\(\int (a g+b g x)^3 (A+B \log (\frac {e (c+d x)^2}{(a+b x)^2})) \, dx\) [202]

Optimal result

Integrand size = 32, antiderivative size = 151 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {B (b c-a d)^3 g^3 x}{2 d^3}-\frac {B (b c-a d)^2 g^3 (a+b x)^2}{4 b d^2}+\frac {B (b c-a d) g^3 (a+b x)^3}{6 b d}-\frac {B (b c-a d)^4 g^3 \log (c+d x)}{2 b d^4}+\frac {g^3 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{4 b} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.81 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {g^3 \left (\frac {B (b c-a d) \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )}{3 d^4}+(a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )\right )}{4 b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.87, 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.

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

\(\Big \downarrow \) 2948

\(\displaystyle \frac {B (b c-a d) \int \frac {g^4 (a+b x)^3}{c+d x}dx}{2 b g}+\frac {g^3 (a+b x)^4 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{4 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B g^3 (b c-a d) \int \frac {(a+b x)^3}{c+d x}dx}{2 b}+\frac {g^3 (a+b x)^4 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{4 b}\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {B g^3 (b c-a d) \int \left (\frac {(a d-b c)^3}{d^3 (c+d x)}+\frac {b (b c-a d)^2}{d^3}+\frac {b (a+b x)^2}{d}-\frac {b (b c-a d) (a+b x)}{d^2}\right )dx}{2 b}+\frac {g^3 (a+b x)^4 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{4 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {g^3 (a+b x)^4 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{4 b}+\frac {B g^3 (b c-a d) \left (-\frac {(b c-a d)^3 \log (c+d x)}{d^4}+\frac {b x (b c-a d)^2}{d^3}-\frac {(a+b x)^2 (b c-a d)}{2 d^2}+\frac {(a+b x)^3}{3 d}\right )}{2 b}\)

Input:

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

Output:

(B*(b*c - a*d)*g^3*((b*(b*c - a*d)^2*x)/d^3 - ((b*c - a*d)*(a + b*x)^2)/(2 
*d^2) + (a + b*x)^3/(3*d) - ((b*c - a*d)^3*Log[c + d*x])/d^4))/(2*b) + (g^ 
3*(a + b*x)^4*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(4*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 = 1.05 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.54

