\(\int (a g+b g x)^4 (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [88]

Optimal result

Integrand size = 30, antiderivative size = 180 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {B (b c-a d)^4 g^4 x}{5 d^4}-\frac {B (b c-a d)^3 g^4 (a+b x)^2}{10 b d^3}+\frac {B (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}-\frac {B (b c-a d) g^4 (a+b x)^4}{20 b d}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 b}-\frac {B (b c-a d)^5 g^4 \log (c+d x)}{5 b d^5} \] Output:

1/5*B*(-a*d+b*c)^4*g^4*x/d^4-1/10*B*(-a*d+b*c)^3*g^4*(b*x+a)^2/b/d^3+1/15* 
B*(-a*d+b*c)^2*g^4*(b*x+a)^3/b/d^2-1/20*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)/(d*x+c)))/b-1/5*B*(-a*d+b*c)^5*g^4*ln(d*x 
+c)/b/d^5
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.79 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^4 \left ((a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-\frac {B (b c-a d) \left (-12 b d (b c-a d)^3 x+6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (-b c+a d) (a+b x)^3+3 d^4 (a+b x)^4+12 (b c-a d)^4 \log (c+d x)\right )}{12 d^5}\right )}{5 b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, 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.

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

\(\Big \downarrow \) 2948

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(g^4*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*b) - (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*L 
og[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. \(443\) vs. \(2(168)=336\).

Time = 1.20 (sec) , antiderivative size = 444, normalized size of antiderivative = 2.47

method	result	size
risch	\(\frac {g^{4} B \ln \left (d x +c \right ) a^{5}}{5 b}+g^{4} b^{3} A a \,x^{4}+\frac {g^{4} b^{3} B a \,x^{4}}{20}-\frac {g^{4} b^{4} B c \,x^{4}}{20 d}+2 g^{4} b^{2} A \,a^{2} x^{3}+\frac {4 g^{4} b^{2} B \,a^{2} x^{3}}{15}+\frac {g^{4} b^{4} B \,c^{2} x^{3}}{15 d^{2}}+2 g^{4} b A \,a^{3} x^{2}+\frac {3 g^{4} b B \,a^{3} x^{2}}{5}-\frac {g^{4} b^{4} B \,c^{3} x^{2}}{10 d^{3}}+g^{4} A \,a^{4} x -\frac {2 g^{4} b B \,a^{3} c x}{d}+\frac {2 g^{4} b^{2} B \,a^{2} c^{2} x}{d^{2}}-\frac {g^{4} b^{3} B a \,c^{3} x}{d^{3}}-\frac {2 g^{4} b^{2} B \ln \left (d x +c \right ) a^{2} c^{3}}{d^{3}}+\frac {g^{4} b^{3} B \ln \left (d x +c \right ) a \,c^{4}}{d^{4}}+\frac {2 g^{4} b B \ln \left (d x +c \right ) a^{3} c^{2}}{d^{2}}-\frac {g^{4} b^{3} B a c \,x^{3}}{3 d}-\frac {g^{4} b^{2} B \,a^{2} c \,x^{2}}{d}+\frac {g^{4} b^{3} B a \,c^{2} x^{2}}{2 d^{2}}+\frac {4 g^{4} B \,a^{4} x}{5}+\frac {g^{4} b^{4} B \,c^{4} x}{5 d^{4}}-\frac {g^{4} B \ln \left (d x +c \right ) a^{4} c}{d}-\frac {g^{4} b^{4} B \ln \left (d x +c \right ) c^{5}}{5 d^{5}}+\frac {\left (b x +a \right )^{5} g^{4} B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{5 b}+\frac {g^{4} b^{4} A \,x^{5}}{5}\)	\(444\)
parallelrisch	\(\frac {120 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{3} d^{5} g^{4}-20 B \,x^{3} a \,b^{4} c \,d^{4} g^{4}+120 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b^{2} d^{5} g^{4}+36 B \,a^{4} b c \,d^{4} g^{4}+60 B \,a^{3} b^{2} c^{2} d^{3} g^{4}-90 B \,a^{2} b^{3} c^{3} d^{2} g^{4}+54 B a \,b^{4} c^{4} d \,g^{4}+60 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} d^{5} g^{4}-12 B \,b^{5} g^{4} c^{5}-60 A \,a^{5} d^{5} g^{4}-180 A \,a^{4} b c \,d^{4} g^{4}-6 B \,x^{2} b^{5} c^{3} d^{2} g^{4}+60 A x \,a^{4} b \,d^{5} g^{4}+48 B x \,a^{4} b \,d^{5} g^{4}+12 B x \,b^{5} c^{4} d \,g^{4}-48 B \,a^{5} d^{5} g^{4}-60 B \,x^{2} a^{2} b^{3} c \,d^{4} g^{4}+12 A \,x^{5} b^{5} d^{5} g^{4}+12 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{5} g^{4}+12 B \ln \left (b x +a \right ) a^{5} d^{5} g^{4}-12 B \ln \left (b x +a \right ) b^{5} c^{5} g^{4}-3 B \,x^{4} b^{5} c \,d^{4} g^{4}+120 A \,x^{3} a^{2} b^{3} d^{5} g^{4}+16 B \,x^{3} a^{2} b^{3} d^{5} g^{4}+4 B \,x^{3} b^{5} c^{2} d^{3} g^{4}+12 B \,x^{5} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} d^{5} g^{4}+60 A \,x^{4} a \,b^{4} d^{5} g^{4}+3 B \,x^{4} a \,b^{4} d^{5} g^{4}+120 A \,x^{2} a^{3} b^{2} d^{5} g^{4}+36 B \,x^{2} a^{3} b^{2} d^{5} g^{4}+30 B \,x^{2} a \,b^{4} c^{2} d^{3} g^{4}+60 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} b \,d^{5} g^{4}-60 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{4} c^{4} d \,g^{4}-60 B \ln \left (b x +a \right ) a^{4} b c \,d^{4} g^{4}+120 B \ln \left (b x +a \right ) a^{3} b^{2} c^{2} d^{3} g^{4}-120 B \ln \left (b x +a \right ) a^{2} b^{3} c^{3} d^{2} g^{4}+60 B \ln \left (b x +a \right ) a \,b^{4} c^{4} d \,g^{4}-120 B x \,a^{3} b^{2} c \,d^{4} g^{4}+120 B x \,a^{2} b^{3} c^{2} d^{3} g^{4}-60 B x a \,b^{4} c^{3} d^{2} g^{4}+60 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} b c \,d^{4} g^{4}-120 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b^{2} c^{2} d^{3} g^{4}+120 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{3} c^{3} d^{2} g^{4}}{60 d^{5} b}\)	\(876\)
parts	\(\text {Expression too large to display}\)	\(1900\)
derivativedivides	\(\text {Expression too large to display}\)	\(1975\)
default	\(\text {Expression too large to display}\)	\(1975\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (168) = 336\).

