\(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(c g+d g x)^5} \, dx\) [37]

Optimal result

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

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

Mathematica [A] (verified)

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

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

Output:

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

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

\(\Big \downarrow \) 2947

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-1/4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(d*g^5*(c + d*x)^4) + (B*(b*c 
- a*d)*n*(1/(4*(b*c - a*d)*(c + d*x)^4) + b/(3*(b*c - a*d)^2*(c + d*x)^3) 
+ b^2/(2*(b*c - a*d)^3*(c + d*x)^2) + b^3/((b*c - a*d)^4*(c + d*x)) + (b^4 
*Log[a + b*x])/(b*c - a*d)^5 - (b^4*Log[c + d*x])/(b*c - a*d)^5))/(4*d*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 = 28.04 (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))/(d*g*x+c*g)^5,x,method=_RETURNVERBOSE)
 

Output:

1/48*(48*A*x*a^5*c^5*d^4*n+48*A*x*a*b^4*c^9*n-12*B*ln(e*((b*x+a)/(d*x+c))^ 
n)*a^5*c^6*d^3*n+48*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^3*c^9*n-3*B*x^4*a^5* 
c^2*d^7*n^2+12*A*x^4*a^5*c^2*d^7*n-12*B*x^3*a^5*c^3*d^6*n^2+48*A*x^3*a^5*c 
^3*d^6*n-18*B*x^2*a^5*c^4*d^5*n^2+72*A*x^2*a^5*c^4*d^5*n-12*B*x*a^5*c^5*d^ 
4*n^2-48*B*x*a*b^4*c^9*n^2+16*B*x^4*a^4*b*c^3*d^6*n^2+288*A*x*a^3*b^2*c^7* 
d^2*n-192*A*x*a^2*b^3*c^8*d*n+48*B*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b*c^7*d^2 
*n-72*B*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^2*c^8*d*n-36*B*x^4*a^3*b^2*c^4*d^5 
*n^2+48*B*x^4*a^2*b^3*c^5*d^4*n^2-25*B*x^4*a*b^4*c^6*d^3*n^2-48*A*x^4*a^4* 
b*c^3*d^6*n+12*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a*b^4*c^6*d^3*n+48*B*x^3*ln 
(e*((b*x+a)/(d*x+c))^n)*a*b^4*c^7*d^2*n+72*B*x^2*ln(e*((b*x+a)/(d*x+c))^n) 
*a*b^4*c^8*d*n-192*A*x*a^4*b*c^6*d^3*n+72*A*x^4*a^3*b^2*c^4*d^5*n-48*A*x^4 
*a^2*b^3*c^5*d^4*n+12*A*x^4*a*b^4*c^6*d^3*n+64*B*x^3*a^4*b*c^4*d^5*n^2-144 
*B*x^3*a^3*b^2*c^5*d^4*n^2+180*B*x^3*a^2*b^3*c^6*d^3*n^2-88*B*x^3*a*b^4*c^ 
7*d^2*n^2-192*A*x^3*a^4*b*c^4*d^5*n+288*A*x^3*a^3*b^2*c^5*d^4*n-192*A*x^3* 
a^2*b^3*c^6*d^3*n+48*A*x^3*a*b^4*c^7*d^2*n+96*B*x^2*a^4*b*c^5*d^4*n^2-210* 
B*x^2*a^3*b^2*c^6*d^3*n^2+240*B*x^2*a^2*b^3*c^7*d^2*n^2-108*B*x^2*a*b^4*c^ 
8*d*n^2-288*A*x^2*a^4*b*c^5*d^4*n+432*A*x^2*a^3*b^2*c^6*d^3*n-288*A*x^2*a^ 
2*b^3*c^7*d^2*n+72*A*x^2*a*b^4*c^8*d*n+48*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a* 
b^4*c^9*n+60*B*x*a^4*b*c^6*d^3*n^2-120*B*x*a^3*b^2*c^7*d^2*n^2+120*B*x*a^2 
*b^3*c^8*d*n^2)/g^5/(d*x+c)^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. 735 vs. \(2 (201) = 402\).

