\(\int \frac {A+B \log (\frac {e (a+b x)^2}{(c+d x)^2})}{(a g+b g x)^3} \, dx\) [125]

Optimal result

Integrand size = 32, antiderivative size = 138 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=-\frac {B}{2 b g^3 (a+b x)^2}+\frac {B d}{b (b c-a d) g^3 (a+b x)}+\frac {B d^2 \log (a+b x)}{b (b c-a d)^2 g^3}-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{2 b g^3 (a+b x)^2}-\frac {B d^2 \log (c+d x)}{b (b c-a d)^2 g^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.79 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+\frac {B \left ((b c-a d) (-3 a d+b (c-2 d x))-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )}{(b c-a d)^2}}{2 b g^3 (a+b x)^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

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

\(\Big \downarrow \) 2948

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 1.35 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.73

method	result	size
parallelrisch	\(-\frac {A \,a^{2} b^{3} d^{3}+A \,b^{5} c^{2} d +3 B \,a^{2} b^{3} d^{3}+B \,b^{5} c^{2} d -B \,x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{5} d^{3}+B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{5} c^{2} d -2 A a \,b^{4} c \,d^{2}-4 B a \,b^{4} c \,d^{2}-2 B x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{4} d^{3}-2 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{4} c \,d^{2}+2 B x a \,b^{4} d^{3}-2 B x \,b^{5} c \,d^{2}}{2 g^{3} \left (b x +a \right )^{2} \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b^{4} d}\)	\(239\)
risch	\(-\frac {B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{2 b \,g^{3} \left (b x +a \right )^{2}}-\frac {-2 B \ln \left (-b x -a \right ) b^{2} d^{2} x^{2}+2 B \ln \left (d x +c \right ) b^{2} d^{2} x^{2}-4 B \ln \left (-b x -a \right ) a b \,d^{2} x +4 B \ln \left (d x +c \right ) a b \,d^{2} x -2 B \,a^{2} \ln \left (-b x -a \right ) d^{2}+2 B \ln \left (d x +c \right ) a^{2} d^{2}+2 B a b \,d^{2} x -2 B \,b^{2} c d x +A \,a^{2} d^{2}-2 A a b c d +A \,b^{2} c^{2}+3 B \,a^{2} d^{2}-4 B a b c d +B \,b^{2} c^{2}}{2 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g^{3} \left (b x +a \right )^{2} b}\)	\(244\)
parts	\(-\frac {A}{2 g^{3} \left (b x +a \right )^{2} b}+\frac {\frac {\left (2 B a d -B b c \right ) x}{a g \left (d a -b c \right )}+\frac {B a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{\left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}+\frac {B c \left (2 d a -b c \right ) \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{2 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {\left (3 B a d -B b c \right ) b \,x^{2}}{2 g \,a^{2} \left (d a -b c \right )}+\frac {B \,d^{2} b \,x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{2 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}}{g^{2} \left (b x +a \right )^{2}}\)	\(251\)
orering	\(\frac {\left (b x +a \right ) \left (8 b \,d^{2} x^{2}+13 a \,d^{2} x +3 b c d x +13 a c d -5 b \,c^{2}\right ) \left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right )}{4 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) \left (b g x +a g \right )^{3}}+\frac {\left (2 b d x +3 d a -b c \right ) \left (b x +a \right )^{2} \left (d x +c \right ) \left (\frac {B \left (\frac {2 e \left (b x +a \right ) b}{\left (d x +c \right )^{2}}-\frac {2 e \left (b x +a \right )^{2} d}{\left (d x +c \right )^{3}}\right ) \left (d x +c \right )^{2}}{e \left (b x +a \right )^{2} \left (b g x +a g \right )^{3}}-\frac {3 \left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right ) b g}{\left (b g x +a g \right )^{4}}\right )}{4 b \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}\)	\(251\)
norman	\(\frac {\frac {\left (A d a -A b c +2 B a d -B b c \right ) x}{a g \left (d a -b c \right )}+\frac {B a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{\left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}+\frac {B c \left (2 d a -b c \right ) \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{2 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}+\frac {\left (A d a -A b c +3 B a d -B b c \right ) b \,x^{2}}{2 a^{2} g \left (d a -b c \right )}+\frac {B \,d^{2} b \,x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{2 \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) g}}{\left (b x +a \right )^{2} g^{2}}\)	\(252\)
derivativedivides	\(-\frac {\frac {d^{3} A \left (\frac {b}{2 \left (d a -b c \right )^{2} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}-\frac {1}{\left (d a -b c \right )^{2} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}\right )}{g^{3}}+\frac {\frac {B \,d^{3}}{g \left (d a -b c \right ) \left (d x +c \right )}+\frac {3 B \,d^{3}}{2 b g \left (d x +c \right )^{2}}-\frac {b B \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}-\frac {B \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (d a -b c \right ) \left (d x +c \right )}}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} g^{2}}}{d}\)	\(286\)
default	\(-\frac {\frac {d^{3} A \left (\frac {b}{2 \left (d a -b c \right )^{2} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}-\frac {1}{\left (d a -b c \right )^{2} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}\right )}{g^{3}}+\frac {\frac {B \,d^{3}}{g \left (d a -b c \right ) \left (d x +c \right )}+\frac {3 B \,d^{3}}{2 b g \left (d x +c \right )^{2}}-\frac {b B \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2 g \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}-\frac {B \,d^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (d a -b c \right ) \left (d x +c \right )}}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} g^{2}}}{d}\)	\(286\)