method	result	size
derivativedivides	\(-\frac {-\frac {g^{3} A \left (b x +a \right )^{4}}{4}+g^{3} B \left (-\frac {\left (b x +a \right )^{4} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{4}-\left (-\frac {d a}{2}+\frac {b c}{2}\right ) \left (\frac {\left (d a -b c \right )^{3} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{4}}+\frac {\left (b x +a \right )^{3}}{3 d}-\frac {\left (-d a +b c \right ) \left (b x +a \right )^{2}}{2 d^{2}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (\frac {1}{b x +a}\right )}{d^{4}}-\frac {\left (-a^{2} d^{2}+2 a c d b -c^{2} b^{2}\right ) \left (b x +a \right )}{d^{3}}\right )\right )}{b}\)	\(233\)
default	\(-\frac {-\frac {g^{3} A \left (b x +a \right )^{4}}{4}+g^{3} B \left (-\frac {\left (b x +a \right )^{4} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{4}-\left (-\frac {d a}{2}+\frac {b c}{2}\right ) \left (\frac {\left (d a -b c \right )^{3} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{4}}+\frac {\left (b x +a \right )^{3}}{3 d}-\frac {\left (-d a +b c \right ) \left (b x +a \right )^{2}}{2 d^{2}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (\frac {1}{b x +a}\right )}{d^{4}}-\frac {\left (-a^{2} d^{2}+2 a c d b -c^{2} b^{2}\right ) \left (b x +a \right )}{d^{3}}\right )\right )}{b}\)	\(233\)
parts	\(\frac {g^{3} A \left (b x +a \right )^{4}}{4 b}-\frac {g^{3} B \left (-\frac {\left (b x +a \right )^{4} \ln \left (\frac {e \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2}}{b^{2}}\right )}{4}-\left (-\frac {d a}{2}+\frac {b c}{2}\right ) \left (\frac {\left (d a -b c \right )^{3} \ln \left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )}{d^{4}}+\frac {\left (b x +a \right )^{3}}{3 d}-\frac {\left (-d a +b c \right ) \left (b x +a \right )^{2}}{2 d^{2}}+\frac {\left (-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \ln \left (\frac {1}{b x +a}\right )}{d^{4}}-\frac {\left (-a^{2} d^{2}+2 a c d b -c^{2} b^{2}\right ) \left (b x +a \right )}{d^{3}}\right )\right )}{b}\)	\(235\)
risch	\(\frac {\left (b x +a \right )^{4} g^{3} B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )}{4 b}+\frac {g^{3} b^{3} A \,x^{4}}{4}+g^{3} b^{2} A a \,x^{3}-\frac {g^{3} b^{2} B a \,x^{3}}{6}+\frac {g^{3} b^{3} B c \,x^{3}}{6 d}+\frac {3 g^{3} b A \,a^{2} x^{2}}{2}-\frac {3 g^{3} b B \,a^{2} x^{2}}{4}+\frac {g^{3} b^{2} B a c \,x^{2}}{d}-\frac {g^{3} b^{3} B \,c^{2} x^{2}}{4 d^{2}}+g^{3} A \,a^{3} x -\frac {g^{3} B \ln \left (d x +c \right ) a^{4}}{2 b}+\frac {2 g^{3} B \ln \left (d x +c \right ) a^{3} c}{d}-\frac {3 g^{3} b B \ln \left (d x +c \right ) a^{2} c^{2}}{d^{2}}+\frac {2 g^{3} b^{2} B \ln \left (d x +c \right ) a \,c^{3}}{d^{3}}-\frac {g^{3} b^{3} B \ln \left (d x +c \right ) c^{4}}{2 d^{4}}-\frac {3 g^{3} B \,a^{3} x}{2}+\frac {3 g^{3} b B \,a^{2} c x}{d}-\frac {2 g^{3} b^{2} B a \,c^{2} x}{d^{2}}+\frac {g^{3} b^{3} B \,c^{3} x}{2 d^{3}}\)	\(317\)
parallelrisch	\(\frac {12 A \,x^{3} a \,b^{3} d^{4} g^{3}-2 B \,x^{3} a \,b^{3} d^{4} g^{3}+2 B \,x^{3} b^{4} c \,d^{3} g^{3}+18 A \,x^{2} a^{2} b^{2} d^{4} g^{3}-9 B \,x^{2} a^{2} b^{2} d^{4} g^{3}-9 B \,a^{3} b c \,d^{3} g^{3}-24 B \,a^{2} b^{2} c^{2} d^{2} g^{3}+21 B a \,b^{3} c^{3} d \,g^{3}+18 B \,a^{4} d^{4} g^{3}-6 B \,b^{4} c^{4} g^{3}+6 B x \,b^{4} c^{3} d \,g^{3}+3 B \,x^{4} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{4} d^{4} g^{3}+12 A x \,a^{3} b \,d^{4} g^{3}-18 B x \,a^{3} b \,d^{4} g^{3}-18 B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a^{2} b^{2} c^{2} d^{2} g^{3}+12 B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a \,b^{3} c^{3} d \,g^{3}-3 B \,x^{2} b^{4} c^{2} d^{2} g^{3}-3 B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) b^{4} c^{4} g^{3}+36 B x \,a^{2} b^{2} c \,d^{3} g^{3}-30 A \,a^{3} b c \,d^{3} g^{3}+3 A \,x^{4} b^{4} d^{4} g^{3}-6 B \ln \left (b x +a \right ) a^{4} d^{4} g^{3}-6 B \ln \left (b x +a \right ) b^{4} c^{4} g^{3}-24 B x a \,b^{3} c^{2} d^{2} g^{3}+24 B \ln \left (b x +a \right ) a^{3} b c \,d^{3} g^{3}-36 B \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d^{2} g^{3}+24 B \ln \left (b x +a \right ) a \,b^{3} c^{3} d \,g^{3}+12 B \,x^{2} a \,b^{3} c \,d^{3} g^{3}-12 A \,a^{4} d^{4} g^{3}+12 B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a^{3} b c \,d^{3} g^{3}+12 B \,x^{3} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a \,b^{3} d^{4} g^{3}+18 B \,x^{2} \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a^{2} b^{2} d^{4} g^{3}+12 B x \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right ) a^{3} b \,d^{4} g^{3}}{12 d^{4} b}\)	\(673\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (141) = 282\).

