\(\int (a g+b g x)^2 (A+B \log (\frac {e (c+d x)^2}{(a+b x)^2})) \, dx\) [203]

Optimal result

Integrand size = 32, antiderivative size = 120 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=-\frac {2 B (b c-a d)^2 g^2 x}{3 d^2}+\frac {B (b c-a d) g^2 (a+b x)^2}{3 b d}+\frac {2 B (b c-a d)^3 g^2 \log (c+d x)}{3 b d^3}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.82 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {g^2 \left (\frac {B (b c-a d) \left (d \left (a^2 d+4 a b d x+b^2 x (-2 c+d x)\right )+2 (b c-a d)^2 \log (c+d x)\right )}{d^3}+(a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )\right )}{3 b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2948, 27, 49, 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.

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

\(\Big \downarrow \) 2948

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[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 = 0.96 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.54

method	result	size
derivativedivides	\(-\frac {-\frac {g^{2} A \left (b x +a \right )^{3}}{3}+g^{2} B \left (-\frac {\left (b x +a \right )^{3} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{3}-\left (-\frac {2 d a}{3}+\frac {2 b c}{3}\right ) \left (\frac {\left (d a -b c \right )^{2} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{3}}+\frac {\left (b x +a \right )^{2}}{2 d}+\frac {\left (-a^{2} d^{2}+2 a c d b -c^{2} b^{2}\right ) \ln \left (\frac {1}{b x +a}\right )}{d^{3}}-\frac {\left (-d a +b c \right ) \left (b x +a \right )}{d^{2}}\right )\right )}{b}\)	\(185\)
default	\(-\frac {-\frac {g^{2} A \left (b x +a \right )^{3}}{3}+g^{2} B \left (-\frac {\left (b x +a \right )^{3} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{3}-\left (-\frac {2 d a}{3}+\frac {2 b c}{3}\right ) \left (\frac {\left (d a -b c \right )^{2} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{3}}+\frac {\left (b x +a \right )^{2}}{2 d}+\frac {\left (-a^{2} d^{2}+2 a c d b -c^{2} b^{2}\right ) \ln \left (\frac {1}{b x +a}\right )}{d^{3}}-\frac {\left (-d a +b c \right ) \left (b x +a \right )}{d^{2}}\right )\right )}{b}\)	\(185\)
parts	\(\frac {g^{2} A \left (b x +a \right )^{3}}{3 b}-\frac {g^{2} B \left (-\frac {\left (b x +a \right )^{3} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{3}-\left (-\frac {2 d a}{3}+\frac {2 b c}{3}\right ) \left (\frac {\left (d a -b c \right )^{2} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{3}}+\frac {\left (b x +a \right )^{2}}{2 d}+\frac {\left (-a^{2} d^{2}+2 a c d b -c^{2} b^{2}\right ) \ln \left (\frac {1}{b x +a}\right )}{d^{3}}-\frac {\left (-d a +b c \right ) \left (b x +a \right )}{d^{2}}\right )\right )}{b}\)	\(187\)
risch	\(\frac {g^{2} \left (b x +a \right )^{3} B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{3 b}+\frac {g^{2} b^{2} A \,x^{3}}{3}+g^{2} b A a \,x^{2}-\frac {g^{2} b B a \,x^{2}}{3}+\frac {g^{2} b^{2} B c \,x^{2}}{3 d}+g^{2} A \,a^{2} x -\frac {2 g^{2} B \ln \left (d x +c \right ) a^{3}}{3 b}+\frac {2 g^{2} B \ln \left (d x +c \right ) a^{2} c}{d}-\frac {2 g^{2} b B \ln \left (d x +c \right ) a \,c^{2}}{d^{2}}+\frac {2 g^{2} b^{2} B \ln \left (d x +c \right ) c^{3}}{3 d^{3}}-\frac {4 g^{2} B \,a^{2} x}{3}+\frac {2 g^{2} b B a c x}{d}-\frac {2 g^{2} b^{2} B \,c^{2} x}{3 d^{2}}\)	\(210\)
parallelrisch	\(\frac {12 B \ln \left (b x +a \right ) a^{2} b c \,d^{2} g^{2}-12 B \ln \left (b x +a \right ) a \,b^{2} c^{2} d \,g^{2}+8 B \,a^{3} d^{3} g^{2}+4 B \,b^{3} c^{3} g^{2}-2 B \,a^{2} b c \,d^{2} g^{2}-10 B a \,b^{2} c^{2} d \,g^{2}-12 A \,a^{2} b c \,d^{2} g^{2}+12 B x a \,b^{2} c \,d^{2} g^{2}+2 A \,x^{3} b^{3} d^{3} g^{2}+6 B x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a^{2} b \,d^{3} g^{2}-8 B x \,a^{2} b \,d^{3} g^{2}-4 B x \,b^{3} c^{2} d \,g^{2}+6 A \,x^{2} a \,b^{2} d^{3} g^{2}-2 B \,x^{2} a \,b^{2} d^{3} g^{2}+2 B \,x^{2} b^{3} c \,d^{2} g^{2}+6 A x \,a^{2} b \,d^{3} g^{2}+2 B \,x^{3} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{3} d^{3} g^{2}+2 B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{3} c^{3} g^{2}+6 B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a^{2} b c \,d^{2} g^{2}-6 B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a \,b^{2} c^{2} d \,g^{2}+6 B \,x^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a \,b^{2} d^{3} g^{2}-4 B \ln \left (b x +a \right ) a^{3} d^{3} g^{2}+4 B \ln \left (b x +a \right ) b^{3} c^{3} g^{2}-6 A \,a^{3} d^{3} g^{2}}{6 b \,d^{3}}\)	\(471\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (112) = 224\).

