\(\int (c g+d g x)^4 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [29]

Optimal result

Integrand size = 33, antiderivative size = 188 \[ \int (c g+d g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {B (b c-a d)^4 g^4 n x}{5 b^4}-\frac {B (b c-a d)^3 g^4 n (c+d x)^2}{10 b^3 d}-\frac {B (b c-a d)^2 g^4 n (c+d x)^3}{15 b^2 d}-\frac {B (b c-a d) g^4 n (c+d x)^4}{20 b d}-\frac {B (b c-a d)^5 g^4 n \log (a+b x)}{5 b^5 d}+\frac {g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.78 \[ \int (c g+d g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^4 \left (-\frac {B (b c-a d) n \left (12 b d (b c-a d)^3 x+6 b^2 (b c-a d)^2 (c+d x)^2+4 b^3 (b c-a d) (c+d x)^3+3 b^4 (c+d x)^4+12 (b c-a d)^4 \log (a+b x)\right )}{12 b^5}+(c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{5 d} \] Input:

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

Output:

(g^4*(-1/12*(B*(b*c - a*d)*n*(12*b*d*(b*c - a*d)^3*x + 6*b^2*(b*c - a*d)^2 
*(c + d*x)^2 + 4*b^3*(b*c - a*d)*(c + d*x)^3 + 3*b^4*(c + d*x)^4 + 12*(b*c 
 - a*d)^4*Log[a + b*x]))/b^5 + (c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d* 
x))^n])))/(5*d)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2947, 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.

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

\(\Big \downarrow \) 2947

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 2947

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + 
 B*Log[e*((a + b*x)/(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] /; Free 
Q[{a, b, c, d, e, f, g, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] 
&& NeQ[m, -2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(863\) vs. \(2(176)=352\).

Time = 11.58 (sec) , antiderivative size = 864, normalized size of antiderivative = 4.60

method	result	size
parallelrisch	\(\frac {60 B \,x^{4} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c \,d^{4} g^{4} n +60 A x \,b^{5} c^{4} d \,g^{4} n +120 A \,x^{3} b^{5} c^{2} d^{3} g^{4} n -60 A \,b^{5} c^{5} g^{4} n -54 B \,a^{4} b c \,d^{4} g^{4} n^{2}+90 B \,a^{3} b^{2} c^{2} d^{3} g^{4} n^{2}-60 B \,a^{2} b^{3} c^{3} d^{2} g^{4} n^{2}-36 B a \,b^{4} c^{4} d \,g^{4} n^{2}+12 B \,x^{5} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} d^{5} g^{4} n +3 B \,x^{4} a \,b^{4} d^{5} g^{4} n^{2}-3 B \,x^{4} b^{5} c \,d^{4} g^{4} n^{2}-4 B \,x^{3} a^{2} b^{3} d^{5} g^{4} n^{2}+120 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c^{2} d^{3} g^{4} n +120 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c^{3} d^{2} g^{4} n +60 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c^{4} d \,g^{4} n -180 A a \,b^{4} c^{4} d \,g^{4} n +60 A \,x^{4} b^{5} c \,d^{4} g^{4} n +120 A \,x^{2} b^{5} c^{3} d^{2} g^{4} n +12 A \,x^{5} b^{5} d^{5} g^{4} n +12 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c^{5} g^{4} n +12 B \ln \left (b x +a \right ) a^{5} d^{5} g^{4} n^{2}-12 B \ln \left (b x +a \right ) b^{5} c^{5} g^{4} n^{2}-16 B \,x^{3} b^{5} c^{2} d^{3} g^{4} n^{2}+6 B \,x^{2} a^{3} b^{2} d^{5} g^{4} n^{2}-36 B \,x^{2} b^{5} c^{3} d^{2} g^{4} n^{2}-12 B x \,a^{4} b \,d^{5} g^{4} n^{2}-48 B x \,b^{5} c^{4} d \,g^{4} n^{2}+20 B \,x^{3} a \,b^{4} c \,d^{4} g^{4} n^{2}-30 B \,x^{2} a^{2} b^{3} c \,d^{4} g^{4} n^{2}+60 B \,x^{2} a \,b^{4} c^{2} d^{3} g^{4} n^{2}+60 B x \,a^{3} b^{2} c \,d^{4} g^{4} n^{2}-120 B x \,a^{2} b^{3} c^{2} d^{3} g^{4} n^{2}+12 B \,a^{5} d^{5} g^{4} n^{2}+48 B \,b^{5} c^{5} g^{4} n^{2}+120 B x a \,b^{4} c^{3} d^{2} g^{4} n^{2}-60 B \ln \left (b x +a \right ) a^{4} b c \,d^{4} g^{4} n^{2}+120 B \ln \left (b x +a \right ) a^{3} b^{2} c^{2} d^{3} g^{4} n^{2}-120 B \ln \left (b x +a \right ) a^{2} b^{3} c^{3} d^{2} g^{4} n^{2}+60 B \ln \left (b x +a \right ) a \,b^{4} c^{4} d \,g^{4} n^{2}}{60 n \,b^{5} d}\)	\(864\)