Time = 0.10 (sec) , antiderivative size = 735, normalized size of antiderivative = 3.42 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^5} \, dx=-\frac {12 \, A b^{4} c^{4} - 48 \, A a b^{3} c^{3} d + 72 \, A a^{2} b^{2} c^{2} d^{2} - 48 \, A a^{3} b c d^{3} + 12 \, A a^{4} d^{4} - 12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} n x^{3} - 6 \, {\left (7 \, B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} n x^{2} - 4 \, {\left (13 \, B b^{4} c^{3} d - 18 \, B a b^{3} c^{2} d^{2} + 6 \, B a^{2} b^{2} c d^{3} - B a^{3} b d^{4}\right )} n x - {\left (25 \, B b^{4} c^{4} - 48 \, B a b^{3} c^{3} d + 36 \, B a^{2} b^{2} c^{2} d^{2} - 16 \, B a^{3} b c d^{3} + 3 \, B a^{4} d^{4}\right )} n + 12 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3} + B a^{4} d^{4}\right )} \log \left (e\right ) - 12 \, {\left (B b^{4} d^{4} n x^{4} + 4 \, B b^{4} c d^{3} n x^{3} + 6 \, B b^{4} c^{2} d^{2} n x^{2} + 4 \, B b^{4} c^{3} d n x + {\left (4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{48 \, {\left ({\left (b^{4} c^{4} d^{5} - 4 \, a b^{3} c^{3} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{7} - 4 \, a^{3} b c d^{8} + a^{4} d^{9}\right )} g^{5} x^{4} + 4 \, {\left (b^{4} c^{5} d^{4} - 4 \, a b^{3} c^{4} d^{5} + 6 \, a^{2} b^{2} c^{3} d^{6} - 4 \, a^{3} b c^{2} d^{7} + a^{4} c d^{8}\right )} g^{5} x^{3} + 6 \, {\left (b^{4} c^{6} d^{3} - 4 \, a b^{3} c^{5} d^{4} + 6 \, a^{2} b^{2} c^{4} d^{5} - 4 \, a^{3} b c^{3} d^{6} + a^{4} c^{2} d^{7}\right )} g^{5} x^{2} + 4 \, {\left (b^{4} c^{7} d^{2} - 4 \, a b^{3} c^{6} d^{3} + 6 \, a^{2} b^{2} c^{5} d^{4} - 4 \, a^{3} b c^{4} d^{5} + a^{4} c^{3} d^{6}\right )} g^{5} x + {\left (b^{4} c^{8} d - 4 \, a b^{3} c^{7} d^{2} + 6 \, a^{2} b^{2} c^{6} d^{3} - 4 \, a^{3} b c^{5} d^{4} + a^{4} c^{4} d^{5}\right )} g^{5}\right )}} \] Input:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (201) = 402\).

