/(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2,x)
 

Output:

int(1/(b*g*x+a*g)^3/(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2,x)
 

Fricas [F]

\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}^{2}} \,d x } \] Input:

integrate(1/(b*g*x+a*g)^3/(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm= 
"fricas")
 

Output:

integral(1/(A^2*b^3*g^3*x^3 + 3*A^2*a*b^2*g^3*x^2 + 3*A^2*a^2*b*g^3*x + A^ 
2*a^3*g^3 + (B^2*b^3*g^3*x^3 + 3*B^2*a*b^2*g^3*x^2 + 3*B^2*a^2*b*g^3*x + B 
^2*a^3*g^3)*log((d^2*e*x^2 + 2*c*d*e*x + c^2*e)/(b^2*x^2 + 2*a*b*x + a^2)) 
^2 + 2*(A*B*b^3*g^3*x^3 + 3*A*B*a*b^2*g^3*x^2 + 3*A*B*a^2*b*g^3*x + A*B*a^ 
3*g^3)*log((d^2*e*x^2 + 2*c*d*e*x + c^2*e)/(b^2*x^2 + 2*a*b*x + a^2))), x)
 

Sympy [F]

\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(1/(b*g*x+a*g)**3/(A+B*ln(e*(d*x+c)**2/(b*x+a)**2))**2,x)
 

Output:

(-c - d*x)/(2*A*B*a**3*d*g**3 - 2*A*B*a**2*b*c*g**3 + 4*A*B*a**2*b*d*g**3* 
x - 4*A*B*a*b**2*c*g**3*x + 2*A*B*a*b**2*d*g**3*x**2 - 2*A*B*b**3*c*g**3*x 
**2 + (2*B**2*a**3*d*g**3 - 2*B**2*a**2*b*c*g**3 + 4*B**2*a**2*b*d*g**3*x 
- 4*B**2*a*b**2*c*g**3*x + 2*B**2*a*b**2*d*g**3*x**2 - 2*B**2*b**3*c*g**3* 
x**2)*log(e*(c + d*x)**2/(a + b*x)**2)) - (Integral(-a*d/(A*a**3 + 3*A*a** 
2*b*x + 3*A*a*b**2*x**2 + A*b**3*x**3 + B*a**3*log(c**2*e/(a**2 + 2*a*b*x 
+ b**2*x**2) + 2*c*d*e*x/(a**2 + 2*a*b*x + b**2*x**2) + d**2*e*x**2/(a**2 
+ 2*a*b*x + b**2*x**2)) + 3*B*a**2*b*x*log(c**2*e/(a**2 + 2*a*b*x + b**2*x 
**2) + 2*c*d*e*x/(a**2 + 2*a*b*x + b**2*x**2) + d**2*e*x**2/(a**2 + 2*a*b* 
x + b**2*x**2)) + 3*B*a*b**2*x**2*log(c**2*e/(a**2 + 2*a*b*x + b**2*x**2) 
+ 2*c*d*e*x/(a**2 + 2*a*b*x + b**2*x**2) + d**2*e*x**2/(a**2 + 2*a*b*x + b 
**2*x**2)) + B*b**3*x**3*log(c**2*e/(a**2 + 2*a*b*x + b**2*x**2) + 2*c*d*e 
*x/(a**2 + 2*a*b*x + b**2*x**2) + d**2*e*x**2/(a**2 + 2*a*b*x + b**2*x**2) 
)), x) + Integral(2*b*c/(A*a**3 + 3*A*a**2*b*x + 3*A*a*b**2*x**2 + A*b**3* 
x**3 + B*a**3*log(c**2*e/(a**2 + 2*a*b*x + b**2*x**2) + 2*c*d*e*x/(a**2 + 
2*a*b*x + b**2*x**2) + d**2*e*x**2/(a**2 + 2*a*b*x + b**2*x**2)) + 3*B*a** 
2*b*x*log(c**2*e/(a**2 + 2*a*b*x + b**2*x**2) + 2*c*d*e*x/(a**2 + 2*a*b*x 
+ b**2*x**2) + d**2*e*x**2/(a**2 + 2*a*b*x + b**2*x**2)) + 3*B*a*b**2*x**2 
*log(c**2*e/(a**2 + 2*a*b*x + b**2*x**2) + 2*c*d*e*x/(a**2 + 2*a*b*x + b** 
2*x**2) + d**2*e*x**2/(a**2 + 2*a*b*x + b**2*x**2)) + B*b**3*x**3*log(c...
 

Maxima [F]

\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}^{2}} \,d x } \] Input:

integrate(1/(b*g*x+a*g)^3/(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm= 
"maxima")
 

Output:

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

Giac [F]

\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\int { \frac {1}{{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}^{2}} \,d x } \] Input:

integrate(1/(b*g*x+a*g)^3/(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm= 
"giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\int \frac {1}{{\left (a\,g+b\,g\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\right )}^2} \,d x \] Input:

int(1/((a*g + b*g*x)^3*(A + B*log((e*(c + d*x)^2)/(a + b*x)^2))^2),x)
 

Output:

int(1/((a*g + b*g*x)^3*(A + B*log((e*(c + d*x)^2)/(a + b*x)^2))^2), x)
 

Reduce [F]

\[ \int \frac {1}{(a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\frac {\int \frac {1}{\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right )^{2} a^{3} b^{2}+3 \mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right )^{2} a^{2} b^{3} x +3 \mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right )^{2} a \,b^{4} x^{2}+\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right )^{2} b^{5} x^{3}+2 \,\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right ) a^{4} b +6 \,\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right ) a^{3} b^{2} x +6 \,\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right ) a^{2} b^{3} x^{2}+2 \,\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right ) a \,b^{4} x^{3}+a^{5}+3 a^{4} b x +3 a^{3} b^{2} x^{2}+a^{2} b^{3} x^{3}}d x}{g^{3}} \] Input:

int(1/(b*g*x+a*g)^3/(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x)
                                                                                    
                                                                                    
 

Output:

int(1/(log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2) 
)**2*a**3*b**2 + 3*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x 
+ b**2*x**2))**2*a**2*b**3*x + 3*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a 
**2 + 2*a*b*x + b**2*x**2))**2*a*b**4*x**2 + log((c**2*e + 2*c*d*e*x + d** 
2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2*b**5*x**3 + 2*log((c**2*e + 2*c 
*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a**4*b + 6*log((c**2*e 
 + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a**3*b**2*x + 6* 
log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a**2* 
b**3*x**2 + 2*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b** 
2*x**2))*a*b**4*x**3 + a**5 + 3*a**4*b*x + 3*a**3*b**2*x**2 + a**2*b**3*x* 
*3),x)/g**3