Integrand size = 43, antiderivative size = 477 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=-\frac {B d^4 n (a+b x)}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 B d^2 n (c+d x)}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {b^3 B d n (c+d x)^2}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 B n (c+d x)^3}{9 (b c-a d)^5 g^4 i^2 (a+b x)^3}+\frac {d^4 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 d^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {2 b^3 d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^5 g^4 i^2 (a+b x)^3}-\frac {4 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^4 i^2}+\frac {2 b B d^3 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^4 i^2} \]
-B*d^4*n*(b*x+a)/(-a*d+b*c)^5/g^4/i^2/(d*x+c)-6*b^2*B*d^2*n*(d*x+c)/(-a*d+ b*c)^5/g^4/i^2/(b*x+a)+b^3*B*d*n*(d*x+c)^2/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^2- 1/9*b^4*B*n*(d*x+c)^3/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^3+d^4*(b*x+a)*(A+B*ln(e *((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^5/g^4/i^2/(d*x+c)-6*b^2*d^2*(d*x+c)*(A+B *ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^5/g^4/i^2/(b*x+a)+2*b^3*d*(d*x+c)^2 *(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^2-1/3*b^4*(d *x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^5/g^4/i^2/(b*x+a)^3-4*b *d^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((b*x+a)/(d*x+c))/(-a*d+b*c)^5/g^4/ i^2+2*b*B*d^3*n*ln((b*x+a)/(d*x+c))^2/(-a*d+b*c)^5/g^4/i^2
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.72 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.15 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=-\frac {\frac {b B (b c-a d)^3 n}{(a+b x)^3}-\frac {6 b B d (b c-a d)^2 n}{(a+b x)^2}+\frac {27 b^2 B c d^2 n}{a+b x}-\frac {27 a b B d^3 n}{a+b x}+\frac {12 b B d^2 (b c-a d) n}{a+b x}-\frac {9 b B c d^3 n}{c+d x}+\frac {9 a B d^4 n}{c+d x}+30 b B d^3 n \log (a+b x)+\frac {3 b (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^3}-\frac {9 b d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2}+\frac {27 b d^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}-\frac {9 d^3 (-b c+a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x}+36 b d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-30 b B d^3 n \log (c+d x)-36 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-18 b B d^3 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+18 b B d^3 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{9 (b c-a d)^5 g^4 i^2} \]
-1/9*((b*B*(b*c - a*d)^3*n)/(a + b*x)^3 - (6*b*B*d*(b*c - a*d)^2*n)/(a + b *x)^2 + (27*b^2*B*c*d^2*n)/(a + b*x) - (27*a*b*B*d^3*n)/(a + b*x) + (12*b* B*d^2*(b*c - a*d)*n)/(a + b*x) - (9*b*B*c*d^3*n)/(c + d*x) + (9*a*B*d^4*n) /(c + d*x) + 30*b*B*d^3*n*Log[a + b*x] + (3*b*(b*c - a*d)^3*(A + B*Log[e*( (a + b*x)/(c + d*x))^n]))/(a + b*x)^3 - (9*b*d*(b*c - a*d)^2*(A + B*Log[e* ((a + b*x)/(c + d*x))^n]))/(a + b*x)^2 + (27*b*d^2*(b*c - a*d)*(A + B*Log[ e*((a + b*x)/(c + d*x))^n]))/(a + b*x) - (9*d^3*(-(b*c) + a*d)*(A + B*Log[ e*((a + b*x)/(c + d*x))^n]))/(c + d*x) + 36*b*d^3*Log[a + b*x]*(A + B*Log[ e*((a + b*x)/(c + d*x))^n]) - 30*b*B*d^3*n*Log[c + d*x] - 36*b*d^3*(A + B* Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - 18*b*B*d^3*n*(Log[a + b*x]* (Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b *x))/(-(b*c) + a*d)]) + 18*b*B*d^3*n*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))/ ((b*c - a*d)^5*g^4*i^2)
Time = 0.47 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.69, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2961, 2772, 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.