Time = 0.07 (sec) , antiderivative size = 652, normalized size of antiderivative = 3.03 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^5} \, dx=\frac {1}{48} \, B n {\left (\frac {12 \, b^{3} d^{3} x^{3} + 25 \, b^{3} c^{3} - 23 \, a b^{2} c^{2} d + 13 \, a^{2} b c d^{2} - 3 \, a^{3} d^{3} + 6 \, {\left (7 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (13 \, b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{{\left (b^{3} c^{3} d^{5} - 3 \, a b^{2} c^{2} d^{6} + 3 \, a^{2} b c d^{7} - a^{3} d^{8}\right )} g^{5} x^{4} + 4 \, {\left (b^{3} c^{4} d^{4} - 3 \, a b^{2} c^{3} d^{5} + 3 \, a^{2} b c^{2} d^{6} - a^{3} c d^{7}\right )} g^{5} x^{3} + 6 \, {\left (b^{3} c^{5} d^{3} - 3 \, a b^{2} c^{4} d^{4} + 3 \, a^{2} b c^{3} d^{5} - a^{3} c^{2} d^{6}\right )} g^{5} x^{2} + 4 \, {\left (b^{3} c^{6} d^{2} - 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{4} d^{4} - a^{3} c^{3} d^{5}\right )} g^{5} x + {\left (b^{3} c^{7} d - 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{5} d^{3} - a^{3} c^{4} d^{4}\right )} g^{5}} + \frac {12 \, b^{4} \log \left (b x + a\right )}{{\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}\right )} g^{5}} - \frac {12 \, b^{4} \log \left (d x + c\right )}{{\left (b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}\right )} g^{5}}\right )} - \frac {B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{4 \, {\left (d^{5} g^{5} x^{4} + 4 \, c d^{4} g^{5} x^{3} + 6 \, c^{2} d^{3} g^{5} x^{2} + 4 \, c^{3} d^{2} g^{5} x + c^{4} d g^{5}\right )}} - \frac {A}{4 \, {\left (d^{5} g^{5} x^{4} + 4 \, c d^{4} g^{5} x^{3} + 6 \, c^{2} d^{3} g^{5} x^{2} + 4 \, c^{3} d^{2} g^{5} x + c^{4} d g^{5}\right )}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 684 vs. \(2 (201) = 402\).

