\(\int \frac {1}{(c g+d g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2} \, dx\) [56]

Optimal result

Integrand size = 35, antiderivative size = 256 \[ \int \frac {1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\frac {b e^{-\frac {A}{B n}} (a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-1/n} \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 (b c-a d)^2 g^3 n^2 (c+d x)}-\frac {2 d e^{-\frac {2 A}{B n}} (a+b x)^2 \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-2/n} \operatorname {ExpIntegralEi}\left (\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2}-\frac {a+b x}{B (b c-a d) g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.12 \[ \int \frac {1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx=\frac {e^{-\frac {2 A}{B n}} (a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-2/n} \left (-B (b c-a d) e^{\frac {2 A}{B n}} n \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{2/n}+b e^{\frac {A}{B n}} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {1}{n}} (c+d x) \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{B n}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 d (a+b x) \operatorname {ExpIntegralEi}\left (\frac {2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.34, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2951, 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.

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

\(\Big \downarrow \) 2951

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

\(\Big \downarrow \) 2757

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

\(\Big \downarrow \) 2737

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

\(\Big \downarrow \) 2609

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

\(\Big \downarrow \) 2767

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(-((b*(a + b*x)*ExpIntegralEi[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(B*n) 
])/(B^2*E^(A/(B*n))*n^2*(e*((a + b*x)/(c + d*x))^n)^n^(-1)*(c + d*x))) + ( 
2*((b*(a + b*x)*ExpIntegralEi[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(B*n) 
])/(B*E^(A/(B*n))*n*(e*((a + b*x)/(c + d*x))^n)^n^(-1)*(c + d*x)) - (d*(a 
+ b*x)^2*ExpIntegralEi[(2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)])/ 
(B*E^((2*A)/(B*n))*n*(e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)^2)))/(B*n 
) - ((a + b*x)*(b - (d*(a + b*x))/(c + d*x)))/(B*n*(c + d*x)*(A + B*Log[e* 
((a + b*x)/(c + d*x))^n])))/((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 2951

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
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] && NeQ[b*c 
- a*d, 0] && IntegersQ[m, p] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, - 
1])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (256) = 512\).

Time = 0.08 (sec) , antiderivative size = 770, normalized size of antiderivative = 3.01 \[ \int \frac {1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx={\left (\frac {b {\rm Ei}\left (\frac {\log \left (e\right )}{n} + \frac {A}{B n} + \log \left (\frac {b x + a}{d x + c}\right )\right ) e^{\left (-\frac {A}{B n}\right )}}{{\left (B^{2} b c g^{3} n^{2} - B^{2} a d g^{3} n^{2}\right )} e^{\left (\frac {1}{n}\right )}} - \frac {2 \, d {\rm Ei}\left (\frac {2 \, \log \left (e\right )}{n} + \frac {2 \, A}{B n} + 2 \, \log \left (\frac {b x + a}{d x + c}\right )\right ) e^{\left (-\frac {2 \, A}{B n}\right )}}{{\left (B^{2} b c g^{3} n^{2} - B^{2} a d g^{3} n^{2}\right )} e^{\frac {2}{n}}} - \frac {\frac {{\left (b x + a\right )} b}{d x + c} - \frac {{\left (b x + a\right )}^{2} d}{{\left (d x + c\right )}^{2}}}{B^{2} b c g^{3} n^{2} \log \left (\frac {b x + a}{d x + c}\right ) - B^{2} a d g^{3} n^{2} \log \left (\frac {b x + a}{d x + c}\right ) + B^{2} b c g^{3} n \log \left (e\right ) - B^{2} a d g^{3} n \log \left (e\right ) + A B b c g^{3} n - A B a d g^{3} n}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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