\(\int \frac {A+B \log (\frac {e (c+d x)}{a+b x})}{(a g+b g x)^3} \, dx\) [179]

Optimal result

Integrand size = 30, antiderivative size = 144 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^3} \, dx=\frac {B}{4 b g^3 (a+b x)^2}-\frac {B d}{2 b (b c-a d) g^3 (a+b x)}-\frac {B d^2 \log (a+b x)}{2 b (b c-a d)^2 g^3}+\frac {B d^2 \log (c+d x)}{2 b (b c-a d)^2 g^3}-\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{2 b g^3 (a+b x)^2} \] Output:

1/4*B/b/g^3/(b*x+a)^2-1/2*B*d/b/(-a*d+b*c)/g^3/(b*x+a)-1/2*B*d^2*ln(b*x+a) 
/b/(-a*d+b*c)^2/g^3+1/2*B*d^2*ln(d*x+c)/b/(-a*d+b*c)^2/g^3-1/2*(A+B*ln(e*( 
d*x+c)/(b*x+a)))/b/g^3/(b*x+a)^2
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.89 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^3} \, dx=-\frac {2 B d^2 (a+b x)^2 \log (a+b x)-2 B d^2 (a+b x)^2 \log (c+d x)+(b c-a d) \left (2 A b c-b B c-2 a A d+3 a B d+2 b B d x+2 B (b c-a d) \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{4 b (b c-a d)^2 g^3 (a+b x)^2} \] Input:

Integrate[(A + B*Log[(e*(c + d*x))/(a + b*x)])/(a*g + b*g*x)^3,x]
 

Output:

-1/4*(2*B*d^2*(a + b*x)^2*Log[a + b*x] - 2*B*d^2*(a + b*x)^2*Log[c + d*x] 
+ (b*c - a*d)*(2*A*b*c - b*B*c - 2*a*A*d + 3*a*B*d + 2*b*B*d*x + 2*B*(b*c 
- a*d)*Log[(e*(c + d*x))/(a + b*x)]))/(b*(b*c - a*d)^2*g^3*(a + b*x)^2)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2948, 27, 54, 2009}

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.

\(\displaystyle \int \frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{(a g+b g x)^3} \, dx\)

\(\Big \downarrow \) 2948

\(\displaystyle -\frac {B (b c-a d) \int \frac {1}{g^2 (a+b x)^3 (c+d x)}dx}{2 b g}-\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{2 b g^3 (a+b x)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {B (b c-a d) \int \frac {1}{(a+b x)^3 (c+d x)}dx}{2 b g^3}-\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{2 b g^3 (a+b x)^2}\)

\(\Big \downarrow \) 54

\(\displaystyle -\frac {B (b c-a d) \int \left (-\frac {d^3}{(b c-a d)^3 (c+d x)}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b}{(b c-a d) (a+b x)^3}\right )dx}{2 b g^3}-\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{2 b g^3 (a+b x)^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{2 b g^3 (a+b x)^2}-\frac {B (b c-a d) \left (\frac {d^2 \log (a+b x)}{(b c-a d)^3}-\frac {d^2 \log (c+d x)}{(b c-a d)^3}+\frac {d}{(a+b x) (b c-a d)^2}-\frac {1}{2 (a+b x)^2 (b c-a d)}\right )}{2 b g^3}\)

Input:

Int[(A + B*Log[(e*(c + d*x))/(a + b*x)])/(a*g + b*g*x)^3,x]
 

Output:

-1/2*(B*(b*c - a*d)*(-1/2*1/((b*c - a*d)*(a + b*x)^2) + d/((b*c - a*d)^2*( 
a + b*x)) + (d^2*Log[a + b*x])/(b*c - a*d)^3 - (d^2*Log[c + d*x])/(b*c - a 
*d)^3))/(b*g^3) - (A + B*Log[(e*(c + d*x))/(a + b*x)])/(2*b*g^3*(a + b*x)^ 
2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2948

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*( 
(A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Simp[B*n*((b*c 
- a*d)/(g*(m + 1)))   Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] / 
; FreeQ[{a, b, c, d, e, f, g, A, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c 
- a*d, 0] && NeQ[m, -1] &&  !(EqQ[m, -2] && IntegerQ[n])
 

Maple [A] (verified)