Time = 0.09 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.04 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {A b^{3} d^{3} g^{2} x^{3} - 2 \, B a^{3} d^{3} g^{2} \log \left (b x + a\right ) + {\left (B b^{3} c d^{2} + {\left (3 \, A - B\right )} a b^{2} d^{3}\right )} g^{2} x^{2} - {\left (2 \, B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} - {\left (3 \, A - 4 \, B\right )} a^{2} b d^{3}\right )} g^{2} x + 2 \, {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} g^{2} \log \left (d x + c\right ) + {\left (B b^{3} d^{3} g^{2} x^{3} + 3 \, B a b^{2} d^{3} g^{2} x^{2} + 3 \, B a^{2} b d^{3} g^{2} x\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{3 \, b d^{3}} \] Input:

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

Output:

1/3*(A*b^3*d^3*g^2*x^3 - 2*B*a^3*d^3*g^2*log(b*x + a) + (B*b^3*c*d^2 + (3* 
A - B)*a*b^2*d^3)*g^2*x^2 - (2*B*b^3*c^2*d - 6*B*a*b^2*c*d^2 - (3*A - 4*B) 
*a^2*b*d^3)*g^2*x + 2*(B*b^3*c^3 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2)*g^2* 
log(d*x + c) + (B*b^3*d^3*g^2*x^3 + 3*B*a*b^2*d^3*g^2*x^2 + 3*B*a^2*b*d^3* 
g^2*x)*log((d^2*e*x^2 + 2*c*d*e*x + c^2*e)/(b^2*x^2 + 2*a*b*x + a^2)))/(b* 
d^3)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (107) = 214\).

Time = 1.51 (sec) , antiderivative size = 517, normalized size of antiderivative = 4.31 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {A b^{2} g^{2} x^{3}}{3} - \frac {2 B a^{3} g^{2} \log {\left (x + \frac {\frac {2 B a^{4} d^{3} g^{2}}{b} + 6 B a^{3} c d^{2} g^{2} - 6 B a^{2} b c^{2} d g^{2} + 2 B a b^{2} c^{3} g^{2}}{2 B a^{3} d^{3} g^{2} + 6 B a^{2} b c d^{2} g^{2} - 6 B a b^{2} c^{2} d g^{2} + 2 B b^{3} c^{3} g^{2}} \right )}}{3 b} + \frac {2 B c g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {8 B a^{3} c d^{2} g^{2} - 6 B a^{2} b c^{2} d g^{2} + 2 B a b^{2} c^{3} g^{2} - 2 B a c g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) + \frac {2 B b c^{2} g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{d}}{2 B a^{3} d^{3} g^{2} + 6 B a^{2} b c d^{2} g^{2} - 6 B a b^{2} c^{2} d g^{2} + 2 B b^{3} c^{3} g^{2}} \right )}}{3 d^{3}} + x^{2} \left (A a b g^{2} - \frac {B a b g^{2}}{3} + \frac {B b^{2} c g^{2}}{3 d}\right ) + x \left (A a^{2} g^{2} - \frac {4 B a^{2} g^{2}}{3} + \frac {2 B a b c g^{2}}{d} - \frac {2 B b^{2} c^{2} g^{2}}{3 d^{2}}\right ) + \left (B a^{2} g^{2} x + B a b g^{2} x^{2} + \frac {B b^{2} g^{2} x^{3}}{3}\right ) \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )} \] Input:

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

Output:

A*b**2*g**2*x**3/3 - 2*B*a**3*g**2*log(x + (2*B*a**4*d**3*g**2/b + 6*B*a** 
3*c*d**2*g**2 - 6*B*a**2*b*c**2*d*g**2 + 2*B*a*b**2*c**3*g**2)/(2*B*a**3*d 
**3*g**2 + 6*B*a**2*b*c*d**2*g**2 - 6*B*a*b**2*c**2*d*g**2 + 2*B*b**3*c**3 
*g**2))/(3*b) + 2*B*c*g**2*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)*log(x + ( 
8*B*a**3*c*d**2*g**2 - 6*B*a**2*b*c**2*d*g**2 + 2*B*a*b**2*c**3*g**2 - 2*B 
*a*c*g**2*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2) + 2*B*b*c**2*g**2*(3*a**2* 
d**2 - 3*a*b*c*d + b**2*c**2)/d)/(2*B*a**3*d**3*g**2 + 6*B*a**2*b*c*d**2*g 
**2 - 6*B*a*b**2*c**2*d*g**2 + 2*B*b**3*c**3*g**2))/(3*d**3) + x**2*(A*a*b 
*g**2 - B*a*b*g**2/3 + B*b**2*c*g**2/(3*d)) + x*(A*a**2*g**2 - 4*B*a**2*g* 
*2/3 + 2*B*a*b*c*g**2/d - 2*B*b**2*c**2*g**2/(3*d**2)) + (B*a**2*g**2*x + 
B*a*b*g**2*x**2 + B*b**2*g**2*x**3/3)*log(e*(c + d*x)**2/(a + b*x)**2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 436 vs. \(2 (112) = 224\).

Time = 0.06 (sec) , antiderivative size = 436, normalized size of antiderivative = 3.63 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {1}{3} \, A b^{2} g^{2} x^{3} + A a b g^{2} x^{2} + {\left (x \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - \frac {2 \, a \log \left (b x + a\right )}{b} + \frac {2 \, c \log \left (d x + c\right )}{d}\right )} B a^{2} g^{2} + {\left (x^{2} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {2 \, c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {2 \, {\left (b c - a d\right )} x}{b d}\right )} B a b g^{2} + \frac {1}{3} \, {\left (x^{3} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} + \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} + \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b^{2} g^{2} + A a^{2} g^{2} x \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (112) = 224\).

Time = 1.09 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.02 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {1}{3} \, A b^{2} g^{2} x^{3} - \frac {2 \, B a^{3} g^{2} \log \left (b x + a\right )}{3 \, b} + \frac {{\left (B b^{2} c g^{2} + 3 \, A a b d g^{2} - B a b d g^{2}\right )} x^{2}}{3 \, d} + \frac {1}{3} \, {\left (B b^{2} g^{2} x^{3} + 3 \, B a b g^{2} x^{2} + 3 \, B a^{2} g^{2} x\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - \frac {{\left (2 \, B b^{2} c^{2} g^{2} - 6 \, B a b c d g^{2} - 3 \, A a^{2} d^{2} g^{2} + 4 \, B a^{2} d^{2} g^{2}\right )} x}{3 \, d^{2}} + \frac {2 \, {\left (B b^{2} c^{3} g^{2} - 3 \, B a b c^{2} d g^{2} + 3 \, B a^{2} c d^{2} g^{2}\right )} \log \left (d x + c\right )}{3 \, d^{3}} \] Input:

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

Output:

1/3*A*b^2*g^2*x^3 - 2/3*B*a^3*g^2*log(b*x + a)/b + 1/3*(B*b^2*c*g^2 + 3*A* 
a*b*d*g^2 - B*a*b*d*g^2)*x^2/d + 1/3*(B*b^2*g^2*x^3 + 3*B*a*b*g^2*x^2 + 3* 
B*a^2*g^2*x)*log((d^2*e*x^2 + 2*c*d*e*x + c^2*e)/(b^2*x^2 + 2*a*b*x + a^2) 
) - 1/3*(2*B*b^2*c^2*g^2 - 6*B*a*b*c*d*g^2 - 3*A*a^2*d^2*g^2 + 4*B*a^2*d^2 
*g^2)*x/d^2 + 2/3*(B*b^2*c^3*g^2 - 3*B*a*b*c^2*d*g^2 + 3*B*a^2*c*d^2*g^2)* 
log(d*x + c)/d^3
 