Input:

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

Output:

1/60*(60*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*b^5*c*d^4*g^4*n+60*A*x*b^5*c^4*d* 
g^4*n+120*A*x^3*b^5*c^2*d^3*g^4*n-60*A*b^5*c^5*g^4*n-54*B*a^4*b*c*d^4*g^4* 
n^2+90*B*a^3*b^2*c^2*d^3*g^4*n^2-60*B*a^2*b^3*c^3*d^2*g^4*n^2-36*B*a*b^4*c 
^4*d*g^4*n^2+12*B*x^5*ln(e*((b*x+a)/(d*x+c))^n)*b^5*d^5*g^4*n+3*B*x^4*a*b^ 
4*d^5*g^4*n^2-3*B*x^4*b^5*c*d^4*g^4*n^2-4*B*x^3*a^2*b^3*d^5*g^4*n^2+120*B* 
x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^5*c^2*d^3*g^4*n+120*B*x^2*ln(e*((b*x+a)/(d 
*x+c))^n)*b^5*c^3*d^2*g^4*n+60*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^5*c^4*d*g^4 
*n-180*A*a*b^4*c^4*d*g^4*n+60*A*x^4*b^5*c*d^4*g^4*n+120*A*x^2*b^5*c^3*d^2* 
g^4*n+12*A*x^5*b^5*d^5*g^4*n+12*B*ln(e*((b*x+a)/(d*x+c))^n)*b^5*c^5*g^4*n+ 
12*B*ln(b*x+a)*a^5*d^5*g^4*n^2-12*B*ln(b*x+a)*b^5*c^5*g^4*n^2-16*B*x^3*b^5 
*c^2*d^3*g^4*n^2+6*B*x^2*a^3*b^2*d^5*g^4*n^2-36*B*x^2*b^5*c^3*d^2*g^4*n^2- 
12*B*x*a^4*b*d^5*g^4*n^2-48*B*x*b^5*c^4*d*g^4*n^2+20*B*x^3*a*b^4*c*d^4*g^4 
*n^2-30*B*x^2*a^2*b^3*c*d^4*g^4*n^2+60*B*x^2*a*b^4*c^2*d^3*g^4*n^2+60*B*x* 
a^3*b^2*c*d^4*g^4*n^2-120*B*x*a^2*b^3*c^2*d^3*g^4*n^2+12*B*a^5*d^5*g^4*n^2 
+48*B*b^5*c^5*g^4*n^2+120*B*x*a*b^4*c^3*d^2*g^4*n^2-60*B*ln(b*x+a)*a^4*b*c 
*d^4*g^4*n^2+120*B*ln(b*x+a)*a^3*b^2*c^2*d^3*g^4*n^2-120*B*ln(b*x+a)*a^2*b 
^3*c^3*d^2*g^4*n^2+60*B*ln(b*x+a)*a*b^4*c^4*d*g^4*n^2)/n/b^5/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (176) = 352\).

