\(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(a g+b g x)^5} \, dx\) [9]

Optimal result

Integrand size = 33, antiderivative size = 215 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^5} \, dx=-\frac {B n}{16 b g^5 (a+b x)^4}+\frac {B d n}{12 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 n}{8 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 n}{4 b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 n \log (a+b x)}{4 b (b c-a d)^4 g^5}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 b g^5 (a+b x)^4}-\frac {B d^4 n \log (c+d x)}{4 b (b c-a d)^4 g^5} \] Output:

-1/16*B*n/b/g^5/(b*x+a)^4+1/12*B*d*n/b/(-a*d+b*c)/g^5/(b*x+a)^3-1/8*B*d^2* 
n/b/(-a*d+b*c)^2/g^5/(b*x+a)^2+1/4*B*d^3*n/b/(-a*d+b*c)^3/g^5/(b*x+a)+1/4* 
B*d^4*n*ln(b*x+a)/b/(-a*d+b*c)^4/g^5-1/4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b 
/g^5/(b*x+a)^4-1/4*B*d^4*n*ln(d*x+c)/b/(-a*d+b*c)^4/g^5
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.75 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^5} \, dx=\frac {-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4}+\frac {B n \left (-\frac {3 (b c-a d)^4}{(a+b x)^4}+\frac {4 d (b c-a d)^3}{(a+b x)^3}-\frac {6 d^2 (b c-a d)^2}{(a+b x)^2}+\frac {12 d^3 (b c-a d)}{a+b x}+12 d^4 \log (a+b x)-12 d^4 \log (c+d x)\right )}{12 (b c-a d)^4}}{4 b g^5} \] Input:

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^5,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2947, 27, 54, 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.

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

\(\Big \downarrow \) 2947

\(\displaystyle \frac {B n (b c-a d) \int \frac {1}{g^4 (a+b x)^5 (c+d x)}dx}{4 b g}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 b g^5 (a+b x)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {B n (b c-a d) \int \frac {1}{(a+b x)^5 (c+d x)}dx}{4 b g^5}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 b g^5 (a+b x)^4}\)

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {B n (b c-a d) \left (\frac {d^4 \log (a+b x)}{(b c-a d)^5}-\frac {d^4 \log (c+d x)}{(b c-a d)^5}+\frac {d^3}{(a+b x) (b c-a d)^4}-\frac {d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac {d}{3 (a+b x)^3 (b c-a d)^2}-\frac {1}{4 (a+b x)^4 (b c-a d)}\right )}{4 b g^5}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 b g^5 (a+b x)^4}\)

Input:

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(a*g + b*g*x)^5,x]
 

Output:

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

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 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[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. \(1042\) vs. \(2(201)=402\).

Time = 27.88 (sec) , antiderivative size = 1043, normalized size of antiderivative = 4.85

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1043\)

Input:

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x,method=_RETURNVERBOSE)
 

Output:

