\(\int \frac {1}{(a g+b g x)^2 (A+B \log (\frac {e (c+d x)^2}{(a+b x)^2}))^2} \, dx\) [227]

Optimal result

Integrand size = 34, antiderivative size = 147 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=-\frac {e^{-\frac {A}{2 B}} (c+d x) \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{2 B}\right )}{4 B^2 (b c-a d) g^2 (a+b x) \sqrt {\frac {e (c+d x)^2}{(a+b x)^2}}}+\frac {c+d x}{2 B (b c-a d) g^2 (a+b x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2952, 2734, 2737, 2609}

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.

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

\(\Big \downarrow \) 2952

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

\(\Big \downarrow \) 2734

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

\(\Big \downarrow \) 2737

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

\(\Big \downarrow \) 2609

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

Input:

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

Output:

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

Defintions of rubi rules used

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 2734

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b 
*Log[c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Simp[1/(b*n*(p + 1))   Int[(a + b 
*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] && Int 
egerQ[2*p]
 

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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(1/(A^2*b^2*g^2*x^2 + 2*A^2*a*b*g^2*x + A^2*a^2*g^2 + (B^2*b^2*g^2 
*x^2 + 2*B^2*a*b*g^2*x + B^2*a^2*g^2)*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^2*g^2*x^2 + 2*A*B*a*b*g^2*x + A*B* 
a^2*g^2)*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]

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

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

Output:

(-c - d*x)/(2*A*B*a**2*d*g**2 - 2*A*B*a*b*c*g**2 + 2*A*B*a*b*d*g**2*x - 2* 
A*B*b**2*c*g**2*x + (2*B**2*a**2*d*g**2 - 2*B**2*a*b*c*g**2 + 2*B**2*a*b*d 
*g**2*x - 2*B**2*b**2*c*g**2*x)*log(e*(c + d*x)**2/(a + b*x)**2)) + Integr 
al(1/(A*a**2 + 2*A*a*b*x + A*b**2*x**2 + B*a**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)) + 2*B*a*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)) + B*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))), x)/(2*B*g**2)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*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**2*b**2*c + 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**2*d*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*a*b**3*c*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*a*b**3*d*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**4*c*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**4*d*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**3*b*c + 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**3*b*d*x + 4*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**2*c*x + 4*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**2*d*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**3*c*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**3*d*x**3 + a**4*c + a**4*d*x + 2*a**3*b*c*x + 2*a**3*b*d*x**2 + a**2 
*b**2*c*x**2 + a**2*b**2*d*x**3),x)*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*d**2 + 4*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**2*b**2*c + 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**2* 
d*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...