\(\int \frac {1}{(a g+b g x)^2 (A+B \log (\frac {e (a+b x)^2}{(c+d x)^2}))} \, dx\) [140]

Optimal result

Integrand size = 34, antiderivative size = 94 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx=\frac {e^{\frac {A}{2 B}} \sqrt {\frac {e (a+b x)^2}{(c+d x)^2}} (c+d x) \operatorname {ExpIntegralEi}\left (\frac {-A-B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 B}\right )}{2 B (b c-a d) g^2 (a+b x)} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2950, 2747, 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.

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

\(\Big \downarrow \) 2950

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

\(\Big \downarrow \) 2747

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

\(\Big \downarrow \) 2609

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

Input:

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

Output:

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

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 2747

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

rule 2950

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/b)^m   Subst[Int[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] & 
& EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && E 
qQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*int(1/(log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2 
*x**2))*a**2*b*c + log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x 
+ d**2*x**2))*a**2*b*d*x + 2*log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 
+ 2*c*d*x + d**2*x**2))*a*b**2*c*x + 2*log((a**2*e + 2*a*b*e*x + b**2*e*x* 
*2)/(c**2 + 2*c*d*x + d**2*x**2))*a*b**2*d*x**2 + log((a**2*e + 2*a*b*e*x 
+ b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))*b**3*c*x**2 + log((a**2*e + 2 
*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))*b**3*d*x**3 + a**3*c 
 + a**3*d*x + 2*a**2*b*c*x + 2*a**2*b*d*x**2 + a*b**2*c*x**2 + a*b**2*d*x* 
*3),x)*a**2*b*d**2 + 4*int(1/(log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 
 + 2*c*d*x + d**2*x**2))*a**2*b*c + log((a**2*e + 2*a*b*e*x + b**2*e*x**2) 
/(c**2 + 2*c*d*x + d**2*x**2))*a**2*b*d*x + 2*log((a**2*e + 2*a*b*e*x + b* 
*2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))*a*b**2*c*x + 2*log((a**2*e + 2*a* 
b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))*a*b**2*d*x**2 + log((a* 
*2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))*b**3*c*x**2 
+ log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))*b** 
3*d*x**3 + a**3*c + a**3*d*x + 2*a**2*b*c*x + 2*a**2*b*d*x**2 + a*b**2*c*x 
**2 + a*b**2*d*x**3),x)*a*b**2*c*d - 2*int(1/(log((a**2*e + 2*a*b*e*x + b* 
*2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))*a**2*b*c + log((a**2*e + 2*a*b*e* 
x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))*a**2*b*d*x + 2*log((a**2*e 
+ 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))*a*b**2*c*x + 2...