Input:

int((A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.72 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=-\frac {{\left (A + B\right )} b^{2} c^{2} - 2 \, {\left (A + 2 \, B\right )} a b c d + {\left (A + 3 \, B\right )} a^{2} d^{2} - 2 \, {\left (B b^{2} c d - B a b d^{2}\right )} x - {\left (B b^{2} d^{2} x^{2} + 2 \, B a b d^{2} x - B b^{2} c^{2} + 2 \, B a b c d\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, {\left ({\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d + a^{3} b^{2} d^{2}\right )} g^{3} x + {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + a^{4} b d^{2}\right )} g^{3}\right )}} \] Input:

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g)^3,x, algorithm="fri 
cas")
 

Output:

-1/2*((A + B)*b^2*c^2 - 2*(A + 2*B)*a*b*c*d + (A + 3*B)*a^2*d^2 - 2*(B*b^2 
*c*d - B*a*b*d^2)*x - (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((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2)))/((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. 418 vs. \(2 (121) = 242\).

Time = 1.10 (sec) , antiderivative size = 418, normalized size of antiderivative = 3.03 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=- \frac {B \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{2 a^{2} b g^{3} + 4 a b^{2} g^{3} x + 2 b^{3} g^{3} x^{2}} - \frac {B d^{2} \log {\left (x + \frac {- \frac {B a^{3} d^{5}}{\left (a d - b c\right )^{2}} + \frac {3 B a^{2} b c d^{4}}{\left (a d - b c\right )^{2}} - \frac {3 B a b^{2} c^{2} d^{3}}{\left (a d - b c\right )^{2}} + B a d^{3} + \frac {B b^{3} c^{3} d^{2}}{\left (a d - b c\right )^{2}} + B b c d^{2}}{2 B b d^{3}} \right )}}{b g^{3} \left (a d - b c\right )^{2}} + \frac {B d^{2} \log {\left (x + \frac {\frac {B a^{3} d^{5}}{\left (a d - b c\right )^{2}} - \frac {3 B a^{2} b c d^{4}}{\left (a d - b c\right )^{2}} + \frac {3 B a b^{2} c^{2} d^{3}}{\left (a d - b c\right )^{2}} + B a d^{3} - \frac {B b^{3} c^{3} d^{2}}{\left (a d - b c\right )^{2}} + B b c d^{2}}{2 B b d^{3}} \right )}}{b g^{3} \left (a d - b c\right )^{2}} + \frac {- A a d + A b c - 3 B a d + B b c - 2 B b d x}{2 a^{3} b d g^{3} - 2 a^{2} b^{2} c g^{3} + x^{2} \cdot \left (2 a b^{3} d g^{3} - 2 b^{4} c g^{3}\right ) + x \left (4 a^{2} b^{2} d g^{3} - 4 a b^{3} c g^{3}\right )} \] Input:

integrate((A+B*ln(e*(b*x+a)**2/(d*x+c)**2))/(b*g*x+a*g)**3,x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (134) = 268\).

