Integrand size = 31, antiderivative size = 195 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx=-\frac {B n}{16 b (a+b x)^4}+\frac {B d n}{12 b (b c-a d) (a+b x)^3}-\frac {B d^2 n}{8 b (b c-a d)^2 (a+b x)^2}+\frac {B d^3 n}{4 b (b c-a d)^3 (a+b x)}+\frac {B d^4 n \log (a+b x)}{4 b (b c-a d)^4}-\frac {B d^4 n \log (c+d x)}{4 b (b c-a d)^4}-\frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{4 b (a+b x)^4} \]
-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
Time = 0.22 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.85 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx=-\frac {\frac {12 A}{(a+b x)^4}+B n \left (\frac {3+\frac {4 d (a+b x)}{-b c+a d}+\frac {6 d^2 (a+b x)^2}{(b c-a d)^2}-\frac {12 d^3 (a+b x)^3}{(b c-a d)^3}}{(a+b x)^4}-\frac {12 d^4 \log (a+b x)}{(b c-a d)^4}+\frac {12 d^4 \log (c+d x)}{(b c-a d)^4}\right )+\frac {12 B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^4}}{48 b} \]
-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
Time = 0.33 (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 |
| \(\displaystyle \frac {B n (b c-a d) \left (\frac {d^4 \log (a+b x)}{(b c-a d)^5}-\frac {d^4 \log (c+d x)}{(b c-a d)^5}+\frac {d^3}{(a+b x) (b c-a d)^4}-\frac {d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac {d}{3 (a+b x)^3 (b c-a d)^2}-\frac {1}{4 (a+b x)^4 (b c-a d)}\right )}{4 b}-\frac {B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A}{4 b (a+b x)^4}\) |
(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)
3.2.55.3.1 Defintions of rubi rules used
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])
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])
Leaf count of result is larger than twice the leaf count of optimal. \(2308\) vs. \(2(184)=368\).
Time = 135.60 (sec) , antiderivative size = 2309, normalized size of antiderivative = 11.84
| method | result | size |
| parallelrisch | \(\text {Expression too large to display}\) | \(2309\) |
| risch | \(\text {Expression too large to display}\) | \(2583\) |
1/48*(48*B*ln(b*x+a)*x^4*a^5*b^4*c^2*d^3*n-288*B*ln(b*x+a)*x^3*a^5*b^4*c^3 *d^2*n+192*B*ln(b*x+a)*x^3*a^4*b^5*c^4*d*n-192*B*ln(d*x+c)*x^3*a^6*b^3*c^2 *d^3*n+288*B*ln(d*x+c)*x^3*a^5*b^4*c^3*d^2*n-192*B*ln(d*x+c)*x^3*a^4*b^5*c ^4*d*n+288*B*ln(b*x+a)*x^2*a^7*b^2*c^2*d^3*n-432*B*ln(b*x+a)*x^2*a^6*b^3*c ^3*d^2*n+288*B*ln(b*x+a)*x^2*a^5*b^4*c^4*d*n-288*B*ln(d*x+c)*x^2*a^7*b^2*c ^2*d^3*n+432*B*ln(d*x+c)*x^2*a^6*b^3*c^3*d^2*n-288*B*ln(d*x+c)*x^2*a^5*b^4 *c^4*d*n+192*B*ln(b*x+a)*x*a^8*b*c^2*d^3*n-288*B*ln(b*x+a)*x*a^7*b^2*c^3*d ^2*n+192*B*ln(b*x+a)*x*a^6*b^3*c^4*d*n-192*B*ln(d*x+c)*x*a^8*b*c^2*d^3*n+2 88*B*ln(d*x+c)*x*a^7*b^2*c^3*d^2*n-192*B*ln(d*x+c)*x*a^6*b^3*c^4*d*n+12*A* x^4*a^2*b^7*c^5+48*A*x^3*a^3*b^6*c^5+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+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-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+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+12*B*x^4*ln(e*( b*x+a)^n/((d*x+c)^n))*a^6*b^3*c*d^4-48*B*x^4*ln(e*(b*x+a)^n/((d*x+c)^n))*a ^5*b^4*c^2*d^3+72*B*x^4*ln(e*(b*x+a)^n/((d*x+c)^n))*a^4*b^5*c^3*d^2-48*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 -192*B*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a^8*b*c^2*d^3+288*B*x*ln(e*(b*x+a)^n/ ((d*x+c)^n))*a^7*b^2*c^3*d^2-192*B*x*ln(e*(b*x+a)^n/((d*x+c)^n))*a^6*b^3*c ^4*d-120*B*x*a^8*b*c^2*d^3*n+120*B*x*a^7*b^2*c^3*d^2*n-60*B*x*a^6*b^3*c...
Leaf count of result is larger than twice the leaf count of optimal. 820 vs. \(2 (181) = 362\).
Time = 0.29 (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=-\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} 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 (b x + a\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 (d x + c\right ) + 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 )}{48 \, {\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} + {\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 )} 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 )} 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 )} 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 )} x\right )}} \]
-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)
Timed out. \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (181) = 362\).