Time = 0.21 (sec) , antiderivative size = 572, normalized size of antiderivative = 3.04 \[ \int (c g+d g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {12 \, A b^{5} d^{5} g^{4} x^{5} - 12 \, B b^{5} c^{5} g^{4} n \log \left (d x + c\right ) + 12 \, {\left (5 \, B a b^{4} c^{4} d - 10 \, B a^{2} b^{3} c^{3} d^{2} + 10 \, B a^{3} b^{2} c^{2} d^{3} - 5 \, B a^{4} b c d^{4} + B a^{5} d^{5}\right )} g^{4} n \log \left (b x + a\right ) + 3 \, {\left (20 \, A b^{5} c d^{4} g^{4} - {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{4} n\right )} x^{4} + 4 \, {\left (30 \, A b^{5} c^{2} d^{3} g^{4} - {\left (4 \, B b^{5} c^{2} d^{3} - 5 \, B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{4} n\right )} x^{3} + 6 \, {\left (20 \, A b^{5} c^{3} d^{2} g^{4} - {\left (6 \, B b^{5} c^{3} d^{2} - 10 \, B a b^{4} c^{2} d^{3} + 5 \, B a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} g^{4} n\right )} x^{2} + 12 \, {\left (5 \, A b^{5} c^{4} d g^{4} - {\left (4 \, B b^{5} c^{4} d - 10 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 5 \, B a^{3} b^{2} c d^{4} + B a^{4} b d^{5}\right )} g^{4} n\right )} x + 12 \, {\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B b^{5} c d^{4} g^{4} x^{4} + 10 \, B b^{5} c^{2} d^{3} g^{4} x^{3} + 10 \, B b^{5} c^{3} d^{2} g^{4} x^{2} + 5 \, B b^{5} c^{4} d g^{4} x\right )} \log \left (e\right ) + 12 \, {\left (B b^{5} d^{5} g^{4} n x^{5} + 5 \, B b^{5} c d^{4} g^{4} n x^{4} + 10 \, B b^{5} c^{2} d^{3} g^{4} n x^{3} + 10 \, B b^{5} c^{3} d^{2} g^{4} n x^{2} + 5 \, B b^{5} c^{4} d g^{4} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{60 \, b^{5} d} \] Input:

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

Output:

1/60*(12*A*b^5*d^5*g^4*x^5 - 12*B*b^5*c^5*g^4*n*log(d*x + c) + 12*(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 + 
 B*a^5*d^5)*g^4*n*log(b*x + a) + 3*(20*A*b^5*c*d^4*g^4 - (B*b^5*c*d^4 - B* 
a*b^4*d^5)*g^4*n)*x^4 + 4*(30*A*b^5*c^2*d^3*g^4 - (4*B*b^5*c^2*d^3 - 5*B*a 
*b^4*c*d^4 + B*a^2*b^3*d^5)*g^4*n)*x^3 + 6*(20*A*b^5*c^3*d^2*g^4 - (6*B*b^ 
5*c^3*d^2 - 10*B*a*b^4*c^2*d^3 + 5*B*a^2*b^3*c*d^4 - B*a^3*b^2*d^5)*g^4*n) 
*x^2 + 12*(5*A*b^5*c^4*d*g^4 - (4*B*b^5*c^4*d - 10*B*a*b^4*c^3*d^2 + 10*B* 
a^2*b^3*c^2*d^3 - 5*B*a^3*b^2*c*d^4 + B*a^4*b*d^5)*g^4*n)*x + 12*(B*b^5*d^ 
5*g^4*x^5 + 5*B*b^5*c*d^4*g^4*x^4 + 10*B*b^5*c^2*d^3*g^4*x^3 + 10*B*b^5*c^ 
3*d^2*g^4*x^2 + 5*B*b^5*c^4*d*g^4*x)*log(e) + 12*(B*b^5*d^5*g^4*n*x^5 + 5* 
B*b^5*c*d^4*g^4*n*x^4 + 10*B*b^5*c^2*d^3*g^4*n*x^3 + 10*B*b^5*c^3*d^2*g^4* 
n*x^2 + 5*B*b^5*c^4*d*g^4*n*x)*log((b*x + a)/(d*x + c)))/(b^5*d)
 

Sympy [F(-1)]

Timed out. \[ \int (c g+d g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 676 vs. \(2 (176) = 352\).