| \(\displaystyle \int \frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{(a g+b g x)^4 (c i+d i x)^2} \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle \frac {\int \frac {(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^4}d\frac {a+b x}{c+d x}}{g^4 i^2 (b c-a d)^5}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {-B n \int \left (d^4-\frac {4 b (c+d x) \log \left (\frac {a+b x}{c+d x}\right ) d^3}{a+b x}-\frac {6 b^2 (c+d x)^2 d^2}{(a+b x)^2}+\frac {2 b^3 (c+d x)^3 d}{(a+b x)^3}-\frac {b^4 (c+d x)^4}{3 (a+b x)^4}\right )d\frac {a+b x}{c+d x}-\frac {b^4 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {2 b^3 d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^2}-\frac {6 b^2 d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+\frac {d^4 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}-4 b d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i^2 (b c-a d)^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b^4 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {2 b^3 d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^2}-\frac {6 b^2 d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+\frac {d^4 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}-4 b d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-B n \left (\frac {b^4 (c+d x)^3}{9 (a+b x)^3}-\frac {b^3 d (c+d x)^2}{(a+b x)^2}+\frac {6 b^2 d^2 (c+d x)}{a+b x}+\frac {d^4 (a+b x)}{c+d x}-2 b d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )\right )}{g^4 i^2 (b c-a d)^5}\) |
((d^4*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x) - (6*b^2 *d^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) + (2*b^3* d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^2 - (b^4*( c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(a + b*x)^3) - 4*b*d ^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)] - B*n*( (d^4*(a + b*x))/(c + d*x) + (6*b^2*d^2*(c + d*x))/(a + b*x) - (b^3*d*(c + d*x)^2)/(a + b*x)^2 + (b^4*(c + d*x)^3)/(9*(a + b*x)^3) - 2*b*d^3*Log[(a + b*x)/(c + d*x)]^2))/((b*c - a*d)^5*g^4*i^2)
3.2.50.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -1])
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol ] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*L og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
Leaf count of result is larger than twice the leaf count of optimal. \(1909\) vs. \(2(473)=946\).
Time = 44.86 (sec) , antiderivative size = 1910, normalized size of antiderivative = 4.00
1/9*(-90*A*x*a^6*b^3*c^5*d^2*n+45*A*x*a^5*b^4*c^6*d*n+54*B*ln(e*((b*x+a)/( d*x+c))^n)*a^7*b^2*c^5*d^2*n-18*B*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^3*c^6*d* n+18*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)^2*a^5*b^4*c^4*d^3-27*B*x^3*a^7*b^2*c^ 2*d^5*n^2+126*B*x^3*a^6*b^3*c^3*d^4*n^2-77*B*x^3*a^5*b^4*c^4*d^3*n^2-27*B* x^3*a^4*b^5*c^5*d^2*n^2+6*B*x^3*a^3*b^6*c^6*d*n^2+108*A*x^3*ln(e*((b*x+a)/ (d*x+c))^n)*a^6*b^3*c^3*d^4+36*A*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b^4*c^4 *d^3+27*A*x^3*a^7*b^2*c^2*d^5*n+63*A*x^3*a^6*b^3*c^3*d^4*n-96*A*x^3*a^5*b^ 4*c^4*d^3*n+9*A*x^3*a^3*b^6*c^6*d*n+9*A*x*a^9*c^2*d^5*n-9*A*x*a^4*b^5*c^7* n+18*B*ln(e*((b*x+a)/(d*x+c))^n)^2*a^8*b*c^4*d^3-9*B*ln(e*((b*x+a)/(d*x+c) )^n)*a^9*c^3*d^4*n+3*B*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b^4*c^7*n+36*A*ln(e*( (b*x+a)/(d*x+c))^n)*a^8*b*c^4*d^3-B*x^3*a^2*b^7*c^7*n^2-3*A*x^3*a^2*b^7*c^ 7*n-3*B*x^2*a^3*b^6*c^7*n^2-9*A*x^2*a^3*b^6*c^7*n-9*B*x*a^9*c^2*d^5*n^2-3* B*x*a^4*b^5*c^7*n^2+54*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^7*b^2*c^3*d^4+5 4*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^6*b^3*c^4*d^3-27*B*x^2*a^8*b*c^2*d^5 *n^2+81*B*x^2*a^7*b^2*c^3*d^4*n^2+27*B*x^2*a^6*b^3*c^4*d^3*n^2-102*B*x^2*a ^5*b^4*c^5*d^2*n^2+24*B*x^2*a^4*b^5*c^6*d*n^2+108*A*x^2*ln(e*((b*x+a)/(d*x +c))^n)*a^7*b^2*c^3*d^4+108*A*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^3*c^4*d^ 3+27*A*x^2*a^8*b*c^2*d^5*n+27*A*x^2*a^7*b^2*c^3*d^4*n-90*A*x^2*a^5*b^4*c^5 *d^2*n+45*A*x^2*a^4*b^5*c^6*d*n+18*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^8*b*c ^3*d^4+54*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^7*b^2*c^4*d^3+9*B*x*a^8*b*c...
