\(\int (a g+b g x)^4 (A+B \log (\frac {e (a+b x)^2}{(c+d x)^2})) \, dx\) [119]

Optimal result

Integrand size = 32, antiderivative size = 182 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {2 B (b c-a d)^4 g^4 x}{5 d^4}-\frac {B (b c-a d)^3 g^4 (a+b x)^2}{5 b d^3}+\frac {2 B (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}-\frac {B (b c-a d) g^4 (a+b x)^4}{10 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 b}-\frac {2 B (b c-a d)^5 g^4 \log (c+d x)}{5 b d^5} \] Output:

2/5*B*(-a*d+b*c)^4*g^4*x/d^4-1/5*B*(-a*d+b*c)^3*g^4*(b*x+a)^2/b/d^3+2/15*B 
*(-a*d+b*c)^2*g^4*(b*x+a)^3/b/d^2-1/10*B*(-a*d+b*c)*g^4*(b*x+a)^4/b/d+1/5* 
g^4*(b*x+a)^5*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/b-2/5*B*(-a*d+b*c)^5*g^4*ln( 
d*x+c)/b/d^5
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.79 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {g^4 \left ((a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+\frac {B (b c-a d) \left (12 b d (b c-a d)^3 x-6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (b c-a d) (a+b x)^3-3 d^4 (a+b x)^4-12 (b c-a d)^4 \log (c+d x)\right )}{6 d^5}\right )}{5 b} \] Input:

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

Output:

(g^4*((a + b*x)^5*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + (B*(b*c - a*d 
)*(12*b*d*(b*c - a*d)^3*x - 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(b*c - 
 a*d)*(a + b*x)^3 - 3*d^4*(a + b*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x]))/(6 
*d^5)))/(5*b)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.86, 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.

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

\(\Big \downarrow \) 2948

\(\displaystyle \frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 b}-\frac {2 B (b c-a d) \int \frac {g^5 (a+b x)^4}{c+d x}dx}{5 b g}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 b}-\frac {2 B g^4 (b c-a d) \int \frac {(a+b x)^4}{c+d x}dx}{5 b}\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 b}-\frac {2 B g^4 (b c-a d) \int \left (\frac {(a d-b c)^4}{d^4 (c+d x)}-\frac {b (b c-a d)^3}{d^4}+\frac {b (a+b x)^3}{d}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (b c-a d)^2 (a+b x)}{d^3}\right )dx}{5 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 b}-\frac {2 B g^4 (b c-a d) \left (\frac {(b c-a d)^4 \log (c+d x)}{d^5}-\frac {b x (b c-a d)^3}{d^4}+\frac {(a+b x)^2 (b c-a d)^2}{2 d^3}-\frac {(a+b x)^3 (b c-a d)}{3 d^2}+\frac {(a+b x)^4}{4 d}\right )}{5 b}\)

Input:

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

Output:

(g^4*(a + b*x)^5*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(5*b) - (2*B*(b 
*c - a*d)*g^4*(-((b*(b*c - a*d)^3*x)/d^4) + ((b*c - a*d)^2*(a + b*x)^2)/(2 
*d^3) - ((b*c - a*d)*(a + b*x)^3)/(3*d^2) + (a + b*x)^4/(4*d) + ((b*c - a* 
d)^4*Log[c + d*x])/d^5))/(5*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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(445\) vs. \(2(170)=340\).