Time = 0.05 (sec) , antiderivative size = 676, normalized size of antiderivative = 3.60 \[ \int (c g+d g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {1}{5} \, B d^{4} g^{4} x^{5} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + \frac {1}{5} \, A d^{4} g^{4} x^{5} + B c d^{3} g^{4} x^{4} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A c d^{3} g^{4} x^{4} + 2 \, B c^{2} d^{2} g^{4} x^{3} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + 2 \, A c^{2} d^{2} g^{4} x^{3} + 2 \, B c^{3} d g^{4} x^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + 2 \, A c^{3} d g^{4} x^{2} + \frac {1}{60} \, B d^{4} g^{4} n {\left (\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 )} - \frac {1}{6} \, B c d^{3} g^{4} n {\left (\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 c^{2} d^{2} g^{4} n {\left (\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 )} - 2 \, B c^{3} d g^{4} n {\left (\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 c^{4} g^{4} n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + B c^{4} g^{4} x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A c^{4} g^{4} x \] Input:

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

Output:

1/5*B*d^4*g^4*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/5*A*d^4*g^4*x 
^5 + B*c*d^3*g^4*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*c*d^3*g^4* 
x^4 + 2*B*c^2*d^2*g^4*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*A*c^2 
*d^2*g^4*x^3 + 2*B*c^3*d*g^4*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 
2*A*c^3*d*g^4*x^2 + 1/60*B*d^4*g^4*n*(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)) - 1/6*B*c*d^3*g^4*n*(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*c^2*d^2*g^4*n*(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)) - 2*B*c^3*d*g^4*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d 
^2 + (b*c - a*d)*x/(b*d)) + B*c^4*g^4*n*(a*log(b*x + a)/b - c*log(d*x + c) 
/d) + B*c^4*g^4*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*c^4*g^4*x
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1876 vs. \(2 (176) = 352\).

Time = 0.85 (sec) , antiderivative size = 1876, normalized size of antiderivative = 9.98 \[ \int (c g+d g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/60*(12*(B*b^6*c^6*g^4*n - 6*B*a*b^5*c^5*d*g^4*n + 15*B*a^2*b^4*c^4*d^2*g 
^4*n - 20*B*a^3*b^3*c^3*d^3*g^4*n + 15*B*a^4*b^2*c^2*d^4*g^4*n - 6*B*a^5*b 
*c*d^5*g^4*n + B*a^6*d^6*g^4*n)*log((b*x + a)/(d*x + c))/(b^5*d - 5*(b*x + 
 a)*b^4*d^2/(d*x + c) + 10*(b*x + a)^2*b^3*d^3/(d*x + c)^2 - 10*(b*x + a)^ 
3*b^2*d^4/(d*x + c)^3 + 5*(b*x + a)^4*b*d^5/(d*x + c)^4 - (b*x + a)^5*d^6/ 
(d*x + c)^5) - (25*B*b^10*c^6*g^4*n - 150*B*a*b^9*c^5*d*g^4*n - 77*(b*x + 
a)*B*b^9*c^6*d*g^4*n/(d*x + c) + 375*B*a^2*b^8*c^4*d^2*g^4*n + 462*(b*x + 
a)*B*a*b^8*c^5*d^2*g^4*n/(d*x + c) + 94*(b*x + a)^2*B*b^8*c^6*d^2*g^4*n/(d 
*x + c)^2 - 500*B*a^3*b^7*c^3*d^3*g^4*n - 1155*(b*x + a)*B*a^2*b^7*c^4*d^3 
*g^4*n/(d*x + c) - 564*(b*x + a)^2*B*a*b^7*c^5*d^3*g^4*n/(d*x + c)^2 - 54* 
(b*x + a)^3*B*b^7*c^6*d^3*g^4*n/(d*x + c)^3 + 375*B*a^4*b^6*c^2*d^4*g^4*n 
+ 1540*(b*x + a)*B*a^3*b^6*c^3*d^4*g^4*n/(d*x + c) + 1410*(b*x + a)^2*B*a^ 
2*b^6*c^4*d^4*g^4*n/(d*x + c)^2 + 324*(b*x + a)^3*B*a*b^6*c^5*d^4*g^4*n/(d 
*x + c)^3 + 12*(b*x + a)^4*B*b^6*c^6*d^4*g^4*n/(d*x + c)^4 - 150*B*a^5*b^5 
*c*d^5*g^4*n - 1155*(b*x + a)*B*a^4*b^5*c^2*d^5*g^4*n/(d*x + c) - 1880*(b* 
x + a)^2*B*a^3*b^5*c^3*d^5*g^4*n/(d*x + c)^2 - 810*(b*x + a)^3*B*a^2*b^5*c 
^4*d^5*g^4*n/(d*x + c)^3 - 72*(b*x + a)^4*B*a*b^5*c^5*d^5*g^4*n/(d*x + c)^ 
4 + 25*B*a^6*b^4*d^6*g^4*n + 462*(b*x + a)*B*a^5*b^4*c*d^6*g^4*n/(d*x + c) 
 + 1410*(b*x + a)^2*B*a^4*b^4*c^2*d^6*g^4*n/(d*x + c)^2 + 1080*(b*x + a)^3 
*B*a^3*b^4*c^3*d^6*g^4*n/(d*x + c)^3 + 180*(b*x + a)^4*B*a^2*b^4*c^4*d^...
 