Leaf count of result is larger than twice the leaf count of optimal. 1458 vs. \(2 (473) = 946\).
Time = 0.43 (sec) , antiderivative size = 1458, normalized size of antiderivative = 3.06 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \]
-1/9*(3*A*b^4*c^4 - 18*A*a*b^3*c^3*d + 54*A*a^2*b^2*c^2*d^2 - 30*A*a^3*b*c *d^3 - 9*A*a^4*d^4 + 6*(6*A*b^4*c*d^3 - 6*A*a*b^3*d^4 + 5*(B*b^4*c*d^3 - B *a*b^3*d^4)*n)*x^3 + 3*(6*A*b^4*c^2*d^2 + 24*A*a*b^3*c*d^3 - 30*A*a^2*b^2* d^4 + (11*B*b^4*c^2*d^2 + 8*B*a*b^3*c*d^3 - 19*B*a^2*b^2*d^4)*n)*x^2 + 18* (B*b^4*d^4*n*x^4 + B*a^3*b*c*d^3*n + (B*b^4*c*d^3 + 3*B*a*b^3*d^4)*n*x^3 + 3*(B*a*b^3*c*d^3 + B*a^2*b^2*d^4)*n*x^2 + (3*B*a^2*b^2*c*d^3 + B*a^3*b*d^ 4)*n*x)*log((b*x + a)/(d*x + c))^2 + (B*b^4*c^4 - 9*B*a*b^3*c^3*d + 54*B*a ^2*b^2*c^2*d^2 - 55*B*a^3*b*c*d^3 + 9*B*a^4*d^4)*n - (6*A*b^4*c^3*d - 54*A *a*b^3*c^2*d^2 - 18*A*a^2*b^2*c*d^3 + 66*A*a^3*b*d^4 + (5*B*b^4*c^3*d - 81 *B*a*b^3*c^2*d^2 + 57*B*a^2*b^2*c*d^3 + 19*B*a^3*b*d^4)*n)*x + 3*(B*b^4*c^ 4 - 6*B*a*b^3*c^3*d + 18*B*a^2*b^2*c^2*d^2 - 10*B*a^3*b*c*d^3 - 3*B*a^4*d^ 4 + 12*(B*b^4*c*d^3 - B*a*b^3*d^4)*x^3 + 6*(B*b^4*c^2*d^2 + 4*B*a*b^3*c*d^ 3 - 5*B*a^2*b^2*d^4)*x^2 - 2*(B*b^4*c^3*d - 9*B*a*b^3*c^2*d^2 - 3*B*a^2*b^ 2*c*d^3 + 11*B*a^3*b*d^4)*x + 12*(B*b^4*d^4*x^4 + B*a^3*b*c*d^3 + (B*b^4*c *d^3 + 3*B*a*b^3*d^4)*x^3 + 3*(B*a*b^3*c*d^3 + B*a^2*b^2*d^4)*x^2 + (3*B*a ^2*b^2*c*d^3 + B*a^3*b*d^4)*x)*log((b*x + a)/(d*x + c)))*log(e) + 3*(12*A* a^3*b*c*d^3 + 2*(5*B*b^4*d^4*n + 6*A*b^4*d^4)*x^4 + 2*(6*A*b^4*c*d^3 + 18* A*a*b^3*d^4 + (11*B*b^4*c*d^3 + 9*B*a*b^3*d^4)*n)*x^3 + 6*(6*A*a*b^3*c*d^3 + 6*A*a^2*b^2*d^4 + (B*b^4*c^2*d^2 + 9*B*a*b^3*c*d^3)*n)*x^2 + (B*b^4*c^4 - 6*B*a*b^3*c^3*d + 18*B*a^2*b^2*c^2*d^2 - 3*B*a^4*d^4)*n + 2*(18*A*a^...
Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2563 vs. \(2 (473) = 946\).
Time = 0.36 (sec) , antiderivative size = 2563, normalized size of antiderivative = 5.37 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \]
-1/3*B*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2*d + 13*a^2*b*c*d^2 + 3*a^3 *d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x^2 - 2*(b^3*c^2*d - 8*a*b^2*c*d^2 - 11 *a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3*b ^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c ^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4*b^3*c*d^4 + 3*a^5*b^2*d^5)*g^4*i^2*x^ 3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2*d^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*i^2*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^4* c^4*d + 14*a^4*b^3*c^3*d^2 - 6*a^5*b^2*c^2*d^3 - a^6*b*c*d^4 + a^7*d^5)*g^ 4*i^2*x + (a^3*b^4*c^5 - 4*a^4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2 - 4*a^6*b*c^2 *d^3 + a^7*c*d^4)*g^4*i^2) + 12*b*d^3*log(b*x + a)/((b^5*c^5 - 5*a*b^4*c^4 *d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^ 4*i^2) - 12*b*d^3*log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3* d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2))*log(e*(b*x/( d*x + c) + a/(d*x + c))^n) - 1/9*(b^4*c^4 - 9*a*b^3*c^3*d + 54*a^2*b^2*c^2 *d^2 - 55*a^3*b*c*d^3 + 9*a^4*d^4 + 30*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 3*(11 *b^4*c^2*d^2 + 8*a*b^3*c*d^3 - 19*a^2*b^2*d^4)*x^2 - 18*(b^4*d^4*x^4 + a^3 *b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x ^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a)^2 - 18*(b^4*d^4*x^4 + a ^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4) *x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(d*x + c)^2 - (5*b^4*c^3*d -...
Time = 180.85 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.84 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=-\frac {1}{18} \, {\left (\frac {6 \, {\left (B b^{2} n - \frac {3 \, {\left (b x + a\right )} B b d n}{d x + c} + \frac {3 \, {\left (b x + a\right )}^{2} B d^{2} n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{3} b^{2} c^{2} g^{4} i^{2}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} a b c d g^{4} i^{2}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b x + a\right )}^{3} a^{2} d^{2} g^{4} i^{2}}{{\left (d x + c\right )}^{3}}} + \frac {2 \, B b^{2} n - \frac {9 \, {\left (b x + a\right )} B b d n}{d x + c} + \frac {18 \, {\left (b x + a\right )}^{2} B d^{2} n}{{\left (d x + c\right )}^{2}} + 6 \, B b^{2} \log \left (e\right ) - \frac {18 \, {\left (b x + a\right )} B b d \log \left (e\right )}{d x + c} + \frac {18 \, {\left (b x + a\right )}^{2} B d^{2} \log \left (e\right )}{{\left (d x + c\right )}^{2}} + 6 \, A b^{2} - \frac {18 \, {\left (b x + a\right )} A b d}{d x + c} + \frac {18 \, {\left (b x + a\right )}^{2} A d^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{3} b^{2} c^{2} g^{4} i^{2}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} a b c d g^{4} i^{2}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b x + a\right )}^{3} a^{2} d^{2} g^{4} i^{2}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \]
