\(\int \frac {1}{(a g+b g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2} \, dx\) [117]

Optimal result

Integrand size = 32, antiderivative size = 103 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=-\frac {e e^{A/B} \operatorname {ExpIntegralEi}\left (-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{B}\right )}{B^2 (b c-a d) g^2}-\frac {c+d x}{B (b c-a d) g^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\frac {e e^{A/B} \operatorname {ExpIntegralEi}\left (-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{B}\right )+\frac {B (c+d x)}{(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}}{B^2 (-b c+a d) g^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2950, 2743, 2746, 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.

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

\(\Big \downarrow \) 2950

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

\(\Big \downarrow \) 2743

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

\(\Big \downarrow \) 2746

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

\(\Big \downarrow \) 2609

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

Input:

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

Output:

(-((e*E^(A/B)*ExpIntegralEi[-((A + B*Log[(e*(a + b*x))/(c + d*x)])/B)])/B^ 
2) - (c + d*x)/(B*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])))/((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 2743

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

rule 2746

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Simp[1/c^ 
(m + 1)   Subst[Int[E^((m + 1)*x)*(a + b*x)^p, x], x, Log[c*x]], x] /; Free 
Q[{a, b, c, p}, x] && IntegerQ[m]
 

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 [A] (verified)

Time = 3.23 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.10

method	result	size
risch	\(\frac {d x +c}{\left (d a -b c \right ) B \left (b x +a \right ) g^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )}-\frac {e \,{\mathrm e}^{\frac {A}{B}} \operatorname {expIntegral}_{1}\left (\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{g^{2} B^{2} \left (d a -b c \right )}\)	\(113\)
derivativedivides	\(-\frac {e \left (-\frac {1}{\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) B \left (A +B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )\right )}+\frac {{\mathrm e}^{\frac {A}{B}} \operatorname {expIntegral}_{1}\left (\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{B^{2}}\right )}{\left (d a -b c \right ) g^{2}}\)	\(132\)
default	\(-\frac {e \left (-\frac {1}{\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) B \left (A +B \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )\right )}+\frac {{\mathrm e}^{\frac {A}{B}} \operatorname {expIntegral}_{1}\left (\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{B^{2}}\right )}{\left (d a -b c \right ) g^{2}}\)	\(132\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.93 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=-\frac {B d x + B c + {\left ({\left (B b e x + B a e\right )} e^{\frac {A}{B}} \log \left (\frac {b e x + a e}{d x + c}\right ) + {\left (A b e x + A a e\right )} e^{\frac {A}{B}}\right )} \operatorname {log\_integral}\left (\frac {{\left (d x + c\right )} e^{\left (-\frac {A}{B}\right )}}{b e x + a e}\right )}{{\left (A B^{2} b^{2} c - A B^{2} a b d\right )} g^{2} x + {\left (A B^{2} a b c - A B^{2} a^{2} d\right )} g^{2} + {\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} g^{2} x + {\left (B^{3} a b c - B^{3} a^{2} d\right )} g^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-(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*lo 
g(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*l 
og(e))*B^2)*x + ((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) - ((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/(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 + (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) - (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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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