Time = 1.20 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.56

method	result	size
parts	\(-\frac {A}{2 g^{3} \left (b x +a \right )^{2} b}-\frac {B b \left (\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2} \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )}{2}-\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}}{4}-\frac {d e \left (\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )+\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}-\frac {d e}{b}\right )}{b}\right )}{g^{3} e^{2} \left (d a -b c \right )^{2}}\)	\(225\)
norman	\(\frac {\frac {B a \,d^{2} x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{\left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}-\frac {2 A a b d -2 A \,b^{2} c -3 B a b d +B \,b^{2} c}{4 g \,b^{2} \left (d a -b c \right )}+\frac {B d x}{2 g \left (d a -b c \right )}+\frac {B c \left (2 d a -b c \right ) \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{2 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {d^{2} B b \,x^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{2 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}}{g^{2} \left (b x +a \right )^{2}}\)	\(228\)
parallelrisch	\(-\frac {2 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{5} c^{2} d -4 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a \,b^{4} c \,d^{2}+2 A \,a^{2} b^{3} d^{3}+2 A \,b^{5} c^{2} d -3 B \,a^{2} b^{3} d^{3}-B \,b^{5} c^{2} d -2 B \,x^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) b^{5} d^{3}-4 A a \,b^{4} c \,d^{2}+4 B a \,b^{4} c \,d^{2}-2 B x a \,b^{4} d^{3}+2 B x \,b^{5} c \,d^{2}-4 B x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a \,b^{4} d^{3}}{4 g^{3} \left (b x +a \right )^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b^{4} d}\)	\(235\)
orering	\(\frac {\left (b x +a \right ) \left (8 b \,d^{2} x^{2}+13 a \,d^{2} x +3 b c d x +13 a c d -5 b \,c^{2}\right ) \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right )}{4 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (b g x +a g \right )^{3}}+\frac {\left (2 b d x +3 d a -b c \right ) \left (b x +a \right )^{2} \left (d x +c \right ) \left (\frac {B \left (\frac {e d}{b x +a}-\frac {e \left (d x +c \right ) b}{\left (b x +a \right )^{2}}\right ) \left (b x +a \right )}{e \left (d x +c \right ) \left (b g x +a g \right )^{3}}-\frac {3 \left (A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )\right ) b g}{\left (b g x +a g \right )^{4}}\right )}{4 b \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}\)	\(237\)
risch	\(-\frac {B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{2 b \,g^{3} \left (b x +a \right )^{2}}-\frac {2 B \ln \left (b x +a \right ) b^{2} d^{2} x^{2}-2 B \ln \left (-d x -c \right ) b^{2} d^{2} x^{2}+4 B \ln \left (b x +a \right ) a b \,d^{2} x -4 B \ln \left (-d x -c \right ) a b \,d^{2} x +2 B \,a^{2} \ln \left (b x +a \right ) d^{2}-2 B \ln \left (-d x -c \right ) a^{2} d^{2}-2 B a b \,d^{2} x +2 B \,b^{2} c d x +2 A \,a^{2} d^{2}-4 A a b c d +2 A \,b^{2} c^{2}-3 B \,a^{2} d^{2}+4 B a b c d -B \,b^{2} c^{2}}{4 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g^{3} \left (b x +a \right )^{2} b}\)	\(245\)
derivativedivides	\(\frac {e \left (d a -b c \right ) \left (-\frac {b^{3} A \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}}{2 \left (d a -b c \right )^{3} e^{3} g^{3}}+\frac {b^{2} A d \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )}{\left (d a -b c \right )^{3} e^{2} g^{3}}-\frac {b^{3} B \left (\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2} \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )}{2}-\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}}{4}\right )}{\left (d a -b c \right )^{3} e^{3} g^{3}}+\frac {b^{2} B d \left (\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )+\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}-\frac {d e}{b}\right )}{\left (d a -b c \right )^{3} e^{2} g^{3}}\right )}{b^{2}}\)	\(341\)
default	\(\frac {e \left (d a -b c \right ) \left (-\frac {b^{3} A \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}}{2 \left (d a -b c \right )^{3} e^{3} g^{3}}+\frac {b^{2} A d \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )}{\left (d a -b c \right )^{3} e^{2} g^{3}}-\frac {b^{3} B \left (\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2} \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )}{2}-\frac {\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )^{2}}{4}\right )}{\left (d a -b c \right )^{3} e^{3} g^{3}}+\frac {b^{2} B d \left (\left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}\right )+\frac {e \left (d a -b c \right )}{b \left (b x +a \right )}-\frac {d e}{b}\right )}{\left (d a -b c \right )^{3} e^{2} g^{3}}\right )}{b^{2}}\)	\(341\)