-1/18*(6*(B*b^2*n - 3*(b*x + a)*B*b*d*n/(d*x + c) + 3*(b*x + a)^2*B*d^2*n/ (d*x + c)^2)*log((b*x + a)/(d*x + c))/((b*x + a)^3*b^2*c^2*g^4*i^2/(d*x + c)^3 - 2*(b*x + a)^3*a*b*c*d*g^4*i^2/(d*x + c)^3 + (b*x + a)^3*a^2*d^2*g^4 *i^2/(d*x + c)^3) + (2*B*b^2*n - 9*(b*x + a)*B*b*d*n/(d*x + c) + 18*(b*x + a)^2*B*d^2*n/(d*x + c)^2 + 6*B*b^2*log(e) - 18*(b*x + a)*B*b*d*log(e)/(d* x + c) + 18*(b*x + a)^2*B*d^2*log(e)/(d*x + c)^2 + 6*A*b^2 - 18*(b*x + a)* A*b*d/(d*x + c) + 18*(b*x + a)^2*A*d^2/(d*x + c)^2)/((b*x + a)^3*b^2*c^2*g ^4*i^2/(d*x + c)^3 - 2*(b*x + a)^3*a*b*c*d*g^4*i^2/(d*x + c)^3 + (b*x + a) ^3*a^2*d^2*g^4*i^2/(d*x + c)^3))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)^2
Time = 6.58 (sec) , antiderivative size = 1665, normalized size of antiderivative = 3.49 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)^2} \, dx=\text {Too large to display} \]
(2*B*b*d^3*log(e*((a + b*x)/(c + d*x))^n)^2)/(g^4*i^2*n*(a*d - b*c)^3*(a^2 *d^2 + b^2*c^2 - 2*a*b*c*d)) - log(e*((a + b*x)/(c + d*x))^n)*(((B*(3*a*d + b*c))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (4*B*b*d*x)/(3*(a^2*d^2 + b^ 2*c^2 - 2*a*b*c*d)))/(x^3*(b^3*c*g^4*i^2 + 3*a*b^2*d*g^4*i^2) + x^2*(3*a*b ^2*c*g^4*i^2 + 3*a^2*b*d*g^4*i^2) + x*(a^3*d*g^4*i^2 + 3*a^2*b*c*g^4*i^2) + a^3*c*g^4*i^2 + b^3*d*g^4*i^2*x^4) + (4*B*b*d^3*(x*((a*d + b*c)*((a*g^4* i^2*n*(a*d - b*c))/(2*d) + (g^4*i^2*n*(a*d - b*c)*(2*a*d - b*c))/(2*d^2)) + (a*b*c*g^4*i^2*n*(a*d - b*c))/d) + x^2*(b*d*((a*g^4*i^2*n*(a*d - b*c))/( 2*d) + (g^4*i^2*n*(a*d - b*c)*(2*a*d - b*c))/(2*d^2)) + (b*g^4*i^2*n*(a*d + b*c)*(a*d - b*c))/d) + a*c*((a*g^4*i^2*n*(a*d - b*c))/(2*d) + (g^4*i^2*n *(a*d - b*c)*(2*a*d - b*c))/(2*d^2)) + b^2*g^4*i^2*n*x^3*(a*d - b*c)))/(g^ 4*i^2*n*(a*d - b*c)^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(x^3*(b^3*c*g^4*i^2 + 3*a*b^2*d*g^4*i^2) + x^2*(3*a*b^2*c*g^4*i^2 + 3*a^2*b*d*g^4*i^2) + x*(a^ 3*d*g^4*i^2 + 3*a^2*b*c*g^4*i^2) + a^3*c*g^4*i^2 + b^3*d*g^4*i^2*x^4))) - (b*d^3*atan((b*d^3*((a^5*d^5*g^4*i^2 + b^5*c^5*g^4*i^2 - 3*a*b^4*c^4*d*g^4 *i^2 - 3*a^4*b*c*d^4*g^4*i^2 + 2*a^2*b^3*c^3*d^2*g^4*i^2 + 2*a^3*b^2*c^2*d ^3*g^4*i^2)/(a^4*d^4*g^4*i^2 + b^4*c^4*g^4*i^2 - 4*a*b^3*c^3*d*g^4*i^2 - 4 *a^3*b*c*d^3*g^4*i^2 + 6*a^2*b^2*c^2*d^2*g^4*i^2) + 2*b*d*x)*(6*A + 5*B*n) *(a^4*d^4*g^4*i^2 + b^4*c^4*g^4*i^2 - 4*a*b^3*c^3*d*g^4*i^2 - 4*a^3*b*c*d^ 3*g^4*i^2 + 6*a^2*b^2*c^2*d^2*g^4*i^2)*2i)/(g^4*i^2*(12*A*b*d^3 + 10*B*...