1/48*(-60*B*x*a^6*b^3*c^4*d*n^2-192*A*x*a^8*b*c^2*d^3*n-288*A*x^2*a^7*b^2* 
c^2*d^3*n+432*A*x^2*a^6*b^3*c^3*d^2*n-288*A*x^2*a^5*b^4*c^4*d*n+48*B*x*ln( 
e*((b*x+a)/(d*x+c))^n)*a^9*c*d^4*n-120*B*x*a^8*b*c^2*d^3*n^2+120*B*x*a^7*b 
^2*c^3*d^2*n^2+3*B*x^4*a^2*b^7*c^5*n^2+12*A*x^4*a^2*b^7*c^5*n+12*B*x^3*a^3 
*b^6*c^5*n^2+48*A*x^3*a^3*b^6*c^5*n+18*B*x^2*a^4*b^5*c^5*n^2+72*A*x^2*a^4* 
b^5*c^5*n+48*B*x*a^9*c*d^4*n^2+12*B*x*a^5*b^4*c^5*n^2+48*A*x*a^9*c*d^4*n+4 
8*A*x*a^5*b^4*c^5*n+48*B*ln(e*((b*x+a)/(d*x+c))^n)*a^9*c^2*d^3*n-12*B*ln(e 
*((b*x+a)/(d*x+c))^n)*a^6*b^3*c^5*n+288*A*x*a^7*b^2*c^3*d^2*n-192*A*x*a^6* 
b^3*c^4*d*n-72*B*ln(e*((b*x+a)/(d*x+c))^n)*a^8*b*c^3*d^2*n+48*B*ln(e*((b*x 
+a)/(d*x+c))^n)*a^7*b^2*c^4*d*n+12*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^3 
*c*d^4*n+48*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b^2*c*d^4*n+72*B*x^2*ln(e* 
((b*x+a)/(d*x+c))^n)*a^8*b*c*d^4*n+25*B*x^4*a^6*b^3*c*d^4*n^2-48*B*x^4*a^5 
*b^4*c^2*d^3*n^2+36*B*x^4*a^4*b^5*c^3*d^2*n^2-16*B*x^4*a^3*b^6*c^4*d*n^2+1 
2*A*x^4*a^6*b^3*c*d^4*n-48*A*x^4*a^5*b^4*c^2*d^3*n+72*A*x^4*a^4*b^5*c^3*d^ 
2*n-48*A*x^4*a^3*b^6*c^4*d*n+88*B*x^3*a^7*b^2*c*d^4*n^2-180*B*x^3*a^6*b^3* 
c^2*d^3*n^2+144*B*x^3*a^5*b^4*c^3*d^2*n^2-64*B*x^3*a^4*b^5*c^4*d*n^2+48*A* 
x^3*a^7*b^2*c*d^4*n-192*A*x^3*a^6*b^3*c^2*d^3*n+288*A*x^3*a^5*b^4*c^3*d^2* 
n-192*A*x^3*a^4*b^5*c^4*d*n+108*B*x^2*a^8*b*c*d^4*n^2-240*B*x^2*a^7*b^2*c^ 
2*d^3*n^2+210*B*x^2*a^6*b^3*c^3*d^2*n^2-96*B*x^2*a^5*b^4*c^4*d*n^2+72*A*x^ 
2*a^8*b*c*d^4*n)/g^5/(b*x+a)^4/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 733 vs. \(2 (201) = 402\).

Time = 0.13 (sec) , antiderivative size = 733, normalized size of antiderivative = 3.41 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^5} \, dx=-\frac {12 \, A b^{4} c^{4} - 48 \, A a b^{3} c^{3} d + 72 \, A a^{2} b^{2} c^{2} d^{2} - 48 \, A a^{3} b c d^{3} + 12 \, A a^{4} d^{4} - 12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} n x^{3} + 6 \, {\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} n x^{2} - 4 \, {\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} n x + {\left (3 \, B b^{4} c^{4} - 16 \, B a b^{3} c^{3} d + 36 \, B a^{2} b^{2} c^{2} d^{2} - 48 \, B a^{3} b c d^{3} + 25 \, B a^{4} d^{4}\right )} n + 12 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3} + B a^{4} d^{4}\right )} \log \left (e\right ) - 12 \, {\left (B b^{4} d^{4} n x^{4} + 4 \, B a b^{3} d^{4} n x^{3} + 6 \, B a^{2} b^{2} d^{4} n x^{2} + 4 \, B a^{3} b d^{4} n x - {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3}\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{48 \, {\left ({\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x + {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\right )}} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x, algorithm="fri 
cas")
 

Output:

-1/48*(12*A*b^4*c^4 - 48*A*a*b^3*c^3*d + 72*A*a^2*b^2*c^2*d^2 - 48*A*a^3*b 
*c*d^3 + 12*A*a^4*d^4 - 12*(B*b^4*c*d^3 - B*a*b^3*d^4)*n*x^3 + 6*(B*b^4*c^ 
2*d^2 - 8*B*a*b^3*c*d^3 + 7*B*a^2*b^2*d^4)*n*x^2 - 4*(B*b^4*c^3*d - 6*B*a* 
b^3*c^2*d^2 + 18*B*a^2*b^2*c*d^3 - 13*B*a^3*b*d^4)*n*x + (3*B*b^4*c^4 - 16 
*B*a*b^3*c^3*d + 36*B*a^2*b^2*c^2*d^2 - 48*B*a^3*b*c*d^3 + 25*B*a^4*d^4)*n 
 + 12*(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 
 + B*a^4*d^4)*log(e) - 12*(B*b^4*d^4*n*x^4 + 4*B*a*b^3*d^4*n*x^3 + 6*B*a^2 
*b^2*d^4*n*x^2 + 4*B*a^3*b*d^4*n*x - (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)*n)*log((b*x + a)/(d*x + c)))/((b^9*c^4 - 
4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*g^5*x^4 
 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + 
a^5*b^4*d^4)*g^5*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^ 
2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*g^5*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^ 
3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d^4)*g^5*x + (a^4*b^5* 
c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4)*g 
^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^5} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**5,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 651 vs. \(2 (201) = 402\).