Input:

int((A+B*ln(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/2*A/g^3/(b*x+a)^2/b-B/g^3*b/e^2/(a*d-b*c)^2*(1/2*(d*e/b-e*(a*d-b*c)/b/( 
b*x+a))^2*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))-1/4*(d*e/b-e*(a*d-b*c)/b/(b*x+a) 
)^2-d*e/b*((d*e/b-e*(a*d-b*c)/b/(b*x+a))*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))+e 
*(a*d-b*c)/b/(b*x+a)-d*e/b))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.53 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^3} \, dx=-\frac {{\left (2 \, A - B\right )} b^{2} c^{2} - 4 \, {\left (A - B\right )} a b c d + {\left (2 \, A - 3 \, B\right )} a^{2} d^{2} + 2 \, {\left (B b^{2} c d - B a b d^{2}\right )} x - 2 \, {\left (B b^{2} d^{2} x^{2} + 2 \, B a b d^{2} x - B b^{2} c^{2} + 2 \, B a b c d\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{4 \, {\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} g^{3} x + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} g^{3}\right )}} \] Input:

integrate((A+B*log(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^3,x, algorithm="fricas" 
)
 

Output:

-1/4*((2*A - B)*b^2*c^2 - 4*(A - B)*a*b*c*d + (2*A - 3*B)*a^2*d^2 + 2*(B*b 
^2*c*d - B*a*b*d^2)*x - 2*(B*b^2*d^2*x^2 + 2*B*a*b*d^2*x - B*b^2*c^2 + 2*B 
*a*b*c*d)*log((d*e*x + c*e)/(b*x + a)))/((b^5*c^2 - 2*a*b^4*c*d + a^2*b^3* 
d^2)*g^3*x^2 + 2*(a*b^4*c^2 - 2*a^2*b^3*c*d + a^3*b^2*d^2)*g^3*x + (a^2*b^ 
3*c^2 - 2*a^3*b^2*c*d + a^4*b*d^2)*g^3)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (122) = 244\).

Time = 1.17 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.93 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^3} \, dx=- \frac {B \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}}{2 a^{2} b g^{3} + 4 a b^{2} g^{3} x + 2 b^{3} g^{3} x^{2}} + \frac {B d^{2} \log {\left (x + \frac {- \frac {B a^{3} d^{5}}{\left (a d - b c\right )^{2}} + \frac {3 B a^{2} b c d^{4}}{\left (a d - b c\right )^{2}} - \frac {3 B a b^{2} c^{2} d^{3}}{\left (a d - b c\right )^{2}} + B a d^{3} + \frac {B b^{3} c^{3} d^{2}}{\left (a d - b c\right )^{2}} + B b c d^{2}}{2 B b d^{3}} \right )}}{2 b g^{3} \left (a d - b c\right )^{2}} - \frac {B d^{2} \log {\left (x + \frac {\frac {B a^{3} d^{5}}{\left (a d - b c\right )^{2}} - \frac {3 B a^{2} b c d^{4}}{\left (a d - b c\right )^{2}} + \frac {3 B a b^{2} c^{2} d^{3}}{\left (a d - b c\right )^{2}} + B a d^{3} - \frac {B b^{3} c^{3} d^{2}}{\left (a d - b c\right )^{2}} + B b c d^{2}}{2 B b d^{3}} \right )}}{2 b g^{3} \left (a d - b c\right )^{2}} + \frac {- 2 A a d + 2 A b c + 3 B a d - B b c + 2 B b d x}{4 a^{3} b d g^{3} - 4 a^{2} b^{2} c g^{3} + x^{2} \cdot \left (4 a b^{3} d g^{3} - 4 b^{4} c g^{3}\right ) + x \left (8 a^{2} b^{2} d g^{3} - 8 a b^{3} c g^{3}\right )} \] Input:

integrate((A+B*ln(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)**3,x)
 

Output:

-B*log(e*(c + d*x)/(a + b*x))/(2*a**2*b*g**3 + 4*a*b**2*g**3*x + 2*b**3*g* 
*3*x**2) + B*d**2*log(x + (-B*a**3*d**5/(a*d - b*c)**2 + 3*B*a**2*b*c*d**4 
/(a*d - b*c)**2 - 3*B*a*b**2*c**2*d**3/(a*d - b*c)**2 + B*a*d**3 + B*b**3* 
c**3*d**2/(a*d - b*c)**2 + B*b*c*d**2)/(2*B*b*d**3))/(2*b*g**3*(a*d - b*c) 
**2) - B*d**2*log(x + (B*a**3*d**5/(a*d - b*c)**2 - 3*B*a**2*b*c*d**4/(a*d 
 - b*c)**2 + 3*B*a*b**2*c**2*d**3/(a*d - b*c)**2 + B*a*d**3 - B*b**3*c**3* 
d**2/(a*d - b*c)**2 + B*b*c*d**2)/(2*B*b*d**3))/(2*b*g**3*(a*d - b*c)**2) 
+ (-2*A*a*d + 2*A*b*c + 3*B*a*d - B*b*c + 2*B*b*d*x)/(4*a**3*b*d*g**3 - 4* 
a**2*b**2*c*g**3 + x**2*(4*a*b**3*d*g**3 - 4*b**4*c*g**3) + x*(8*a**2*b**2 
*d*g**3 - 8*a*b**3*c*g**3))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.77 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^3} \, dx=-\frac {1}{4} \, B {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} + \frac {2 \, \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {A}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \] Input:

integrate((A+B*log(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^3,x, algorithm="maxima" 
)
 

Output:

-1/4*B*((2*b*d*x - b*c + 3*a*d)/((b^4*c - a*b^3*d)*g^3*x^2 + 2*(a*b^3*c - 
a^2*b^2*d)*g^3*x + (a^2*b^2*c - a^3*b*d)*g^3) + 2*log(d*e*x/(b*x + a) + c* 
e/(b*x + a))/(b^3*g^3*x^2 + 2*a*b^2*g^3*x + a^2*b*g^3) + 2*d^2*log(b*x + a 
)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3) - 2*d^2*log(d*x + c)/((b^3*c^2 
 - 2*a*b^2*c*d + a^2*b*d^2)*g^3)) - 1/2*A/(b^3*g^3*x^2 + 2*a*b^2*g^3*x + a 
^2*b*g^3)
 

Giac [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.62 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^3} \, dx=-\frac {1}{4} \, {\left (2 \, {\left (\frac {{\left (d e x + c e\right )}^{2} B b}{{\left (b c e g^{3} - a d e g^{3}\right )} {\left (b x + a\right )}^{2}} - \frac {2 \, {\left (d e x + c e\right )} B d}{{\left (b c g^{3} - a d g^{3}\right )} {\left (b x + a\right )}}\right )} \log \left (\frac {d e x + c e}{b x + a}\right ) + \frac {{\left (d e x + c e\right )}^{2} {\left (2 \, A b - B b\right )}}{{\left (b c e g^{3} - a d e g^{3}\right )} {\left (b x + a\right )}^{2}} - \frac {4 \, {\left (d e x + c e\right )} {\left (A d - B d\right )}}{{\left (b c g^{3} - a d g^{3}\right )} {\left (b x + a\right )}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:

integrate((A+B*log(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^3,x, algorithm="giac")
 

Output:

-1/4*(2*((d*e*x + c*e)^2*B*b/((b*c*e*g^3 - a*d*e*g^3)*(b*x + a)^2) - 2*(d* 
e*x + c*e)*B*d/((b*c*g^3 - a*d*g^3)*(b*x + a)))*log((d*e*x + c*e)/(b*x + a 
)) + (d*e*x + c*e)^2*(2*A*b - B*b)/((b*c*e*g^3 - a*d*e*g^3)*(b*x + a)^2) - 
 4*(d*e*x + c*e)*(A*d - B*d)/((b*c*g^3 - a*d*g^3)*(b*x + a)))*(b*c/((b*c*e 
 - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))
 

Mupad [B] (verification not implemented)

Time = 26.39 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.44 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^3} \, dx=\frac {B\,d^2\,\mathrm {atanh}\left (\frac {2\,b^3\,c^2\,g^3-2\,a^2\,b\,d^2\,g^3}{2\,b\,g^3\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{b\,g^3\,{\left (a\,d-b\,c\right )}^2}-\frac {B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )}{2\,b^2\,g^3\,\left (2\,a\,x+b\,x^2+\frac {a^2}{b}\right )}-\frac {\frac {2\,A\,a\,d-2\,A\,b\,c-3\,B\,a\,d+B\,b\,c}{2\,\left (a\,d-b\,c\right )}-\frac {B\,b\,d\,x}{a\,d-b\,c}}{2\,a^2\,b\,g^3+4\,a\,b^2\,g^3\,x+2\,b^3\,g^3\,x^2} \] Input:

int((A + B*log((e*(c + d*x))/(a + b*x)))/(a*g + b*g*x)^3,x)
 

Output:

(B*d^2*atanh((2*b^3*c^2*g^3 - 2*a^2*b*d^2*g^3)/(2*b*g^3*(a*d - b*c)^2) - ( 
2*b*d*x)/(a*d - b*c)))/(b*g^3*(a*d - b*c)^2) - (B*log((e*(c + d*x))/(a + b 
*x)))/(2*b^2*g^3*(2*a*x + b*x^2 + a^2/b)) - ((2*A*a*d - 2*A*b*c - 3*B*a*d 
+ B*b*c)/(2*(a*d - b*c)) - (B*b*d*x)/(a*d - b*c))/(2*a^2*b*g^3 + 2*b^3*g^3 
*x^2 + 4*a*b^2*g^3*x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 556, normalized size of antiderivative = 3.86 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^3} \, dx=\frac {-2 a^{3} b^{2} c^{2}+2 a^{4} b \,d^{2}+4 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a^{3} b^{2} d^{2} x +2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a^{2} b^{3} d^{2} x^{2}+4 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a \,b^{4} c^{2} x -a^{2} b^{3} d^{2} x^{2}+2 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} c^{2}+2 \,\mathrm {log}\left (b x +a \right ) b^{5} c^{2} x^{2}-2 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} c^{2}-2 \,\mathrm {log}\left (d x +c \right ) b^{5} c^{2} x^{2}-3 a^{3} b^{2} c d +a^{2} b^{3} c^{2}+2 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) b^{5} c^{2} x^{2}+4 a^{4} b c d -8 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a^{2} b^{3} c d x -4 \,\mathrm {log}\left (\frac {d e x +c e}{b x +a}\right ) a \,b^{4} c d \,x^{2}-2 a^{5} d^{2}+a \,b^{4} c d \,x^{2}-8 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} c d x -4 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} c d \,x^{2}+8 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} c d x +4 \,\mathrm {log}\left (d x +c \right ) a \,b^{4} c d \,x^{2}-4 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} c d +4 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} c^{2} x +4 \,\mathrm {log}\left (d x +c \right ) a^{3} b^{2} c d -4 \,\mathrm {log}\left (d x +c \right ) a \,b^{4} c^{2} x}{4 a^{2} b \,g^{3} \left (a^{2} b^{2} d^{2} x^{2}-2 a \,b^{3} c d \,x^{2}+b^{4} c^{2} x^{2}+2 a^{3} b \,d^{2} x -4 a^{2} b^{2} c d x +2 a \,b^{3} c^{2} x +a^{4} d^{2}-2 a^{3} b c d +a^{2} b^{2} c^{2}\right )} \] Input:

int((A+B*log(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^3,x)
 

Output:

( - 4*log(a + b*x)*a**3*b**2*c*d + 2*log(a + b*x)*a**2*b**3*c**2 - 8*log(a 
 + b*x)*a**2*b**3*c*d*x + 4*log(a + b*x)*a*b**4*c**2*x - 4*log(a + b*x)*a* 
b**4*c*d*x**2 + 2*log(a + b*x)*b**5*c**2*x**2 + 4*log(c + d*x)*a**3*b**2*c 
*d - 2*log(c + d*x)*a**2*b**3*c**2 + 8*log(c + d*x)*a**2*b**3*c*d*x - 4*lo 
g(c + d*x)*a*b**4*c**2*x + 4*log(c + d*x)*a*b**4*c*d*x**2 - 2*log(c + d*x) 
*b**5*c**2*x**2 + 4*log((c*e + d*e*x)/(a + b*x))*a**3*b**2*d**2*x - 8*log( 
(c*e + d*e*x)/(a + b*x))*a**2*b**3*c*d*x + 2*log((c*e + d*e*x)/(a + b*x))* 
a**2*b**3*d**2*x**2 + 4*log((c*e + d*e*x)/(a + b*x))*a*b**4*c**2*x - 4*log 
((c*e + d*e*x)/(a + b*x))*a*b**4*c*d*x**2 + 2*log((c*e + d*e*x)/(a + b*x)) 
*b**5*c**2*x**2 - 2*a**5*d**2 + 4*a**4*b*c*d + 2*a**4*b*d**2 - 2*a**3*b**2 
*c**2 - 3*a**3*b**2*c*d + a**2*b**3*c**2 - a**2*b**3*d**2*x**2 + a*b**4*c* 
d*x**2)/(4*a**2*b*g**3*(a**4*d**2 - 2*a**3*b*c*d + 2*a**3*b*d**2*x + a**2* 
b**2*c**2 - 4*a**2*b**2*c*d*x + a**2*b**2*d**2*x**2 + 2*a*b**3*c**2*x - 2* 
a*b**3*c*d*x**2 + b**4*c**2*x**2))