Time = 0.12 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.39 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {12 \, A b^{5} d^{5} g^{4} x^{5} + 12 \, B a^{5} d^{5} g^{4} \log \left (b x + a\right ) - 3 \, {\left (B b^{5} c d^{4} - {\left (20 \, A + B\right )} a b^{4} d^{5}\right )} g^{4} x^{4} + 4 \, {\left (B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} + 2 \, {\left (15 \, A + 2 \, B\right )} a^{2} b^{3} d^{5}\right )} g^{4} x^{3} - 6 \, {\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} + 10 \, B a^{2} b^{3} c d^{4} - 2 \, {\left (10 \, A + 3 \, B\right )} a^{3} b^{2} d^{5}\right )} g^{4} x^{2} + 12 \, {\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 10 \, B a^{3} b^{2} c d^{4} + {\left (5 \, A + 4 \, B\right )} a^{4} b d^{5}\right )} g^{4} x - 12 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4}\right )} g^{4} \log \left (d x + c\right ) + 12 \, {\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B a b^{4} d^{5} g^{4} x^{4} + 10 \, B a^{2} b^{3} d^{5} g^{4} x^{3} + 10 \, B a^{3} b^{2} d^{5} g^{4} x^{2} + 5 \, B a^{4} b d^{5} g^{4} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{60 \, b d^{5}} \] Input:

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

Output:

1/60*(12*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 - (20*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 + 2 
*(15*A + 2*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 
+ 10*B*a^2*b^3*c*d^4 - 2*(10*A + 3*B)*a^3*b^2*d^5)*g^4*x^2 + 12*(B*b^5*c^4 
*d - 5*B*a*b^4*c^3*d^2 + 10*B*a^2*b^3*c^2*d^3 - 10*B*a^3*b^2*c*d^4 + (5*A 
+ 4*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) + 12*(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*e*x + a*e)/(d*x + c)))/( 
b*d^5)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 969 vs. \(2 (155) = 310\).

Time = 3.19 (sec) , antiderivative size = 969, normalized size of antiderivative = 5.38 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 623 vs. \(2 (168) = 336\).