Time = 1.30 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.45

method	result	size
risch	\(\frac {6 g^{4} b B \,a^{3} x^{2}}{5}-\frac {g^{4} b^{4} B \,c^{3} x^{2}}{5 d^{3}}+g^{4} A \,a^{4} x +\frac {8 g^{4} B \,a^{4} x}{5}+\frac {2 g^{4} b^{4} B \,c^{4} x}{5 d^{4}}-\frac {2 g^{4} b^{4} B \ln \left (d x +c \right ) c^{5}}{5 d^{5}}-\frac {2 g^{4} B \ln \left (d x +c \right ) a^{4} c}{d}-\frac {4 g^{4} b B \,a^{3} c x}{d}+\frac {4 g^{4} b^{2} B \,a^{2} c^{2} x}{d^{2}}-\frac {2 g^{4} b^{3} B a \,c^{3} x}{d^{3}}+\frac {4 g^{4} b B \ln \left (d x +c \right ) a^{3} c^{2}}{d^{2}}-\frac {4 g^{4} b^{2} B \ln \left (d x +c \right ) a^{2} c^{3}}{d^{3}}+\frac {2 g^{4} b^{3} B \ln \left (d x +c \right ) a \,c^{4}}{d^{4}}-\frac {2 g^{4} b^{3} B a c \,x^{3}}{3 d}-\frac {2 g^{4} b^{2} B \,a^{2} c \,x^{2}}{d}+\frac {g^{4} b^{3} B a \,c^{2} x^{2}}{d^{2}}-\frac {g^{4} b^{4} B c \,x^{4}}{10 d}+2 g^{4} b^{2} A \,a^{2} x^{3}+\frac {8 g^{4} b^{2} B \,a^{2} x^{3}}{15}+\frac {2 g^{4} b^{4} B \,c^{2} x^{3}}{15 d^{2}}+2 g^{4} b A \,a^{3} x^{2}+\frac {2 g^{4} B \ln \left (d x +c \right ) a^{5}}{5 b}+g^{4} b^{3} A a \,x^{4}+\frac {g^{4} b^{3} B a \,x^{4}}{10}+\frac {g^{4} b^{4} A \,x^{5}}{5}+\frac {\left (b x +a \right )^{5} g^{4} B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{5 b}\)	\(446\)
parallelrisch	\(\frac {30 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{2} b^{4} c \,d^{5} g^{4}+60 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{3} b^{3} c \,d^{5} g^{4}+60 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{4} b^{2} c \,d^{5} g^{4}+30 B x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{5} b c \,d^{5} g^{4}+6 B \,x^{5} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{5} c \,d^{5} g^{4}+48 B x \,a^{5} b c \,d^{5} g^{4}-120 B x \,a^{4} b^{2} c^{2} d^{4} g^{4}+120 B x \,a^{3} b^{3} c^{3} d^{3} g^{4}-60 B x \,a^{2} b^{4} c^{4} d^{2} g^{4}+6 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a \,b^{5} c^{6} g^{4}+12 B \ln \left (b x +a \right ) a^{6} c \,d^{5} g^{4}-12 B \ln \left (b x +a \right ) a \,b^{5} c^{6} g^{4}-60 B \ln \left (b x +a \right ) a^{5} b \,c^{2} d^{4} g^{4}+120 B \ln \left (b x +a \right ) a^{4} b^{2} c^{3} d^{3} g^{4}-120 B \ln \left (b x +a \right ) a^{3} b^{3} c^{4} d^{2} g^{4}+60 B \ln \left (b x +a \right ) a^{2} b^{4} c^{5} d \,g^{4}+6 A \,x^{5} a \,b^{5} c \,d^{5} g^{4}+30 A \,x^{4} a^{2} b^{4} c \,d^{5} g^{4}+3 B \,x^{4} a^{2} b^{4} c \,d^{5} g^{4}-3 B \,x^{4} a \,b^{5} c^{2} d^{4} g^{4}+60 A \,x^{3} a^{3} b^{3} c \,d^{5} g^{4}+16 B \,x^{3} a^{3} b^{3} c \,d^{5} g^{4}-20 B \,x^{3} a^{2} b^{4} c^{2} d^{4} g^{4}+4 B \,x^{3} a \,b^{5} c^{3} d^{3} g^{4}+60 A \,x^{2} a^{4} b^{2} c \,d^{5} g^{4}+36 B \,x^{2} a^{4} b^{2} c \,d^{5} g^{4}-60 B \,x^{2} a^{3} b^{3} c^{2} d^{4} g^{4}+30 B \,x^{2} a^{2} b^{4} c^{3} d^{3} g^{4}-6 B \,x^{2} a \,b^{5} c^{4} d^{2} g^{4}+30 A x \,a^{5} b c \,d^{5} g^{4}+12 B x a \,b^{5} c^{5} d \,g^{4}+30 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{5} b \,c^{2} d^{4} g^{4}-60 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{4} b^{2} c^{3} d^{3} g^{4}+60 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{3} b^{3} c^{4} d^{2} g^{4}-30 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) a^{2} b^{4} c^{5} d \,g^{4}}{30 a b c \,d^{5}}\)	\(839\)
parts	\(\text {Expression too large to display}\)	\(1008\)
derivativedivides	\(\text {Expression too large to display}\)	\(1172\)
default	\(\text {Expression too large to display}\)	\(1172\)