Time = 0.10 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.27 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {3 \, A b^{4} d^{4} g^{3} x^{4} - 6 \, B a^{4} d^{4} g^{3} \log \left (b x + a\right ) + 2 \, {\left (B b^{4} c d^{3} + {\left (6 \, A - B\right )} a b^{3} d^{4}\right )} g^{3} x^{3} - 3 \, {\left (B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} - 3 \, {\left (2 \, A - B\right )} a^{2} b^{2} d^{4}\right )} g^{3} x^{2} + 6 \, {\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 6 \, B a^{2} b^{2} c d^{3} + {\left (2 \, A - 3 \, B\right )} a^{3} b d^{4}\right )} g^{3} x - 6 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3}\right )} g^{3} \log \left (d x + c\right ) + 3 \, {\left (B b^{4} d^{4} g^{3} x^{4} + 4 \, B a b^{3} d^{4} g^{3} x^{3} + 6 \, B a^{2} b^{2} d^{4} g^{3} x^{2} + 4 \, B a^{3} b d^{4} g^{3} x\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{12 \, b d^{4}} \] Input:

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

Output:

1/12*(3*A*b^4*d^4*g^3*x^4 - 6*B*a^4*d^4*g^3*log(b*x + a) + 2*(B*b^4*c*d^3 
+ (6*A - B)*a*b^3*d^4)*g^3*x^3 - 3*(B*b^4*c^2*d^2 - 4*B*a*b^3*c*d^3 - 3*(2 
*A - B)*a^2*b^2*d^4)*g^3*x^2 + 6*(B*b^4*c^3*d - 4*B*a*b^3*c^2*d^2 + 6*B*a^ 
2*b^2*c*d^3 + (2*A - 3*B)*a^3*b*d^4)*g^3*x - 6*(B*b^4*c^4 - 4*B*a*b^3*c^3* 
d + 6*B*a^2*b^2*c^2*d^2 - 4*B*a^3*b*c*d^3)*g^3*log(d*x + c) + 3*(B*b^4*d^4 
*g^3*x^4 + 4*B*a*b^3*d^4*g^3*x^3 + 6*B*a^2*b^2*d^4*g^3*x^2 + 4*B*a^3*b*d^4 
*g^3*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^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 707 vs. \(2 (131) = 262\).