Time = 0.07 (sec) , antiderivative size = 651, normalized size of antiderivative = 3.03 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^5} \, dx=\frac {1}{48} \, B n {\left (\frac {12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \, {\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} g^{5} x + {\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} g^{5}} + \frac {12 \, d^{4} \log \left (b x + a\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}} - \frac {12 \, d^{4} \log \left (d x + c\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}}\right )} - \frac {B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{4 \, {\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\right )}} - \frac {A}{4 \, {\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\right )}} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x, algorithm="max 
ima")
 

Output:

1/48*B*n*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 
25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^ 
2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5 
*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4 
*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b 
^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6 
*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b 
*d^3)*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2 
*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4 - 
 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5)) - 
1/4*B*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^ 
3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) - 1/4*A/(b^5*g^5*x^4 
+ 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (201) = 402\).

Time = 0.65 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.52 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^5} \, dx=-\frac {1}{48} \, {\left (\frac {12 \, {\left (B b^{3} n - \frac {4 \, {\left (b x + a\right )} B b^{2} d n}{d x + c} + \frac {6 \, {\left (b x + a\right )}^{2} B b d^{2} n}{{\left (d x + c\right )}^{2}} - \frac {4 \, {\left (b x + a\right )}^{3} B d^{3} n}{{\left (d x + c\right )}^{3}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{4} b^{3} c^{3} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {3 \, {\left (b x + a\right )}^{4} a b^{2} c^{2} d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {3 \, {\left (b x + a\right )}^{4} a^{2} b c d^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a^{3} d^{3} g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {3 \, B b^{3} n - \frac {16 \, {\left (b x + a\right )} B b^{2} d n}{d x + c} + \frac {36 \, {\left (b x + a\right )}^{2} B b d^{2} n}{{\left (d x + c\right )}^{2}} - \frac {48 \, {\left (b x + a\right )}^{3} B d^{3} n}{{\left (d x + c\right )}^{3}} + 12 \, B b^{3} \log \left (e\right ) - \frac {48 \, {\left (b x + a\right )} B b^{2} d \log \left (e\right )}{d x + c} + \frac {72 \, {\left (b x + a\right )}^{2} B b d^{2} \log \left (e\right )}{{\left (d x + c\right )}^{2}} - \frac {48 \, {\left (b x + a\right )}^{3} B d^{3} \log \left (e\right )}{{\left (d x + c\right )}^{3}} + 12 \, A b^{3} - \frac {48 \, {\left (b x + a\right )} A b^{2} d}{d x + c} + \frac {72 \, {\left (b x + a\right )}^{2} A b d^{2}}{{\left (d x + c\right )}^{2}} - \frac {48 \, {\left (b x + a\right )}^{3} A d^{3}}{{\left (d x + c\right )}^{3}}}{\frac {{\left (b x + a\right )}^{4} b^{3} c^{3} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {3 \, {\left (b x + a\right )}^{4} a b^{2} c^{2} d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {3 \, {\left (b x + a\right )}^{4} a^{2} b c d^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a^{3} d^{3} g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x, algorithm="gia 
c")
 

Output:

-1/48*(12*(B*b^3*n - 4*(b*x + a)*B*b^2*d*n/(d*x + c) + 6*(b*x + a)^2*B*b*d 
^2*n/(d*x + c)^2 - 4*(b*x + a)^3*B*d^3*n/(d*x + c)^3)*log((b*x + a)/(d*x + 
 c))/((b*x + a)^4*b^3*c^3*g^5/(d*x + c)^4 - 3*(b*x + a)^4*a*b^2*c^2*d*g^5/ 
(d*x + c)^4 + 3*(b*x + a)^4*a^2*b*c*d^2*g^5/(d*x + c)^4 - (b*x + a)^4*a^3* 
d^3*g^5/(d*x + c)^4) + (3*B*b^3*n - 16*(b*x + a)*B*b^2*d*n/(d*x + c) + 36* 
(b*x + a)^2*B*b*d^2*n/(d*x + c)^2 - 48*(b*x + a)^3*B*d^3*n/(d*x + c)^3 + 1 
2*B*b^3*log(e) - 48*(b*x + a)*B*b^2*d*log(e)/(d*x + c) + 72*(b*x + a)^2*B* 
b*d^2*log(e)/(d*x + c)^2 - 48*(b*x + a)^3*B*d^3*log(e)/(d*x + c)^3 + 12*A* 
b^3 - 48*(b*x + a)*A*b^2*d/(d*x + c) + 72*(b*x + a)^2*A*b*d^2/(d*x + c)^2 
- 48*(b*x + a)^3*A*d^3/(d*x + c)^3)/((b*x + a)^4*b^3*c^3*g^5/(d*x + c)^4 - 
 3*(b*x + a)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*x + a)^4*a^2*b*c*d^2*g^5 
/(d*x + c)^4 - (b*x + a)^4*a^3*d^3*g^5/(d*x + c)^4))*(b*c/(b*c - a*d)^2 - 
a*d/(b*c - a*d)^2)
 

Mupad [B] (verification not implemented)

Time = 26.50 (sec) , antiderivative size = 603, normalized size of antiderivative = 2.80 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^5} \, dx=-\frac {\frac {12\,A\,a^3\,d^3-12\,A\,b^3\,c^3+25\,B\,a^3\,d^3\,n-3\,B\,b^3\,c^3\,n+36\,A\,a\,b^2\,c^2\,d-36\,A\,a^2\,b\,c\,d^2+13\,B\,a\,b^2\,c^2\,d\,n-23\,B\,a^2\,b\,c\,d^2\,n}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {d\,x\,\left (13\,B\,n\,a^2\,b\,d^2-5\,B\,n\,a\,b^2\,c\,d+B\,n\,b^3\,c^2\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {d^2\,x^2\,\left (B\,b^3\,c\,n-7\,B\,a\,b^2\,d\,n\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B\,b^3\,d^3\,n\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,a^4\,b\,g^5+16\,a^3\,b^2\,g^5\,x+24\,a^2\,b^3\,g^5\,x^2+16\,a\,b^4\,g^5\,x^3+4\,b^5\,g^5\,x^4}-\frac {B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{4\,b\,\left (a^4\,g^5+4\,a^3\,b\,g^5\,x+6\,a^2\,b^2\,g^5\,x^2+4\,a\,b^3\,g^5\,x^3+b^4\,g^5\,x^4\right )}-\frac {B\,d^4\,n\,\mathrm {atanh}\left (\frac {-4\,a^4\,b\,d^4\,g^5+8\,a^3\,b^2\,c\,d^3\,g^5-8\,a\,b^4\,c^3\,d\,g^5+4\,b^5\,c^4\,g^5}{4\,b\,g^5\,{\left (a\,d-b\,c\right )}^4}-\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,b\,g^5\,{\left (a\,d-b\,c\right )}^4} \] Input:

int((A + B*log(e*((a + b*x)/(c + d*x))^n))/(a*g + b*g*x)^5,x)
 

Output:

- ((12*A*a^3*d^3 - 12*A*b^3*c^3 + 25*B*a^3*d^3*n - 3*B*b^3*c^3*n + 36*A*a* 
b^2*c^2*d - 36*A*a^2*b*c*d^2 + 13*B*a*b^2*c^2*d*n - 23*B*a^2*b*c*d^2*n)/(1 
2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (d*x*(B*b^3*c^2*n 
 + 13*B*a^2*b*d^2*n - 5*B*a*b^2*c*d*n))/(3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^ 
2*d - 3*a^2*b*c*d^2)) - (d^2*x^2*(B*b^3*c*n - 7*B*a*b^2*d*n))/(2*(a^3*d^3 
- b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (B*b^3*d^3*n*x^3)/(a^3*d^3 - 
 b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(4*a^4*b*g^5 + 4*b^5*g^5*x^4 + 
16*a^3*b^2*g^5*x + 16*a*b^4*g^5*x^3 + 24*a^2*b^3*g^5*x^2) - (B*log(e*((a + 
 b*x)/(c + d*x))^n))/(4*b*(a^4*g^5 + b^4*g^5*x^4 + 4*a*b^3*g^5*x^3 + 6*a^2 
*b^2*g^5*x^2 + 4*a^3*b*g^5*x)) - (B*d^4*n*atanh((4*b^5*c^4*g^5 - 4*a^4*b*d 
^4*g^5 - 8*a*b^4*c^3*d*g^5 + 8*a^3*b^2*c*d^3*g^5)/(4*b*g^5*(a*d - b*c)^4) 
- (2*b*d*x*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*d - b*c 
)^4))/(2*b*g^5*(a*d - b*c)^4)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 934, normalized size of antiderivative = 4.34 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^5,x)
 

Output:

(12*log(a + b*x)*a**5*b*d**4*n + 48*log(a + b*x)*a**4*b**2*d**4*n*x + 72*l 
og(a + b*x)*a**3*b**3*d**4*n*x**2 + 48*log(a + b*x)*a**2*b**4*d**4*n*x**3 
+ 12*log(a + b*x)*a*b**5*d**4*n*x**4 - 12*log(c + d*x)*a**5*b*d**4*n - 48* 
log(c + d*x)*a**4*b**2*d**4*n*x - 72*log(c + d*x)*a**3*b**3*d**4*n*x**2 - 
48*log(c + d*x)*a**2*b**4*d**4*n*x**3 - 12*log(c + d*x)*a*b**5*d**4*n*x**4 
 - 12*log(((a + b*x)**n*e)/(c + d*x)**n)*a**5*b*d**4 + 48*log(((a + b*x)** 
n*e)/(c + d*x)**n)*a**4*b**2*c*d**3 - 72*log(((a + b*x)**n*e)/(c + d*x)**n 
)*a**3*b**3*c**2*d**2 + 48*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**4*c* 
*3*d - 12*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**5*c**4 - 12*a**6*d**4 + 
48*a**5*b*c*d**3 - 22*a**5*b*d**4*n - 72*a**4*b**2*c**2*d**2 + 45*a**4*b** 
2*c*d**3*n - 40*a**4*b**2*d**4*n*x + 48*a**3*b**3*c**3*d - 36*a**3*b**3*c* 
*2*d**2*n + 60*a**3*b**3*c*d**3*n*x - 24*a**3*b**3*d**4*n*x**2 - 12*a**2*b 
**4*c**4 + 16*a**2*b**4*c**3*d*n - 24*a**2*b**4*c**2*d**2*n*x + 30*a**2*b* 
*4*c*d**3*n*x**2 - 3*a*b**5*c**4*n + 4*a*b**5*c**3*d*n*x - 6*a*b**5*c**2*d 
**2*n*x**2 + 3*a*b**5*d**4*n*x**4 - 3*b**6*c*d**3*n*x**4)/(48*a*b*g**5*(a* 
*8*d**4 - 4*a**7*b*c*d**3 + 4*a**7*b*d**4*x + 6*a**6*b**2*c**2*d**2 - 16*a 
**6*b**2*c*d**3*x + 6*a**6*b**2*d**4*x**2 - 4*a**5*b**3*c**3*d + 24*a**5*b 
**3*c**2*d**2*x - 24*a**5*b**3*c*d**3*x**2 + 4*a**5*b**3*d**4*x**3 + a**4* 
b**4*c**4 - 16*a**4*b**4*c**3*d*x + 36*a**4*b**4*c**2*d**2*x**2 - 16*a**4* 
b**4*c*d**3*x**3 + a**4*b**4*d**4*x**4 + 4*a**3*b**5*c**4*x - 24*a**3*b...