Time = 0.05 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.22 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=\frac {1}{2} \, B {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} - \frac {\log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {A}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} \] Input:

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g)^3,x, algorithm="max 
ima")
 

Output:

1/2*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) - log(b^2*e*x^2/(d^2*x^2 + 2* 
c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c* 
d*x + c^2))/(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 = 0.15 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.94 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=\frac {B d^{2} \log \left (b x + a\right )}{b^{3} c^{2} g^{3} - 2 \, a b^{2} c d g^{3} + a^{2} b d^{2} g^{3}} - \frac {B d^{2} \log \left (d x + c\right )}{b^{3} c^{2} g^{3} - 2 \, a b^{2} c d g^{3} + a^{2} b d^{2} g^{3}} - \frac {B \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\right )}} + \frac {2 \, B b d x - A b c - B b c + A a d + 3 \, B a d}{2 \, {\left (b^{4} c g^{3} x^{2} - a b^{3} d g^{3} x^{2} + 2 \, a b^{3} c g^{3} x - 2 \, a^{2} b^{2} d g^{3} x + a^{2} b^{2} c g^{3} - a^{3} b d g^{3}\right )}} \] Input:

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(b*g*x+a*g)^3,x, algorithm="gia 
c")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 26.17 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.49 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=-\frac {\frac {A\,a\,d-A\,b\,c+3\,B\,a\,d-B\,b\,c}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b\,d\,x}{a\,d-b\,c}}{a^2\,b\,g^3+2\,a\,b^2\,g^3\,x+b^3\,g^3\,x^2}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}{2\,b^2\,g^3\,\left (2\,a\,x+b\,x^2+\frac {a^2}{b}\right )}-\frac {2\,B\,d^2\,\mathrm {atanh}\left (\frac {b^3\,c^2\,g^3-a^2\,b\,d^2\,g^3}{b\,g^3\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{b\,g^3\,{\left (a\,d-b\,c\right )}^2} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 693, normalized size of antiderivative = 5.02 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a g+b g x)^3} \, dx=\frac {-4 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a^{2} b^{3} c d x -2 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a \,b^{4} c d \,x^{2}+\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a^{2} b^{3} d^{2} x^{2}-a^{3} b^{2} c^{2}-2 a^{4} b \,d^{2}+\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) b^{5} c^{2} x^{2}+2 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a^{3} b^{2} d^{2} x +2 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a \,b^{4} c^{2} x +a^{2} b^{3} d^{2} x^{2}-2 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} c^{2}-2 \,\mathrm {log}\left (b x +a \right ) b^{5} c^{2} x^{2}+2 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} c^{2}+2 \,\mathrm {log}\left (d x +c \right ) b^{5} c^{2} x^{2}+3 a^{3} b^{2} c d -a^{2} b^{3} c^{2}+2 a^{4} b c d -a^{5} d^{2}-a \,b^{4} c d \,x^{2}+8 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{3} c d x +4 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} c d \,x^{2}-8 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} c d x -4 \,\mathrm {log}\left (d x +c \right ) a \,b^{4} c d \,x^{2}+4 \,\mathrm {log}\left (b x +a \right ) a^{3} b^{2} c d -4 \,\mathrm {log}\left (b x +a \right ) a \,b^{4} c^{2} x -4 \,\mathrm {log}\left (d x +c \right ) a^{3} b^{2} c d +4 \,\mathrm {log}\left (d x +c \right ) a \,b^{4} c^{2} x}{2 a^{2} b \,g^{3} \left (a^{2} b^{2} d^{2} x^{2}-2 a \,b^{3} c d \,x^{2}+b^{4} c^{2} x^{2}+2 a^{3} b \,d^{2} x -4 a^{2} b^{2} c d x +2 a \,b^{3} c^{2} x +a^{4} d^{2}-2 a^{3} b c d +a^{2} b^{2} c^{2}\right )} \] Input:

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