Input:

int((b*g*x+a*g)^4*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2)),x,method=_RETURNVERBOSE)
 

Output:

6/5*g^4*b*B*a^3*x^2-1/5*g^4/d^3*b^4*B*c^3*x^2+g^4*A*a^4*x+8/5*g^4*B*a^4*x+ 
2/5*g^4/d^4*b^4*B*c^4*x-2/5*g^4/d^5*b^4*B*ln(d*x+c)*c^5-2*g^4/d*B*ln(d*x+c 
)*a^4*c-4*g^4/d*b*B*a^3*c*x+4*g^4/d^2*b^2*B*a^2*c^2*x-2*g^4/d^3*b^3*B*a*c^ 
3*x+4*g^4/d^2*b*B*ln(d*x+c)*a^3*c^2-4*g^4/d^3*b^2*B*ln(d*x+c)*a^2*c^3+2*g^ 
4/d^4*b^3*B*ln(d*x+c)*a*c^4-2/3*g^4/d*b^3*B*a*c*x^3-2*g^4/d*b^2*B*a^2*c*x^ 
2+g^4/d^2*b^3*B*a*c^2*x^2-1/10*g^4/d*b^4*B*c*x^4+2*g^4*b^2*A*a^2*x^3+8/15* 
g^4*b^2*B*a^2*x^3+2/15*g^4/d^2*b^4*B*c^2*x^3+2*g^4*b*A*a^3*x^2+2/5*g^4/b*B 
*ln(d*x+c)*a^5+g^4*b^3*A*a*x^4+1/10*g^4*b^3*B*a*x^4+1/5*g^4*b^4*A*x^5+1/5* 
(b*x+a)^5*g^4*B/b*ln(e*(b*x+a)^2/(d*x+c)^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (170) = 340\).

Time = 0.12 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.49 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {6 \, A b^{5} d^{5} g^{4} x^{5} + 12 \, B a^{5} d^{5} g^{4} \log \left (b x + a\right ) - 3 \, {\left (B b^{5} c d^{4} - {\left (10 \, A + B\right )} a b^{4} d^{5}\right )} g^{4} x^{4} + 4 \, {\left (B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} + {\left (15 \, A + 4 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{4} x^{3} - 6 \, {\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} + 10 \, B a^{2} b^{3} c d^{4} - 2 \, {\left (5 \, A + 3 \, B\right )} a^{3} b^{2} d^{5}\right )} g^{4} x^{2} + 6 \, {\left (2 \, B b^{5} c^{4} d - 10 \, B a b^{4} c^{3} d^{2} + 20 \, B a^{2} b^{3} c^{2} d^{3} - 20 \, B a^{3} b^{2} c d^{4} + {\left (5 \, A + 8 \, B\right )} a^{4} b d^{5}\right )} g^{4} x - 12 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4}\right )} g^{4} \log \left (d x + c\right ) + 6 \, {\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B a b^{4} d^{5} g^{4} x^{4} + 10 \, B a^{2} b^{3} d^{5} g^{4} x^{3} + 10 \, B a^{3} b^{2} d^{5} g^{4} x^{2} + 5 \, B a^{4} b d^{5} g^{4} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{30 \, b d^{5}} \] Input:

integrate((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)^2/(d*x+c)^2)),x, algorithm="fri 
cas")
 

Output:

1/30*(6*A*b^5*d^5*g^4*x^5 + 12*B*a^5*d^5*g^4*log(b*x + a) - 3*(B*b^5*c*d^4 
 - (10*A + B)*a*b^4*d^5)*g^4*x^4 + 4*(B*b^5*c^2*d^3 - 5*B*a*b^4*c*d^4 + (1 
5*A + 4*B)*a^2*b^3*d^5)*g^4*x^3 - 6*(B*b^5*c^3*d^2 - 5*B*a*b^4*c^2*d^3 + 1 
0*B*a^2*b^3*c*d^4 - 2*(5*A + 3*B)*a^3*b^2*d^5)*g^4*x^2 + 6*(2*B*b^5*c^4*d 
- 10*B*a*b^4*c^3*d^2 + 20*B*a^2*b^3*c^2*d^3 - 20*B*a^3*b^2*c*d^4 + (5*A + 
8*B)*a^4*b*d^5)*g^4*x - 12*(B*b^5*c^5 - 5*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3 
*d^2 - 10*B*a^3*b^2*c^2*d^3 + 5*B*a^4*b*c*d^4)*g^4*log(d*x + c) + 6*(B*b^5 
*d^5*g^4*x^5 + 5*B*a*b^4*d^5*g^4*x^4 + 10*B*a^2*b^3*d^5*g^4*x^3 + 10*B*a^3 
*b^2*d^5*g^4*x^2 + 5*B*a^4*b*d^5*g^4*x)*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*d^5)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 998 vs. \(2 (163) = 326\).

Time = 3.34 (sec) , antiderivative size = 998, normalized size of antiderivative = 5.48 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx =\text {Too large to display} \] Input:

integrate((b*g*x+a*g)**4*(A+B*ln(e*(b*x+a)**2/(d*x+c)**2)),x)
 

Output:

A*b**4*g**4*x**5/5 + 2*B*a**5*g**4*log(x + (2*B*a**6*d**5*g**4/b + 10*B*a* 
*5*c*d**4*g**4 - 20*B*a**4*b*c**2*d**3*g**4 + 20*B*a**3*b**2*c**3*d**2*g** 
4 - 10*B*a**2*b**3*c**4*d*g**4 + 2*B*a*b**4*c**5*g**4)/(2*B*a**5*d**5*g**4 
 + 10*B*a**4*b*c*d**4*g**4 - 20*B*a**3*b**2*c**2*d**3*g**4 + 20*B*a**2*b** 
3*c**3*d**2*g**4 - 10*B*a*b**4*c**4*d*g**4 + 2*B*b**5*c**5*g**4))/(5*b) - 
2*B*c*g**4*(5*a**4*d**4 - 10*a**3*b*c*d**3 + 10*a**2*b**2*c**2*d**2 - 5*a* 
b**3*c**3*d + b**4*c**4)*log(x + (12*B*a**5*c*d**4*g**4 - 20*B*a**4*b*c**2 
*d**3*g**4 + 20*B*a**3*b**2*c**3*d**2*g**4 - 10*B*a**2*b**3*c**4*d*g**4 + 
2*B*a*b**4*c**5*g**4 - 2*B*a*c*g**4*(5*a**4*d**4 - 10*a**3*b*c*d**3 + 10*a 
**2*b**2*c**2*d**2 - 5*a*b**3*c**3*d + b**4*c**4) + 2*B*b*c**2*g**4*(5*a** 
4*d**4 - 10*a**3*b*c*d**3 + 10*a**2*b**2*c**2*d**2 - 5*a*b**3*c**3*d + b** 
4*c**4)/d)/(2*B*a**5*d**5*g**4 + 10*B*a**4*b*c*d**4*g**4 - 20*B*a**3*b**2* 
c**2*d**3*g**4 + 20*B*a**2*b**3*c**3*d**2*g**4 - 10*B*a*b**4*c**4*d*g**4 + 
 2*B*b**5*c**5*g**4))/(5*d**5) + x**4*(A*a*b**3*g**4 + B*a*b**3*g**4/10 - 
B*b**4*c*g**4/(10*d)) + x**3*(2*A*a**2*b**2*g**4 + 8*B*a**2*b**2*g**4/15 - 
 2*B*a*b**3*c*g**4/(3*d) + 2*B*b**4*c**2*g**4/(15*d**2)) + x**2*(2*A*a**3* 
b*g**4 + 6*B*a**3*b*g**4/5 - 2*B*a**2*b**2*c*g**4/d + B*a*b**3*c**2*g**4/d 
**2 - B*b**4*c**3*g**4/(5*d**3)) + x*(A*a**4*g**4 + 8*B*a**4*g**4/5 - 4*B* 
a**3*b*c*g**4/d + 4*B*a**2*b**2*c**2*g**4/d**2 - 2*B*a*b**3*c**3*g**4/d**3 
 + 2*B*b**4*c**4*g**4/(5*d**4)) + (B*a**4*g**4*x + 2*B*a**3*b*g**4*x**2...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 885 vs. \(2 (170) = 340\).