Time = 0.67 (sec) , antiderivative size = 684, normalized size of antiderivative = 3.18 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^5} \, dx=\frac {1}{48} \, {\left (12 \, {\left (\frac {4 \, {\left (b x + a\right )} B b^{3} n}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}} - \frac {6 \, {\left (b x + a\right )}^{2} B b^{2} d n}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{2}} + \frac {4 \, {\left (b x + a\right )}^{3} B b d^{2} n}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{4} B d^{3} n}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{4}}\right )} \log \left (\frac {b x + a}{d x + c}\right ) + \frac {3 \, {\left (B d^{3} n - 4 \, B d^{3} \log \left (e\right ) - 4 \, A d^{3}\right )} {\left (b x + a\right )}^{4}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{4}} - \frac {16 \, {\left (B b d^{2} n - 3 \, B b d^{2} \log \left (e\right ) - 3 \, A b d^{2}\right )} {\left (b x + a\right )}^{3}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{3}} + \frac {36 \, {\left (B b^{2} d n - 2 \, B b^{2} d \log \left (e\right ) - 2 \, A b^{2} d\right )} {\left (b x + a\right )}^{2}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}^{2}} - \frac {48 \, {\left (B b^{3} n - B b^{3} \log \left (e\right ) - A b^{3}\right )} {\left (b x + a\right )}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (d x + c\right )}}\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))/(d*g*x+c*g)^5,x, algorithm="gia 
c")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 26.67 (sec) , antiderivative size = 603, normalized size of antiderivative = 2.80 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^5} \, dx=\frac {B\,b^4\,n\,\mathrm {atanh}\left (\frac {4\,a^4\,d^5\,g^5-8\,a^3\,b\,c\,d^4\,g^5+8\,a\,b^3\,c^3\,d^2\,g^5-4\,b^4\,c^4\,d\,g^5}{4\,d\,g^5\,{\left (a\,d-b\,c\right )}^4}+\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,d\,g^5\,{\left (a\,d-b\,c\right )}^4}-\frac {B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{4\,d\,\left (c^4\,g^5+4\,c^3\,d\,g^5\,x+6\,c^2\,d^2\,g^5\,x^2+4\,c\,d^3\,g^5\,x^3+d^4\,g^5\,x^4\right )}-\frac {\frac {12\,A\,a^3\,d^3-12\,A\,b^3\,c^3-3\,B\,a^3\,d^3\,n+25\,B\,b^3\,c^3\,n+36\,A\,a\,b^2\,c^2\,d-36\,A\,a^2\,b\,c\,d^2-23\,B\,a\,b^2\,c^2\,d\,n+13\,B\,a^2\,b\,c\,d^2\,n}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b\,x\,\left (B\,n\,a^2\,d^3-5\,B\,n\,a\,b\,c\,d^2+13\,B\,n\,b^2\,c^2\,d\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {b^2\,x^2\,\left (B\,a\,d^3\,n-7\,B\,b\,c\,d^2\,n\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B\,b^3\,d^3\,n\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,c^4\,d\,g^5+16\,c^3\,d^2\,g^5\,x+24\,c^2\,d^3\,g^5\,x^2+16\,c\,d^4\,g^5\,x^3+4\,d^5\,g^5\,x^4} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 929, normalized size of antiderivative = 4.32 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(12*log(a + b*x)*b**5*c**5*n + 48*log(a + b*x)*b**5*c**4*d*n*x + 72*log(a 
+ b*x)*b**5*c**3*d**2*n*x**2 + 48*log(a + b*x)*b**5*c**2*d**3*n*x**3 + 12* 
log(a + b*x)*b**5*c*d**4*n*x**4 - 12*log(c + d*x)*b**5*c**5*n - 48*log(c + 
 d*x)*b**5*c**4*d*n*x - 72*log(c + d*x)*b**5*c**3*d**2*n*x**2 - 48*log(c + 
 d*x)*b**5*c**2*d**3*n*x**3 - 12*log(c + d*x)*b**5*c*d**4*n*x**4 - 12*log( 
((a + b*x)**n*e)/(c + d*x)**n)*a**4*b*c*d**4 + 48*log(((a + b*x)**n*e)/(c 
+ d*x)**n)*a**3*b**2*c**2*d**3 - 72*log(((a + b*x)**n*e)/(c + d*x)**n)*a** 
2*b**3*c**3*d**2 + 48*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**4*c**4*d - 1 
2*log(((a + b*x)**n*e)/(c + d*x)**n)*b**5*c**5 - 12*a**5*c*d**4 + 48*a**4* 
b*c**2*d**3 + 3*a**4*b*c*d**4*n - 72*a**3*b**2*c**3*d**2 - 16*a**3*b**2*c* 
*2*d**3*n - 4*a**3*b**2*c*d**4*n*x + 48*a**2*b**3*c**4*d + 36*a**2*b**3*c* 
*3*d**2*n + 24*a**2*b**3*c**2*d**3*n*x + 6*a**2*b**3*c*d**4*n*x**2 - 12*a* 
b**4*c**5 - 45*a*b**4*c**4*d*n - 60*a*b**4*c**3*d**2*n*x - 30*a*b**4*c**2* 
d**3*n*x**2 + 3*a*b**4*d**5*n*x**4 + 22*b**5*c**5*n + 40*b**5*c**4*d*n*x + 
 24*b**5*c**3*d**2*n*x**2 - 3*b**5*c*d**4*n*x**4)/(48*c*d*g**5*(a**4*c**4* 
d**4 + 4*a**4*c**3*d**5*x + 6*a**4*c**2*d**6*x**2 + 4*a**4*c*d**7*x**3 + a 
**4*d**8*x**4 - 4*a**3*b*c**5*d**3 - 16*a**3*b*c**4*d**4*x - 24*a**3*b*c** 
3*d**5*x**2 - 16*a**3*b*c**2*d**6*x**3 - 4*a**3*b*c*d**7*x**4 + 6*a**2*b** 
2*c**6*d**2 + 24*a**2*b**2*c**5*d**3*x + 36*a**2*b**2*c**4*d**4*x**2 + 24* 
a**2*b**2*c**3*d**5*x**3 + 6*a**2*b**2*c**2*d**6*x**4 - 4*a*b**3*c**7*d...