\(\int \frac {A+B \log (e (a+b x)^n (c+d x)^{-n})}{(a+b x)^5} \, dx\) [155]

Optimal result

Integrand size = 31, antiderivative size = 195 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx=-\frac {B n}{16 b (a+b x)^4}+\frac {B d n}{12 b (b c-a d) (a+b x)^3}-\frac {B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}+\frac {B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac {B d^4 n \log (a+b x)}{4 b (b c-a d)^4}-\frac {B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.85 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx=-\frac {\frac {12 A}{(a+b x)^4}+B n \left (\frac {3+\frac {4 d (a+b x)}{-b c+a d}+\frac {6 d^2 (a+b x)^2}{(b c-a d)^2}-\frac {12 d^3 (a+b x)^3}{(b c-a d)^3}}{(a+b x)^4}-\frac {12 d^4 \log (a+b x)}{(b c-a d)^4}+\frac {12 d^4 \log (c+d x)}{(b c-a d)^4}\right )+\frac {12 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}}{48 b} \] Input:

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

Output:

-1/48*((12*A)/(a + b*x)^4 + B*n*((3 + (4*d*(a + b*x))/(-(b*c) + a*d) + (6* 
d^2*(a + b*x)^2)/(b*c - a*d)^2 - (12*d^3*(a + b*x)^3)/(b*c - a*d)^3)/(a + 
b*x)^4 - (12*d^4*Log[a + b*x])/(b*c - a*d)^4 + (12*d^4*Log[c + d*x])/(b*c 
- a*d)^4) + (12*B*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(a + b*x)^4)/b
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2948, 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.

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

\(\Big \downarrow \) 2948

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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. \(2308\) vs. \(2(181)=362\).

Time = 96.59 (sec) , antiderivative size = 2309, normalized size of antiderivative = 11.84

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(2309\)
risch	\(\text {Expression too large to display}\)	\(2583\)

Input:

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

Output:

1/48*(48*A*x^3*a^7*b^2*c*d^4-192*A*x^3*a^6*b^3*c^2*d^3+288*A*x^3*a^5*b^4*c 
^3*d^2-192*A*x^3*a^4*b^5*c^4*d+72*A*x^2*a^8*b*c*d^4-288*A*x^2*a^7*b^2*c^2* 
d^3+432*A*x^2*a^6*b^3*c^3*d^2+72*A*x^4*a^4*b^5*c^3*d^2-48*A*x^4*a^3*b^6*c^ 
4*d+72*B*x^2*ln(e*(b*x+a)^n/((d*x+c)^n))*a^4*b^5*c^5+18*B*x^2*a^4*b^5*c^5* 
n-48*B*x^4*ln(e*(b*x+a)^n/((d*x+c)^n))*a^3*b^6*c^4*d+25*B*x^4*a^6*b^3*c*d^ 
4*n-48*B*x^4*a^5*b^4*c^2*d^3*n+36*B*x^4*a^4*b^5*c^3*d^2*n-16*B*x^4*a^3*b^6 
*c^4*d*n+48*B*x^3*ln(e*(b*x+a)^n/((d*x+c)^n))*a^7*b^2*c*d^4-192*B*x^3*ln(e 
*(b*x+a)^n/((d*x+c)^n))*a^6*b^3*c^2*d^3+288*B*x^3*ln(e*(b*x+a)^n/((d*x+c)^ 
n))*a^5*b^4*c^3*d^2-192*B*x^3*ln(e*(b*x+a)^n/((d*x+c)^n))*a^4*b^5*c^4*d-19 
2*B*ln(d*x+c)*x*a^8*b*c^2*d^3*n+48*B*ln(b*x+a)*a^9*c^2*d^3*n-12*B*ln(b*x+a 
)*a^6*b^3*c^5*n-288*A*x^2*a^5*b^4*c^4*d-192*A*x*a^8*b*c^2*d^3+288*A*x*a^7* 
b^2*c^3*d^2-192*A*x*a^6*b^3*c^4*d+12*A*x^4*a^6*b^3*c*d^4-48*A*x^4*a^5*b^4* 
c^2*d^3+72*A*x^2*a^4*b^5*c^5+48*A*x*a^9*c*d^4+48*A*x*a^5*b^4*c^5+12*A*x^4* 
a^2*b^7*c^5+48*A*x^3*a^3*b^6*c^5-48*B*ln(d*x+c)*a^9*c^2*d^3*n+12*B*ln(d*x+ 
c)*a^6*b^3*c^5*n+48*B*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a^9*c*d^4+48*B*x*ln(e* 
(b*x+a)^n/((d*x+c)^n))*a^5*b^4*c^5+48*B*x*a^9*c*d^4*n+12*B*x*a^5*b^4*c^5*n 
+12*B*x^4*ln(e*(b*x+a)^n/((d*x+c)^n))*a^2*b^7*c^5+3*B*x^4*a^2*b^7*c^5*n+48 
*B*x^3*ln(e*(b*x+a)^n/((d*x+c)^n))*a^3*b^6*c^5+12*B*x^3*a^3*b^6*c^5*n-48*B 
*ln(b*x+a)*x^3*a^3*b^6*c^5*n+48*B*ln(d*x+c)*x^3*a^3*b^6*c^5*n-72*B*ln(b*x+ 
a)*x^2*a^4*b^5*c^5*n+72*B*ln(d*x+c)*x^2*a^4*b^5*c^5*n-12*B*ln(b*x+a)*x^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 820 vs. \(2 (181) = 362\).