Time = 0.05 (sec) , antiderivative size = 623, normalized size of antiderivative = 3.46 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {1}{5} \, A b^{4} g^{4} x^{5} + A a b^{3} g^{4} x^{4} + 2 \, A a^{2} b^{2} g^{4} x^{3} + 2 \, A a^{3} b g^{4} x^{2} + {\left (x \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B a^{4} g^{4} + 2 \, {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a^{3} b g^{4} + {\left (2 \, x^{3} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B a^{2} b^{2} g^{4} + \frac {1}{6} \, {\left (6 \, x^{4} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} + \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} - \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} B a b^{3} g^{4} + \frac {1}{60} \, {\left (12 \, x^{5} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {12 \, a^{5} \log \left (b x + a\right )}{b^{5}} - \frac {12 \, c^{5} \log \left (d x + c\right )}{d^{5}} - \frac {3 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} B b^{4} g^{4} + A a^{4} g^{4} x \] Input:

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

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*e*x/(d*x + c) + a*e/(d*x + c)) + a*log(b*x + a)/b - c*log(d 
*x + c)/d)*B*a^4*g^4 + 2*(x^2*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*l 
og(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/(b*d))*B*a^3*b*g^4 
+ (2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 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 + 1/6*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c 
)) - 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 + 1/60*(12*x^5*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 
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. 4036 vs. \(2 (168) = 336\).

Time = 0.32 (sec) , antiderivative size = 4036, normalized size of antiderivative = 22.42 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/60*(12*(B*b^10*c^6*e^6*g^4 - 6*B*a*b^9*c^5*d*e^6*g^4 + 15*B*a^2*b^8*c^4* 
d^2*e^6*g^4 - 20*B*a^3*b^7*c^3*d^3*e^6*g^4 + 15*B*a^4*b^6*c^2*d^4*e^6*g^4 
- 6*B*a^5*b^5*c*d^5*e^6*g^4 + B*a^6*b^4*d^6*e^6*g^4 - 5*(b*e*x + a*e)*B*b^ 
9*c^6*d*e^5*g^4/(d*x + c) + 30*(b*e*x + a*e)*B*a*b^8*c^5*d^2*e^5*g^4/(d*x 
+ c) - 75*(b*e*x + a*e)*B*a^2*b^7*c^4*d^3*e^5*g^4/(d*x + c) + 100*(b*e*x + 
 a*e)*B*a^3*b^6*c^3*d^4*e^5*g^4/(d*x + c) - 75*(b*e*x + a*e)*B*a^4*b^5*c^2 
*d^5*e^5*g^4/(d*x + c) + 30*(b*e*x + a*e)*B*a^5*b^4*c*d^6*e^5*g^4/(d*x + c 
) - 5*(b*e*x + a*e)*B*a^6*b^3*d^7*e^5*g^4/(d*x + c) + 10*(b*e*x + a*e)^2*B 
*b^8*c^6*d^2*e^4*g^4/(d*x + c)^2 - 60*(b*e*x + a*e)^2*B*a*b^7*c^5*d^3*e^4* 
g^4/(d*x + c)^2 + 150*(b*e*x + a*e)^2*B*a^2*b^6*c^4*d^4*e^4*g^4/(d*x + c)^ 
2 - 200*(b*e*x + a*e)^2*B*a^3*b^5*c^3*d^5*e^4*g^4/(d*x + c)^2 + 150*(b*e*x 
 + a*e)^2*B*a^4*b^4*c^2*d^6*e^4*g^4/(d*x + c)^2 - 60*(b*e*x + a*e)^2*B*a^5 
*b^3*c*d^7*e^4*g^4/(d*x + c)^2 + 10*(b*e*x + a*e)^2*B*a^6*b^2*d^8*e^4*g^4/ 
(d*x + c)^2 - 10*(b*e*x + a*e)^3*B*b^7*c^6*d^3*e^3*g^4/(d*x + c)^3 + 60*(b 
*e*x + a*e)^3*B*a*b^6*c^5*d^4*e^3*g^4/(d*x + c)^3 - 150*(b*e*x + a*e)^3*B* 
a^2*b^5*c^4*d^5*e^3*g^4/(d*x + c)^3 + 200*(b*e*x + a*e)^3*B*a^3*b^4*c^3*d^ 
6*e^3*g^4/(d*x + c)^3 - 150*(b*e*x + a*e)^3*B*a^4*b^3*c^2*d^7*e^3*g^4/(d*x 
 + c)^3 + 60*(b*e*x + a*e)^3*B*a^5*b^2*c*d^8*e^3*g^4/(d*x + c)^3 - 10*(b*e 
*x + a*e)^3*B*a^6*b*d^9*e^3*g^4/(d*x + c)^3 + 5*(b*e*x + a*e)^4*B*b^6*c^6* 
d^4*e^2*g^4/(d*x + c)^4 - 30*(b*e*x + a*e)^4*B*a*b^5*c^5*d^5*e^2*g^4/(d...
 

