\(\int \frac {1}{(a g+b g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))} \, dx\) [112]

Optimal result

Integrand size = 32, antiderivative size = 50 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )} \, dx=\frac {e e^{A/B} \operatorname {ExpIntegralEi}\left (-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{B}\right )}{B (b c-a d) g^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )} \, dx=\frac {e e^{A/B} \operatorname {ExpIntegralEi}\left (-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{B}\right )}{b B c g^2-a B d g^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2950, 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.

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

\(\Big \downarrow \) 2950

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

\(\Big \downarrow \) 2746

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

\(\Big \downarrow \) 2609

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

Input:

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

Output:

(e*E^(A/B)*ExpIntegralEi[-((A + B*Log[(e*(a + b*x))/(c + d*x)])/B)])/(B*(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 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 = 2.43 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.22

method	result	size
derivativedivides	\(\frac {e \,{\mathrm e}^{\frac {A}{B}} \operatorname {expIntegral}_{1}\left (\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{\left (d a -b c \right ) g^{2} B}\)	\(61\)
default	\(\frac {e \,{\mathrm e}^{\frac {A}{B}} \operatorname {expIntegral}_{1}\left (\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{\left (d a -b c \right ) g^{2} B}\)	\(61\)
risch	\(\frac {e \,{\mathrm e}^{\frac {A}{B}} \operatorname {expIntegral}_{1}\left (\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )+\frac {A}{B}\right )}{\left (d a -b c \right ) g^{2} B}\)	\(61\)

Input:

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

Output:

e/(a*d-b*c)/g^2/B*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 = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )} \, dx=\frac {e e^{\frac {A}{B}} \operatorname {log\_integral}\left (\frac {{\left (d x + c\right )} e^{\left (-\frac {A}{B}\right )}}{b e x + a e}\right )}{{\left (B b c - B a d\right )} g^{2}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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)/g**2
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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