Mupad [B] (verification not implemented)

Time = 26.09 (sec) , antiderivative size = 1045, normalized size of antiderivative = 5.56 \[ \int (c g+d g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

x^2*(((5*a*d + 5*b*c)*((((d^3*g^4*(5*A*a*d + 25*A*b*c + B*a*d*n - B*b*c*n) 
)/(5*b) - (A*d^3*g^4*(5*a*d + 5*b*c))/(5*b))*(5*a*d + 5*b*c))/(5*b*d) - (c 
*d^2*g^4*(5*A*a*d + 10*A*b*c + B*a*d*n - B*b*c*n))/b + (A*a*c*d^3*g^4)/b)) 
/(10*b*d) - (a*c*((d^3*g^4*(5*A*a*d + 25*A*b*c + B*a*d*n - B*b*c*n))/(5*b) 
 - (A*d^3*g^4*(5*a*d + 5*b*c))/(5*b)))/(2*b*d) + (c^2*d*g^4*(5*A*a*d + 5*A 
*b*c + B*a*d*n - B*b*c*n))/b) - x^3*((((d^3*g^4*(5*A*a*d + 25*A*b*c + B*a* 
d*n - B*b*c*n))/(5*b) - (A*d^3*g^4*(5*a*d + 5*b*c))/(5*b))*(5*a*d + 5*b*c) 
)/(15*b*d) - (c*d^2*g^4*(5*A*a*d + 10*A*b*c + B*a*d*n - B*b*c*n))/(3*b) + 
(A*a*c*d^3*g^4)/(3*b)) + x^4*((d^3*g^4*(5*A*a*d + 25*A*b*c + B*a*d*n - B*b 
*c*n))/(20*b) - (A*d^3*g^4*(5*a*d + 5*b*c))/(20*b)) + log(e*((a + b*x)/(c 
+ d*x))^n)*((B*d^4*g^4*x^5)/5 + B*c^4*g^4*x + 2*B*c^3*d*g^4*x^2 + B*c*d^3* 
g^4*x^4 + 2*B*c^2*d^2*g^4*x^3) + x*((c^3*g^4*(10*A*a*d + 5*A*b*c + 2*B*a*d 
*n - 2*B*b*c*n))/b - ((5*a*d + 5*b*c)*(((5*a*d + 5*b*c)*((((d^3*g^4*(5*A*a 
*d + 25*A*b*c + B*a*d*n - B*b*c*n))/(5*b) - (A*d^3*g^4*(5*a*d + 5*b*c))/(5 
*b))*(5*a*d + 5*b*c))/(5*b*d) - (c*d^2*g^4*(5*A*a*d + 10*A*b*c + B*a*d*n - 
 B*b*c*n))/b + (A*a*c*d^3*g^4)/b))/(5*b*d) - (a*c*((d^3*g^4*(5*A*a*d + 25* 
A*b*c + B*a*d*n - B*b*c*n))/(5*b) - (A*d^3*g^4*(5*a*d + 5*b*c))/(5*b)))/(b 
*d) + (2*c^2*d*g^4*(5*A*a*d + 5*A*b*c + B*a*d*n - B*b*c*n))/b))/(5*b*d) + 
(a*c*((((d^3*g^4*(5*A*a*d + 25*A*b*c + B*a*d*n - B*b*c*n))/(5*b) - (A*d^3* 
g^4*(5*a*d + 5*b*c))/(5*b))*(5*a*d + 5*b*c))/(5*b*d) - (c*d^2*g^4*(5*A*...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 689, normalized size of antiderivative = 3.66 \[ \int (c g+d g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

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