Time = 0.21 (sec) , antiderivative size = 618, normalized size of antiderivative = 3.17 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx=\frac {{\left (\frac {12 \, d^{4} e n \log \left (b x + a\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac {12 \, d^{4} e n \log \left (d x + c\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} + \frac {12 \, b^{3} d^{3} e n x^{3} - 3 \, b^{3} c^{3} e n + 13 \, a b^{2} c^{2} d e n - 23 \, a^{2} b c d^{2} e n + 25 \, a^{3} d^{3} e n - 6 \, {\left (b^{3} c d^{2} e n - 7 \, a b^{2} d^{3} e n\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d e n - 5 \, a b^{2} c d^{2} e n + 13 \, a^{2} b d^{3} e n\right )} x}{a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{4} + 4 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} x}\right )} B}{48 \, e} - \frac {B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {A}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} \]
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)
Leaf count of result is larger than twice the leaf count of optimal. 718 vs. \(2 (181) = 362\).
Time = 0.31 (sec) , antiderivative size = 718, normalized size of antiderivative = 3.68 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx=\frac {B d^{4} n \log \left (b x + a\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} - \frac {B d^{4} n \log \left (d x + c\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} - \frac {B n \log \left (b x + a\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {B n \log \left (d x + c\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {12 \, B b^{3} d^{3} n x^{3} - 6 \, B b^{3} c d^{2} n x^{2} + 42 \, B a b^{2} d^{3} n x^{2} + 4 \, B b^{3} c^{2} d n x - 20 \, B a b^{2} c d^{2} n x + 52 \, B a^{2} b d^{3} n x - 3 \, B b^{3} c^{3} n + 13 \, B a b^{2} c^{2} d n - 23 \, B a^{2} b c d^{2} n + 25 \, B a^{3} d^{3} n - 12 \, B b^{3} c^{3} \log \left (e\right ) + 36 \, B a b^{2} c^{2} d \log \left (e\right ) - 36 \, B a^{2} b c d^{2} \log \left (e\right ) + 12 \, B a^{3} d^{3} \log \left (e\right ) - 12 \, A b^{3} c^{3} + 36 \, A a b^{2} c^{2} d - 36 \, A a^{2} b c d^{2} + 12 \, A a^{3} d^{3}}{48 \, {\left (b^{8} c^{3} x^{4} - 3 \, a b^{7} c^{2} d x^{4} + 3 \, a^{2} b^{6} c d^{2} x^{4} - a^{3} b^{5} d^{3} x^{4} + 4 \, a b^{7} c^{3} x^{3} - 12 \, a^{2} b^{6} c^{2} d x^{3} + 12 \, a^{3} b^{5} c d^{2} x^{3} - 4 \, a^{4} b^{4} d^{3} x^{3} + 6 \, a^{2} b^{6} c^{3} x^{2} - 18 \, a^{3} b^{5} c^{2} d x^{2} + 18 \, a^{4} b^{4} c d^{2} x^{2} - 6 \, a^{5} b^{3} d^{3} x^{2} + 4 \, a^{3} b^{5} c^{3} x - 12 \, a^{4} b^{4} c^{2} d x + 12 \, a^{5} b^{3} c d^{2} x - 4 \, a^{6} b^{2} d^{3} x + a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )}} \]
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)
Time = 1.96 (sec) , antiderivative size = 555, normalized size of antiderivative = 2.85 \[ \int \frac {A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x)^5} \, dx=-\frac {\frac {12\,A\,a^3\,d^3-12\,A\,b^3\,c^3+25\,B\,a^3\,d^3\,n-3\,B\,b^3\,c^3\,n+36\,A\,a\,b^2\,c^2\,d-36\,A\,a^2\,b\,c\,d^2+13\,B\,a\,b^2\,c^2\,d\,n-23\,B\,a^2\,b\,c\,d^2\,n}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {d\,x\,\left (13\,B\,n\,a^2\,b\,d^2-5\,B\,n\,a\,b^2\,c\,d+B\,n\,b^3\,c^2\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {d^2\,x^2\,\left (B\,b^3\,c\,n-7\,B\,a\,b^2\,d\,n\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B\,b^3\,d^3\,n\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,a^4\,b+16\,a^3\,b^2\,x+24\,a^2\,b^3\,x^2+16\,a\,b^4\,x^3+4\,b^5\,x^4}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}{4\,b\,\left (a^4+4\,a^3\,b\,x+6\,a^2\,b^2\,x^2+4\,a\,b^3\,x^3+b^4\,x^4\right )}-\frac {B\,d^4\,n\,\mathrm {atanh}\left (\frac {-4\,a^4\,b\,d^4+8\,a^3\,b^2\,c\,d^3-8\,a\,b^4\,c^3\,d+4\,b^5\,c^4}{4\,b\,{\left (a\,d-b\,c\right )}^4}-\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,b\,{\left (a\,d-b\,c\right )}^4} \]
- ((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)