Mupad [B] (verification not implemented)

Time = 27.42 (sec) , antiderivative size = 1009, normalized size of antiderivative = 5.61 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

log((e*(a + b*x))/(c + d*x))*((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^3*((((b^3*g^4*(25*A*a 
*d + 5*A*b*c + B*a*d - 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 + B*a*d - B*b*c 
))/(3*d) + (A*a*b^3*c*g^4)/(3*d)) + x^2*(((5*a*d + 5*b*c)*((((b^3*g^4*(25* 
A*a*d + 5*A*b*c + B*a*d - 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 + B*a*d - 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 + B* 
a*d - B*b*c))/d - (a*c*((b^3*g^4*(25*A*a*d + 5*A*b*c + B*a*d - B*b*c))/(5* 
d) - (A*b^3*g^4*(5*a*d + 5*b*c))/(5*d)))/(2*b*d)) + x*((a^3*g^4*(5*A*a*d + 
 10*A*b*c + 2*B*a*d - 2*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 + B*a*d - 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 
+ B*a*d - 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 + B*a*d - B*b*c))/d - (a*c*((b^3*g^4*(25*A*a*d + 5*A*b*c + B*a*d 
 - 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 + B*a*d - 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 + B*a*d - B*b*c))/d + (A*a*b^3*c*g^4)/d))/(b*d)) + x^4*((b^3*g^4*(2 
5*A*a*d + 5*A*b*c + B*a*d - B*b*c))/(20*d) - (A*b^3*g^4*(5*a*d + 5*b*c)...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 525, normalized size of antiderivative = 2.92 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^{4} \left (12 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) b^{5} d^{5} x^{5}+48 a^{4} b \,d^{5} x +36 a^{3} b^{2} d^{5} x^{2}+16 a^{2} b^{3} d^{5} x^{3}+3 a \,b^{4} d^{5} x^{4}+12 b^{5} c^{4} d x -6 b^{5} c^{3} d^{2} x^{2}+4 b^{5} c^{2} d^{3} x^{3}-3 b^{5} c \,d^{4} x^{4}+60 a^{5} d^{5} x +30 a \,b^{4} c^{2} d^{3} x^{2}-20 a \,b^{4} c \,d^{4} x^{3}+12 \,\mathrm {log}\left (d x +c \right ) a^{5} d^{5}-12 \,\mathrm {log}\left (d x +c \right ) b^{5} c^{5}+12 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{5} d^{5}+120 a^{4} b \,d^{5} x^{2}+120 a^{3} b^{2} d^{5} x^{3}+60 a^{2} b^{3} d^{5} x^{4}+12 a \,b^{4} d^{5} x^{5}-60 \,\mathrm {log}\left (d x +c \right ) a^{4} b c \,d^{4}+120 \,\mathrm {log}\left (d x +c \right ) a^{3} b^{2} c^{2} d^{3}-120 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{3} c^{3} d^{2}+60 \,\mathrm {log}\left (d x +c \right ) a \,b^{4} c^{4} d +60 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{4} b \,d^{5} x +120 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{3} b^{2} d^{5} x^{2}+120 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a^{2} b^{3} d^{5} x^{3}+60 \,\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) a \,b^{4} d^{5} x^{4}-120 a^{3} b^{2} c \,d^{4} x +120 a^{2} b^{3} c^{2} d^{3} x -60 a^{2} b^{3} c \,d^{4} x^{2}-60 a \,b^{4} c^{3} d^{2} x \right )}{60 d^{5}} \] Input:

int((b*g*x+a*g)^4*(A+B*log(e*(b*x+a)/(d*x+c))),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 + 12*log((a*e + b*e 
*x)/(c + d*x))*a**5*d**5 + 60*log((a*e + b*e*x)/(c + d*x))*a**4*b*d**5*x + 
 120*log((a*e + b*e*x)/(c + d*x))*a**3*b**2*d**5*x**2 + 120*log((a*e + b*e 
*x)/(c + d*x))*a**2*b**3*d**5*x**3 + 60*log((a*e + b*e*x)/(c + d*x))*a*b** 
4*d**5*x**4 + 12*log((a*e + b*e*x)/(c + d*x))*b**5*d**5*x**5 + 60*a**5*d** 
5*x + 120*a**4*b*d**5*x**2 + 48*a**4*b*d**5*x - 120*a**3*b**2*c*d**4*x + 1 
20*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 + 60*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 + 12*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))/(60*d**5)