Time = 0.08 (sec) , antiderivative size = 885, normalized size of antiderivative = 4.86 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx =\text {Too large to display} \] Input:

integrate((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)^2/(d*x+c)^2)),x, algorithm="max 
ima")
 

Output:

1/5*A*b^4*g^4*x^5 + A*a*b^3*g^4*x^4 + 2*A*a^2*b^2*g^4*x^3 + 2*A*a^3*b*g^4* 
x^2 + (x*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)) + 2*a*log(b*x + a)/b - 2*c 
*log(d*x + c)/d)*B*a^4*g^4 + 2*(x^2*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)) 
 - 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^3*b*g^4 + 2*(x^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)) + 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^2*b^2*g^4 + 1/3*(3*x^4*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)) - 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*a*b^3*g^4 + 1/30*(6*x^5*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)) + 12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3* 
(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c 
^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*B*b^4*g^4 + A 
*a^4*g^4*x
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (170) = 340\).

Time = 37.13 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.69 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {1}{5} \, A b^{4} g^{4} x^{5} + \frac {2 \, B a^{5} g^{4} \log \left (b x + a\right )}{5 \, b} - \frac {{\left (B b^{4} c g^{4} - 10 \, A a b^{3} d g^{4} - B a b^{3} d g^{4}\right )} x^{4}}{10 \, d} + \frac {2 \, {\left (B b^{4} c^{2} g^{4} - 5 \, B a b^{3} c d g^{4} + 15 \, A a^{2} b^{2} d^{2} g^{4} + 4 \, B a^{2} b^{2} d^{2} g^{4}\right )} x^{3}}{15 \, d^{2}} + \frac {1}{5} \, {\left (B b^{4} g^{4} x^{5} + 5 \, B a b^{3} g^{4} x^{4} + 10 \, B a^{2} b^{2} g^{4} x^{3} + 10 \, B a^{3} b g^{4} x^{2} + 5 \, B a^{4} g^{4} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {{\left (B b^{4} c^{3} g^{4} - 5 \, B a b^{3} c^{2} d g^{4} + 10 \, B a^{2} b^{2} c d^{2} g^{4} - 10 \, A a^{3} b d^{3} g^{4} - 6 \, B a^{3} b d^{3} g^{4}\right )} x^{2}}{5 \, d^{3}} + \frac {{\left (2 \, B b^{4} c^{4} g^{4} - 10 \, B a b^{3} c^{3} d g^{4} + 20 \, B a^{2} b^{2} c^{2} d^{2} g^{4} - 20 \, B a^{3} b c d^{3} g^{4} + 5 \, A a^{4} d^{4} g^{4} + 8 \, B a^{4} d^{4} g^{4}\right )} x}{5 \, d^{4}} - \frac {2 \, {\left (B b^{4} c^{5} g^{4} - 5 \, B a b^{3} c^{4} d g^{4} + 10 \, B a^{2} b^{2} c^{3} d^{2} g^{4} - 10 \, B a^{3} b c^{2} d^{3} g^{4} + 5 \, B a^{4} c d^{4} g^{4}\right )} \log \left (-d x - c\right )}{5 \, d^{5}} \] Input:

integrate((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)^2/(d*x+c)^2)),x, algorithm="gia 
c")
 

Output:

1/5*A*b^4*g^4*x^5 + 2/5*B*a^5*g^4*log(b*x + a)/b - 1/10*(B*b^4*c*g^4 - 10* 
A*a*b^3*d*g^4 - B*a*b^3*d*g^4)*x^4/d + 2/15*(B*b^4*c^2*g^4 - 5*B*a*b^3*c*d 
*g^4 + 15*A*a^2*b^2*d^2*g^4 + 4*B*a^2*b^2*d^2*g^4)*x^3/d^2 + 1/5*(B*b^4*g^ 
4*x^5 + 5*B*a*b^3*g^4*x^4 + 10*B*a^2*b^2*g^4*x^3 + 10*B*a^3*b*g^4*x^2 + 5* 
B*a^4*g^4*x)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2) 
) - 1/5*(B*b^4*c^3*g^4 - 5*B*a*b^3*c^2*d*g^4 + 10*B*a^2*b^2*c*d^2*g^4 - 10 
*A*a^3*b*d^3*g^4 - 6*B*a^3*b*d^3*g^4)*x^2/d^3 + 1/5*(2*B*b^4*c^4*g^4 - 10* 
B*a*b^3*c^3*d*g^4 + 20*B*a^2*b^2*c^2*d^2*g^4 - 20*B*a^3*b*c*d^3*g^4 + 5*A* 
a^4*d^4*g^4 + 8*B*a^4*d^4*g^4)*x/d^4 - 2/5*(B*b^4*c^5*g^4 - 5*B*a*b^3*c^4* 
d*g^4 + 10*B*a^2*b^2*c^3*d^2*g^4 - 10*B*a^3*b*c^2*d^3*g^4 + 5*B*a^4*c*d^4* 
g^4)*log(-d*x - c)/d^5
 

Mupad [B] (verification not implemented)

Time = 26.03 (sec) , antiderivative size = 1025, normalized size of antiderivative = 5.63 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx =\text {Too large to display} \] Input:

int((a*g + b*g*x)^4*(A + B*log((e*(a + b*x)^2)/(c + d*x)^2)),x)
 

Output:

x^2*(((5*a*d + 5*b*c)*((((b^3*g^4*(25*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c) 
)/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*(5*a*d + 5*b*c))/(5*b*d) - (a 
*b^2*g^4*(10*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/d + (A*a*b^3*c*g^4)/d)) 
/(10*b*d) + (a^2*b*g^4*(5*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/d - (a*c*( 
(b^3*g^4*(25*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/(5*d) - (A*b^3*g^4*(5*a 
*d + 5*b*c))/(5*d)))/(2*b*d)) - x^3*((((b^3*g^4*(25*A*a*d + 5*A*b*c + 2*B* 
a*d - 2*B*b*c))/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*(5*a*d + 5*b*c) 
)/(15*b*d) - (a*b^2*g^4*(10*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/(3*d) + 
(A*a*b^3*c*g^4)/(3*d)) + x*((a^3*g^4*(5*A*a*d + 10*A*b*c + 4*B*a*d - 4*B*b 
*c))/d - ((5*a*d + 5*b*c)*(((5*a*d + 5*b*c)*((((b^3*g^4*(25*A*a*d + 5*A*b* 
c + 2*B*a*d - 2*B*b*c))/(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d))*(5*a*d 
+ 5*b*c))/(5*b*d) - (a*b^2*g^4*(10*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/d 
 + (A*a*b^3*c*g^4)/d))/(5*b*d) + (2*a^2*b*g^4*(5*A*a*d + 5*A*b*c + 2*B*a*d 
 - 2*B*b*c))/d - (a*c*((b^3*g^4*(25*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/ 
(5*d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d)))/(b*d)))/(5*b*d) + (a*c*((((b^3 
*g^4*(25*A*a*d + 5*A*b*c + 2*B*a*d - 2*B*b*c))/(5*d) - (A*b^3*g^4*(5*a*d + 
 5*b*c))/(5*d))*(5*a*d + 5*b*c))/(5*b*d) - (a*b^2*g^4*(10*A*a*d + 5*A*b*c 
+ 2*B*a*d - 2*B*b*c))/d + (A*a*b^3*c*g^4)/d))/(b*d)) + log((e*(a + b*x)^2) 
/(c + d*x)^2)*((B*b^4*g^4*x^5)/5 + B*a^4*g^4*x + 2*B*a^3*b*g^4*x^2 + B*a*b 
^3*g^4*x^4 + 2*B*a^2*b^2*g^4*x^3) + x^4*((b^3*g^4*(25*A*a*d + 5*A*b*c +...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 663, normalized size of antiderivative = 3.64 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {g^{4} \left (48 a^{4} b \,d^{5} x +36 a^{3} b^{2} d^{5} x^{2}+16 a^{2} b^{3} d^{5} x^{3}+3 a \,b^{4} d^{5} x^{4}+12 b^{5} c^{4} d x -6 b^{5} c^{3} d^{2} x^{2}+4 b^{5} c^{2} d^{3} x^{3}-3 b^{5} c \,d^{4} x^{4}+6 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a^{5} d^{5}+30 a^{5} d^{5} x +30 a \,b^{4} c^{2} d^{3} x^{2}-20 a \,b^{4} c \,d^{4} x^{3}+12 \,\mathrm {log}\left (d x +c \right ) a^{5} d^{5}-12 \,\mathrm {log}\left (d x +c \right ) b^{5} c^{5}+60 a^{4} b \,d^{5} x^{2}+60 a^{3} b^{2} d^{5} x^{3}+30 a^{2} b^{3} d^{5} x^{4}+6 a \,b^{4} d^{5} x^{5}-60 \,\mathrm {log}\left (d x +c \right ) a^{4} b c \,d^{4}+120 \,\mathrm {log}\left (d x +c \right ) a^{3} b^{2} c^{2} d^{3}-120 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} c^{3} d^{2}+60 \,\mathrm {log}\left (d x +c \right ) a \,b^{4} c^{4} d -120 a^{3} b^{2} c \,d^{4} x +120 a^{2} b^{3} c^{2} d^{3} x -60 a^{2} b^{3} c \,d^{4} x^{2}-60 a \,b^{4} c^{3} d^{2} x +30 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a^{4} b \,d^{5} x +60 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a^{3} b^{2} d^{5} x^{2}+60 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a^{2} b^{3} d^{5} x^{3}+30 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) a \,b^{4} d^{5} x^{4}+6 \,\mathrm {log}\left (\frac {b^{2} e \,x^{2}+2 a b e x +a^{2} e}{d^{2} x^{2}+2 c d x +c^{2}}\right ) b^{5} d^{5} x^{5}\right )}{30 d^{5}} \] Input:

int((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)^2/(d*x+c)^2)),x)
 

Output:

(g**4*(12*log(c + d*x)*a**5*d**5 - 60*log(c + d*x)*a**4*b*c*d**4 + 120*log 
(c + d*x)*a**3*b**2*c**2*d**3 - 120*log(c + d*x)*a**2*b**3*c**3*d**2 + 60* 
log(c + d*x)*a*b**4*c**4*d - 12*log(c + d*x)*b**5*c**5 + 6*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**5*d**5 + 30*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**4*b*d** 
5*x + 60*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**5*x**2 + 60*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**5*x**3 + 30*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*d**5*x**4 + 6*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*d**5*x**5 
 + 30*a**5*d**5*x + 60*a**4*b*d**5*x**2 + 48*a**4*b*d**5*x - 120*a**3*b**2 
*c*d**4*x + 60*a**3*b**2*d**5*x**3 + 36*a**3*b**2*d**5*x**2 + 120*a**2*b** 
3*c**2*d**3*x - 60*a**2*b**3*c*d**4*x**2 + 30*a**2*b**3*d**5*x**4 + 16*a** 
2*b**3*d**5*x**3 - 60*a*b**4*c**3*d**2*x + 30*a*b**4*c**2*d**3*x**2 - 20*a 
*b**4*c*d**4*x**3 + 6*a*b**4*d**5*x**5 + 3*a*b**4*d**5*x**4 + 12*b**5*c**4 
*d*x - 6*b**5*c**3*d**2*x**2 + 4*b**5*c**2*d**3*x**3 - 3*b**5*c*d**4*x**4) 
)/(30*d**5)