Time = 2.18 (sec) , antiderivative size = 707, normalized size of antiderivative = 4.68 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {A b^{3} g^{3} x^{4}}{4} - \frac {B a^{4} g^{3} \log {\left (x + \frac {\frac {B a^{5} d^{4} g^{3}}{b} + 4 B a^{4} c d^{3} g^{3} - 6 B a^{3} b c^{2} d^{2} g^{3} + 4 B a^{2} b^{2} c^{3} d g^{3} - B a b^{3} c^{4} g^{3}}{B a^{4} d^{4} g^{3} + 4 B a^{3} b c d^{3} g^{3} - 6 B a^{2} b^{2} c^{2} d^{2} g^{3} + 4 B a b^{3} c^{3} d g^{3} - B b^{4} c^{4} g^{3}} \right )}}{2 b} + \frac {B c g^{3} \cdot \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {5 B a^{4} c d^{3} g^{3} - 6 B a^{3} b c^{2} d^{2} g^{3} + 4 B a^{2} b^{2} c^{3} d g^{3} - B a b^{3} c^{4} g^{3} - B a c g^{3} \cdot \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right ) + \frac {B b c^{2} g^{3} \cdot \left (2 a d - b c\right ) \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{d}}{B a^{4} d^{4} g^{3} + 4 B a^{3} b c d^{3} g^{3} - 6 B a^{2} b^{2} c^{2} d^{2} g^{3} + 4 B a b^{3} c^{3} d g^{3} - B b^{4} c^{4} g^{3}} \right )}}{2 d^{4}} + x^{3} \left (A a b^{2} g^{3} - \frac {B a b^{2} g^{3}}{6} + \frac {B b^{3} c g^{3}}{6 d}\right ) + x^{2} \cdot \left (\frac {3 A a^{2} b g^{3}}{2} - \frac {3 B a^{2} b g^{3}}{4} + \frac {B a b^{2} c g^{3}}{d} - \frac {B b^{3} c^{2} g^{3}}{4 d^{2}}\right ) + x \left (A a^{3} g^{3} - \frac {3 B a^{3} g^{3}}{2} + \frac {3 B a^{2} b c g^{3}}{d} - \frac {2 B a b^{2} c^{2} g^{3}}{d^{2}} + \frac {B b^{3} c^{3} g^{3}}{2 d^{3}}\right ) + \left (B a^{3} g^{3} x + \frac {3 B a^{2} b g^{3} x^{2}}{2} + B a b^{2} g^{3} x^{3} + \frac {B b^{3} g^{3} x^{4}}{4}\right ) \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )} \] Input:

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

Output:

A*b**3*g**3*x**4/4 - B*a**4*g**3*log(x + (B*a**5*d**4*g**3/b + 4*B*a**4*c* 
d**3*g**3 - 6*B*a**3*b*c**2*d**2*g**3 + 4*B*a**2*b**2*c**3*d*g**3 - B*a*b* 
*3*c**4*g**3)/(B*a**4*d**4*g**3 + 4*B*a**3*b*c*d**3*g**3 - 6*B*a**2*b**2*c 
**2*d**2*g**3 + 4*B*a*b**3*c**3*d*g**3 - B*b**4*c**4*g**3))/(2*b) + B*c*g* 
*3*(2*a*d - b*c)*(2*a**2*d**2 - 2*a*b*c*d + b**2*c**2)*log(x + (5*B*a**4*c 
*d**3*g**3 - 6*B*a**3*b*c**2*d**2*g**3 + 4*B*a**2*b**2*c**3*d*g**3 - B*a*b 
**3*c**4*g**3 - B*a*c*g**3*(2*a*d - b*c)*(2*a**2*d**2 - 2*a*b*c*d + b**2*c 
**2) + B*b*c**2*g**3*(2*a*d - b*c)*(2*a**2*d**2 - 2*a*b*c*d + b**2*c**2)/d 
)/(B*a**4*d**4*g**3 + 4*B*a**3*b*c*d**3*g**3 - 6*B*a**2*b**2*c**2*d**2*g** 
3 + 4*B*a*b**3*c**3*d*g**3 - B*b**4*c**4*g**3))/(2*d**4) + x**3*(A*a*b**2* 
g**3 - B*a*b**2*g**3/6 + B*b**3*c*g**3/(6*d)) + x**2*(3*A*a**2*b*g**3/2 - 
3*B*a**2*b*g**3/4 + B*a*b**2*c*g**3/d - B*b**3*c**2*g**3/(4*d**2)) + x*(A* 
a**3*g**3 - 3*B*a**3*g**3/2 + 3*B*a**2*b*c*g**3/d - 2*B*a*b**2*c**2*g**3/d 
**2 + B*b**3*c**3*g**3/(2*d**3)) + (B*a**3*g**3*x + 3*B*a**2*b*g**3*x**2/2 
 + B*a*b**2*g**3*x**3 + B*b**3*g**3*x**4/4)*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. 645 vs. \(2 (141) = 282\).