Mupad [B] (verification not implemented)

Time = 26.45 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.47 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=x^2\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{6\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{6\,d}\right )-x\,\left (\frac {\left (3\,a\,d+3\,b\,c\right )\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{3\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{3\,d}\right )}{3\,b\,d}-\frac {a\,g^2\,\left (3\,A\,a\,d+3\,A\,b\,c-2\,B\,a\,d+2\,B\,b\,c\right )}{d}+\frac {A\,a\,b\,c\,g^2}{d}\right )+\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\,\left (B\,a^2\,g^2\,x+B\,a\,b\,g^2\,x^2+\frac {B\,b^2\,g^2\,x^3}{3}\right )+\frac {\ln \left (c+d\,x\right )\,\left (6\,B\,a^2\,c\,d^2\,g^2-6\,B\,a\,b\,c^2\,d\,g^2+2\,B\,b^2\,c^3\,g^2\right )}{3\,d^3}+\frac {A\,b^2\,g^2\,x^3}{3}-\frac {2\,B\,a^3\,g^2\,\ln \left (a+b\,x\right )}{3\,b} \] Input:

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

Output:

x^2*((b*g^2*(9*A*a*d + 3*A*b*c - 2*B*a*d + 2*B*b*c))/(6*d) - (A*b*g^2*(3*a 
*d + 3*b*c))/(6*d)) - x*(((3*a*d + 3*b*c)*((b*g^2*(9*A*a*d + 3*A*b*c - 2*B 
*a*d + 2*B*b*c))/(3*d) - (A*b*g^2*(3*a*d + 3*b*c))/(3*d)))/(3*b*d) - (a*g^ 
2*(3*A*a*d + 3*A*b*c - 2*B*a*d + 2*B*b*c))/d + (A*a*b*c*g^2)/d) + log((e*( 
c + d*x)^2)/(a + b*x)^2)*((B*b^2*g^2*x^3)/3 + B*a^2*g^2*x + B*a*b*g^2*x^2) 
 + (log(c + d*x)*(2*B*b^2*c^3*g^2 + 6*B*a^2*c*d^2*g^2 - 6*B*a*b*c^2*d*g^2) 
)/(3*d^3) + (A*b^2*g^2*x^3)/3 - (2*B*a^3*g^2*log(a + b*x))/(3*b)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.95 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {g^{2} \left (-2 \,\mathrm {log}\left (d x +c \right ) a^{3} d^{3}+6 \,\mathrm {log}\left (d x +c \right ) a^{2} b c \,d^{2}-6 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} c^{2} d +2 \,\mathrm {log}\left (d x +c \right ) b^{3} c^{3}+\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right ) a^{3} d^{3}+3 \,\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right ) a^{2} b \,d^{3} x +3 \,\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right ) a \,b^{2} d^{3} x^{2}+\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right ) b^{3} d^{3} x^{3}+3 a^{3} d^{3} x +3 a^{2} b \,d^{3} x^{2}-4 a^{2} b \,d^{3} x +6 a \,b^{2} c \,d^{2} x +a \,b^{2} d^{3} x^{3}-a \,b^{2} d^{3} x^{2}-2 b^{3} c^{2} d x +b^{3} c \,d^{2} x^{2}\right )}{3 d^{3}} \] Input:

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

Output:

(g**2*( - 2*log(c + d*x)*a**3*d**3 + 6*log(c + d*x)*a**2*b*c*d**2 - 6*log( 
c + d*x)*a*b**2*c**2*d + 2*log(c + d*x)*b**3*c**3 + log((c**2*e + 2*c*d*e* 
x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a**3*d**3 + 3*log((c**2*e + 
 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a**2*b*d**3*x + 3* 
log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a*b** 
2*d**3*x**2 + log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b** 
2*x**2))*b**3*d**3*x**3 + 3*a**3*d**3*x + 3*a**2*b*d**3*x**2 - 4*a**2*b*d* 
*3*x + 6*a*b**2*c*d**2*x + a*b**2*d**3*x**3 - a*b**2*d**3*x**2 - 2*b**3*c* 
*2*d*x + b**3*c*d**2*x**2))/(3*d**3)