Time = 0.11 (sec) , antiderivative size = 820, normalized size of antiderivative = 4.21 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx =\text {Too large to display} \] Input:

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

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*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) + 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(d*x + c) + 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))/(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 + (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)*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)*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)*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)*x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (181) = 362\).

Time = 0.06 (sec) , antiderivative size = 618, normalized size of antiderivative = 3.17 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx=\frac {{\left (\frac {12 \, d^{4} e n \log \left (b x + a\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac {12 \, d^{4} e n \log \left (d x + c\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac {12 \, b^{3} d^{3} e n x^{3} - 3 \, b^{3} c^{3} e n + 13 \, a b^{2} c^{2} d e n - 23 \, a^{2} b c d^{2} e n + 25 \, a^{3} d^{3} e n - 6 \, {\left (b^{3} c d^{2} e n - 7 \, a b^{2} d^{3} e n\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d e n - 5 \, a b^{2} c d^{2} e n + 13 \, a^{2} b d^{3} e n\right )} x}{a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{4} + 4 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} x}\right )} B}{48 \, e} - \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {A}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 718 vs. \(2 (181) = 362\).

Time = 0.13 (sec) , antiderivative size = 718, normalized size of antiderivative = 3.68 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx=\frac {B d^{4} n \log \left (b x + a\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} - \frac {B d^{4} n \log \left (d x + c\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} - \frac {B n \log \left (b x + a\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {B n \log \left (d x + c\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {12 \, B b^{3} d^{3} n x^{3} - 6 \, B b^{3} c d^{2} n x^{2} + 42 \, B a b^{2} d^{3} n x^{2} + 4 \, B b^{3} c^{2} d n x - 20 \, B a b^{2} c d^{2} n x + 52 \, B a^{2} b d^{3} n x - 3 \, B b^{3} c^{3} n + 13 \, B a b^{2} c^{2} d n - 23 \, B a^{2} b c d^{2} n + 25 \, B a^{3} d^{3} n - 12 \, B b^{3} c^{3} \log \left (e\right ) + 36 \, B a b^{2} c^{2} d \log \left (e\right ) - 36 \, B a^{2} b c d^{2} \log \left (e\right ) + 12 \, B a^{3} d^{3} \log \left (e\right ) - 12 \, A b^{3} c^{3} + 36 \, A a b^{2} c^{2} d - 36 \, A a^{2} b c d^{2} + 12 \, A a^{3} d^{3}}{48 \, {\left (b^{8} c^{3} x^{4} - 3 \, a b^{7} c^{2} d x^{4} + 3 \, a^{2} b^{6} c d^{2} x^{4} - a^{3} b^{5} d^{3} x^{4} + 4 \, a b^{7} c^{3} x^{3} - 12 \, a^{2} b^{6} c^{2} d x^{3} + 12 \, a^{3} b^{5} c d^{2} x^{3} - 4 \, a^{4} b^{4} d^{3} x^{3} + 6 \, a^{2} b^{6} c^{3} x^{2} - 18 \, a^{3} b^{5} c^{2} d x^{2} + 18 \, a^{4} b^{4} c d^{2} x^{2} - 6 \, a^{5} b^{3} d^{3} x^{2} + 4 \, a^{3} b^{5} c^{3} x - 12 \, a^{4} b^{4} c^{2} d x + 12 \, a^{5} b^{3} c d^{2} x - 4 \, a^{6} b^{2} d^{3} x + a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )}} \] Input:

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

Output:

1/4*B*d^4*n*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) - 1/4*B*d^4*n*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) - 1/4*B*n*log(b*x 
+ a)/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b) + 1/4*B 
*n*log(d*x + c)/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4 
*b) + 1/48*(12*B*b^3*d^3*n*x^3 - 6*B*b^3*c*d^2*n*x^2 + 42*B*a*b^2*d^3*n*x^ 
2 + 4*B*b^3*c^2*d*n*x - 20*B*a*b^2*c*d^2*n*x + 52*B*a^2*b*d^3*n*x - 3*B*b^ 
3*c^3*n + 13*B*a*b^2*c^2*d*n - 23*B*a^2*b*c*d^2*n + 25*B*a^3*d^3*n - 12*B* 
b^3*c^3*log(e) + 36*B*a*b^2*c^2*d*log(e) - 36*B*a^2*b*c*d^2*log(e) + 12*B* 
a^3*d^3*log(e) - 12*A*b^3*c^3 + 36*A*a*b^2*c^2*d - 36*A*a^2*b*c*d^2 + 12*A 
*a^3*d^3)/(b^8*c^3*x^4 - 3*a*b^7*c^2*d*x^4 + 3*a^2*b^6*c*d^2*x^4 - a^3*b^5 
*d^3*x^4 + 4*a*b^7*c^3*x^3 - 12*a^2*b^6*c^2*d*x^3 + 12*a^3*b^5*c*d^2*x^3 - 
 4*a^4*b^4*d^3*x^3 + 6*a^2*b^6*c^3*x^2 - 18*a^3*b^5*c^2*d*x^2 + 18*a^4*b^4 
*c*d^2*x^2 - 6*a^5*b^3*d^3*x^2 + 4*a^3*b^5*c^3*x - 12*a^4*b^4*c^2*d*x + 12 
*a^5*b^3*c*d^2*x - 4*a^6*b^2*d^3*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)
 

Mupad [B] (verification not implemented)

Time = 26.01 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.85 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx=-\frac {\frac {12\,A\,a^3\,d^3-12\,A\,b^3\,c^3+25\,B\,a^3\,d^3\,n-3\,B\,b^3\,c^3\,n+36\,A\,a\,b^2\,c^2\,d-36\,A\,a^2\,b\,c\,d^2+13\,B\,a\,b^2\,c^2\,d\,n-23\,B\,a^2\,b\,c\,d^2\,n}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {d\,x\,\left (13\,B\,n\,a^2\,b\,d^2-5\,B\,n\,a\,b^2\,c\,d+B\,n\,b^3\,c^2\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {d^2\,x^2\,\left (B\,b^3\,c\,n-7\,B\,a\,b^2\,d\,n\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B\,b^3\,d^3\,n\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,a^4\,b+16\,a^3\,b^2\,x+24\,a^2\,b^3\,x^2+16\,a\,b^4\,x^3+4\,b^5\,x^4}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{4\,b\,\left (a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4\right )}-\frac {B\,d^4\,n\,\mathrm {atanh}\left (\frac {-4\,a^4\,b\,d^4+8\,a^3\,b^2\,c\,d^3-8\,a\,b^4\,c^3\,d+4\,b^5\,c^4}{4\,b\,{\left (a\,d-b\,c\right )}^4}-\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,b\,{\left (a\,d-b\,c\right )}^4} \] Input:

int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))/(a + b*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 + 4*b^5*x^4 + 16*a^3*b 
^2*x + 16*a*b^4*x^3 + 24*a^2*b^3*x^2) - (B*log((e*(a + b*x)^n)/(c + d*x)^n 
))/(4*b*(a^4 + b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x)) - (B*d^ 
4*n*atanh((4*b^5*c^4 - 4*a^4*b*d^4 + 8*a^3*b^2*c*d^3 - 8*a*b^4*c^3*d)/(4*b 
*(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*(a*d - b*c)^4)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 931, normalized size of antiderivative = 4.77 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx =\text {Too large to display} \] Input:

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