\(\int \frac {1}{(a g+b g x)^2 (A+B \log (\frac {e (c+d x)}{a+b x}))^2} \, dx\) [199]

Optimal result

Integrand size = 32, antiderivative size = 104 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=-\frac {e^{-\frac {A}{B}} \operatorname {ExpIntegralEi}\left (\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{B}\right )}{B^2 (b c-a d) e g^2}+\frac {c+d x}{B (b c-a d) g^2 (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\frac {\frac {e^{-\frac {A}{B}} \operatorname {ExpIntegralEi}\left (\frac {A}{B}+\log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{e}-\frac {B (c+d x)}{(a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}}{B^2 (-b c+a d) g^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2952, 2734, 2736, 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.

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

\(\Big \downarrow \) 2952

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

\(\Big \downarrow \) 2734

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

\(\Big \downarrow \) 2736

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

\(\Big \downarrow \) 2609

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

Input:

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

Output:

-((ExpIntegralEi[(A + B*Log[(e*(c + d*x))/(a + b*x)])/B]/(B^2*e*E^(A/B)) - 
 (c + d*x)/(B*(a + b*x)*(A + B*Log[(e*(c + d*x))/(a + b*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 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 2736

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

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 [A] (verified)

Time = 4.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.03

method	result	size
risch	\(-\frac {d x +c}{\left (d a -b c \right ) B \left (b x +a \right ) g^{2} \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right )}-\frac {{\mathrm e}^{-\frac {A}{B}} \operatorname {expIntegral}_{1}\left (-\ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )-\frac {A}{B}\right )}{g^{2} B^{2} e \left (d a -b c \right )}\)	\(107\)
derivativedivides	\(\frac {-\frac {\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}}{\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-{\mathrm e}^{-\frac {A}{B}} \operatorname {expIntegral}_{1}\left (-\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )}{e \left (d a -b c \right ) g^{2} B^{2}}\)	\(138\)
default	\(\frac {-\frac {\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}}{\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )+\frac {A}{B}}-{\mathrm e}^{-\frac {A}{B}} \operatorname {expIntegral}_{1}\left (-\ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )-\frac {A}{B}\right )}{e \left (d a -b c \right ) g^{2} B^{2}}\)	\(138\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (103) = 206\).

Time = 0.09 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.00 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\frac {{\left (B d e x + B c e\right )} e^{\frac {A}{B}} - {\left (A b x + A a + {\left (B b x + B a\right )} \log \left (\frac {d e x + c e}{b x + a}\right )\right )} \operatorname {log\_integral}\left (\frac {{\left (d e x + c e\right )} e^{\frac {A}{B}}}{b x + a}\right )}{{\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} e g^{2} x + {\left (B^{3} a b c - B^{3} a^{2} d\right )} e g^{2}\right )} e^{\frac {A}{B}} \log \left (\frac {d e x + c e}{b x + a}\right ) + {\left ({\left (A B^{2} b^{2} c - A B^{2} a b d\right )} e g^{2} x + {\left (A B^{2} a b c - A B^{2} a^{2} d\right )} e g^{2}\right )} e^{\frac {A}{B}}} \] Input:

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

Output:

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

Sympy [F]

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

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

Maxima [F]

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

integrate(1/(b*g*x+a*g)^2/(A+B*log(e*(d*x+c)/(b*x+a)))^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*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*lo 
g(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 [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx={\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} {\left (\frac {d e x + c e}{{\left (B^{2} g^{2} \log \left (\frac {d e x + c e}{b x + a}\right ) + A B g^{2}\right )} {\left (b x + a\right )}} - \frac {{\rm Ei}\left (\frac {A}{B} + \log \left (\frac {d e x + c e}{b x + a}\right )\right ) e^{\left (-\frac {A}{B}\right )}}{B^{2} g^{2}}\right )} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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