Time = 0.06 (sec) , antiderivative size = 645, normalized size of antiderivative = 4.27 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {1}{4} \, A b^{3} g^{3} x^{4} + A a b^{2} g^{3} x^{3} + \frac {3}{2} \, A a^{2} b g^{3} x^{2} + {\left (x \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - \frac {2 \, a \log \left (b x + a\right )}{b} + \frac {2 \, c \log \left (d x + c\right )}{d}\right )} B a^{3} g^{3} + \frac {3}{2} \, {\left (x^{2} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {2 \, c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {2 \, {\left (b c - a d\right )} x}{b d}\right )} B a^{2} b g^{3} + {\left (x^{3} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} + \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} + \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a b^{2} g^{3} + \frac {1}{12} \, {\left (3 \, x^{4} \log \left (\frac {d^{2} e x^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d e x}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B b^{3} g^{3} + A a^{3} g^{3} x \] Input:

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

Output:

1/4*A*b^3*g^3*x^4 + A*a*b^2*g^3*x^3 + 3/2*A*a^2*b*g^3*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 
^3*g^3 + 3/2*(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^2*b*g^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*a*b^2*g^3 + 1/12*(3*x^4*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)) 
+ 6*a^4*log(b*x + a)/b^4 - 6*c^4*log(d*x + c)/d^4 + (2*(b^3*c*d^2 - a*b^2* 
d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d 
^3))*B*b^3*g^3 + A*a^3*g^3*x
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (141) = 282\).

Time = 5.91 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.37 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {1}{4} \, A b^{3} g^{3} x^{4} - \frac {B a^{4} g^{3} \log \left (b x + a\right )}{2 \, b} + \frac {{\left (B b^{3} c g^{3} + 6 \, A a b^{2} d g^{3} - B a b^{2} d g^{3}\right )} x^{3}}{6 \, d} + \frac {1}{4} \, {\left (B b^{3} g^{3} x^{4} + 4 \, B a b^{2} g^{3} x^{3} + 6 \, B a^{2} b g^{3} x^{2} + 4 \, B a^{3} g^{3} x\right )} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - \frac {{\left (B b^{3} c^{2} g^{3} - 4 \, B a b^{2} c d g^{3} - 6 \, A a^{2} b d^{2} g^{3} + 3 \, B a^{2} b d^{2} g^{3}\right )} x^{2}}{4 \, d^{2}} + \frac {{\left (B b^{3} c^{3} g^{3} - 4 \, B a b^{2} c^{2} d g^{3} + 6 \, B a^{2} b c d^{2} g^{3} + 2 \, A a^{3} d^{3} g^{3} - 3 \, B a^{3} d^{3} g^{3}\right )} x}{2 \, d^{3}} - \frac {{\left (B b^{3} c^{4} g^{3} - 4 \, B a b^{2} c^{3} d g^{3} + 6 \, B a^{2} b c^{2} d^{2} g^{3} - 4 \, B a^{3} c d^{3} g^{3}\right )} \log \left (-d x - c\right )}{2 \, d^{4}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 26.19 (sec) , antiderivative size = 567, normalized size of antiderivative = 3.75 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\,\left (B\,a^3\,g^3\,x+\frac {3\,B\,a^2\,b\,g^3\,x^2}{2}+B\,a\,b^2\,g^3\,x^3+\frac {B\,b^3\,g^3\,x^4}{4}\right )-x^2\,\left (\frac {\left (\frac {b^2\,g^3\,\left (8\,A\,a\,d+2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{2\,d}-\frac {A\,b^2\,g^3\,\left (2\,a\,d+2\,b\,c\right )}{2\,d}\right )\,\left (2\,a\,d+2\,b\,c\right )}{4\,b\,d}-\frac {a\,b\,g^3\,\left (3\,A\,a\,d+2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{d}+\frac {A\,a\,b^2\,c\,g^3}{2\,d}\right )+x\,\left (\frac {\left (2\,a\,d+2\,b\,c\right )\,\left (\frac {\left (\frac {b^2\,g^3\,\left (8\,A\,a\,d+2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{2\,d}-\frac {A\,b^2\,g^3\,\left (2\,a\,d+2\,b\,c\right )}{2\,d}\right )\,\left (2\,a\,d+2\,b\,c\right )}{2\,b\,d}-\frac {2\,a\,b\,g^3\,\left (3\,A\,a\,d+2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{d}+\frac {A\,a\,b^2\,c\,g^3}{d}\right )}{2\,b\,d}+\frac {a^2\,g^3\,\left (4\,A\,a\,d+6\,A\,b\,c-3\,B\,a\,d+3\,B\,b\,c\right )}{d}-\frac {a\,c\,\left (\frac {b^2\,g^3\,\left (8\,A\,a\,d+2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{2\,d}-\frac {A\,b^2\,g^3\,\left (2\,a\,d+2\,b\,c\right )}{2\,d}\right )}{b\,d}\right )+x^3\,\left (\frac {b^2\,g^3\,\left (8\,A\,a\,d+2\,A\,b\,c-B\,a\,d+B\,b\,c\right )}{6\,d}-\frac {A\,b^2\,g^3\,\left (2\,a\,d+2\,b\,c\right )}{6\,d}\right )-\frac {\ln \left (c+d\,x\right )\,\left (-4\,B\,a^3\,c\,d^3\,g^3+6\,B\,a^2\,b\,c^2\,d^2\,g^3-4\,B\,a\,b^2\,c^3\,d\,g^3+B\,b^3\,c^4\,g^3\right )}{2\,d^4}+\frac {A\,b^3\,g^3\,x^4}{4}-\frac {B\,a^4\,g^3\,\ln \left (a+b\,x\right )}{2\,b} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.32 \[ \int (a g+b g x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx=\frac {g^{3} \left (-6 \,\mathrm {log}\left (d x +c \right ) a^{4} d^{4}+24 \,\mathrm {log}\left (d x +c \right ) a^{3} b c \,d^{3}-36 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} c^{2} d^{2}+24 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c^{3} d -6 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{4}+3 \,\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right ) a^{4} d^{4}+12 \,\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right ) a^{3} b \,d^{4} x +18 \,\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right ) a^{2} b^{2} d^{4} x^{2}+12 \,\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right ) a \,b^{3} d^{4} x^{3}+3 \,\mathrm {log}\left (\frac {d^{2} e \,x^{2}+2 c d e x +c^{2} e}{b^{2} x^{2}+2 a b x +a^{2}}\right ) b^{4} d^{4} x^{4}+12 a^{4} d^{4} x +18 a^{3} b \,d^{4} x^{2}-18 a^{3} b \,d^{4} x +36 a^{2} b^{2} c \,d^{3} x +12 a^{2} b^{2} d^{4} x^{3}-9 a^{2} b^{2} d^{4} x^{2}-24 a \,b^{3} c^{2} d^{2} x +12 a \,b^{3} c \,d^{3} x^{2}+3 a \,b^{3} d^{4} x^{4}-2 a \,b^{3} d^{4} x^{3}+6 b^{4} c^{3} d x -3 b^{4} c^{2} d^{2} x^{2}+2 b^{4} c \,d^{3} x^{3}\right )}{12 d^{4}} \] Input:

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

Output:

(g**3*( - 6*log(c + d*x)*a**4*d**4 + 24*log(c + d*x)*a**3*b*c*d**3 - 36*lo 
g(c + d*x)*a**2*b**2*c**2*d**2 + 24*log(c + d*x)*a*b**3*c**3*d - 6*log(c + 
 d*x)*b**4*c**4 + 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**4*d**4 + 12*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*b*d**4*x + 18*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**2*d**4*x**2 + 12*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**3*d**4*x** 
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)) 
*b**4*d**4*x**4 + 12*a**4*d**4*x + 18*a**3*b*d**4*x**2 - 18*a**3*b*d**4*x 
+ 36*a**2*b**2*c*d**3*x + 12*a**2*b**2*d**4*x**3 - 9*a**2*b**2*d**4*x**2 - 
 24*a*b**3*c**2*d**2*x + 12*a*b**3*c*d**3*x**2 + 3*a*b**3*d**4*x**4 - 2*a* 
b**3*d**4*x**3 + 6*b**4*c**3*d*x - 3*b**4*c**2*d**2*x**2 + 2*b**4*c*d**3*x 
**3))/(12*d**4)