Integrand size = 43, antiderivative size = 381 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\frac {B d^3 n (a+b x)^2}{4 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {3 b B d^2 n (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 B n (c+d x)}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {d^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^4 g^2 i^3}+\frac {3 b^2 B d n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^2 i^3} \]
1/4*B*d^3*n*(b*x+a)^2/(-a*d+b*c)^4/g^2/i^3/(d*x+c)^2-3*b*B*d^2*n*(b*x+a)/( -a*d+b*c)^4/g^2/i^3/(d*x+c)-b^3*B*n*(d*x+c)/(-a*d+b*c)^4/g^2/i^3/(b*x+a)-1 /2*d^3*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^2/i^3/(d*x +c)^2+3*b*d^2*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^2/i^3 /(d*x+c)-b^3*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^2/i^3/ (b*x+a)-3*b^2*d*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((b*x+a)/(d*x+c))/(-a*d+ b*c)^4/g^2/i^3+3/2*b^2*B*d*n*ln((b*x+a)/(d*x+c))^2/(-a*d+b*c)^4/g^2/i^3
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.39 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.25 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\frac {-\frac {4 b^3 B c n}{a+b x}+\frac {4 a b^2 B d n}{a+b x}+\frac {B d (b c-a d)^2 n}{(c+d x)^2}+\frac {8 b^2 B c d n}{c+d x}-\frac {8 a b B d^2 n}{c+d x}+\frac {2 b B d (b c-a d) n}{c+d x}+6 b^2 B d n \log (a+b x)-\frac {4 b^2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}-\frac {2 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2}-\frac {8 b d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x}-12 b^2 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 b^2 B d n \log (c+d x)+12 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+6 b^2 B d n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-6 b^2 B d n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{4 (b c-a d)^4 g^2 i^3} \]
((-4*b^3*B*c*n)/(a + b*x) + (4*a*b^2*B*d*n)/(a + b*x) + (B*d*(b*c - a*d)^2 *n)/(c + d*x)^2 + (8*b^2*B*c*d*n)/(c + d*x) - (8*a*b*B*d^2*n)/(c + d*x) + (2*b*B*d*(b*c - a*d)*n)/(c + d*x) + 6*b^2*B*d*n*Log[a + b*x] - (4*b^2*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) - (2*d*(b*c - a* d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x)^2 - (8*b*d*(b*c - a *d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x) - 12*b^2*d*Log[a + b *x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 6*b^2*B*d*n*Log[c + d*x] + 12 *b^2*d*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 6*b^2*B*d*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)]) - 6*b^2*B*d*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)]))/(4*(b*c - a*d)^4*g^2*i^3)
Time = 0.44 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.70, 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.
| \(\displaystyle \int \frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{(a g+b g x)^2 (c i+d i x)^3} \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle \frac {\int \frac {(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2}d\frac {a+b x}{c+d x}}{g^2 i^3 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {-B n \int \left (-\frac {(c+d x)^2 b^3}{(a+b x)^2}-\frac {3 d (c+d x) \log \left (\frac {a+b x}{c+d x}\right ) b^2}{a+b x}+3 d^2 b-\frac {d^3 (a+b x)}{2 (c+d x)}\right )d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-3 b^2 d \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-\frac {d^3 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}}{g^2 i^3 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b^3 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-3 b^2 d \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-\frac {d^3 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}-B n \left (\frac {b^3 (c+d x)}{a+b x}-\frac {3}{2} b^2 d \log ^2\left (\frac {a+b x}{c+d x}\right )-\frac {d^3 (a+b x)^2}{4 (c+d x)^2}+\frac {3 b d^2 (a+b x)}{c+d x}\right )}{g^2 i^3 (b c-a d)^4}\) |
(-1/2*(d^3*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x)^2 + (3*b*d^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x) - (b^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) - 3*b^2*d *(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)] - B*n*(-1 /4*(d^3*(a + b*x)^2)/(c + d*x)^2 + (3*b*d^2*(a + b*x))/(c + d*x) + (b^3*(c + d*x))/(a + b*x) - (3*b^2*d*Log[(a + b*x)/(c + d*x)]^2)/2))/((b*c - a*d) ^4*g^2*i^3)
3.2.56.3.1 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. \(961\) vs. \(2(375)=750\).
Time = 28.55 (sec) , antiderivative size = 962, normalized size of antiderivative = 2.52
| method | result | size |
| parallelrisch | \(-\frac {-24 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} c \,d^{6} n +6 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{6} d^{7}+12 A \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} d^{7}-18 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} d^{7} n +12 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a \,b^{5} c \,d^{6}-6 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{4} d^{7} n +12 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} c^{2} d^{5} n -6 B x a \,b^{5} c \,d^{6} n^{2}+24 A x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} c \,d^{6}-12 A x a \,b^{5} c \,d^{6} n -12 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{4} c \,d^{6} n -6 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} d^{7} n +6 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a \,b^{5} d^{7}+12 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{6} c \,d^{6}+6 B \,x^{2} a \,b^{5} d^{7} n^{2}-6 B \,x^{2} b^{6} c \,d^{6} n^{2}+12 A \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} d^{7}+24 A \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} c \,d^{6}-12 A \,x^{2} a \,b^{5} d^{7} n +12 A \,x^{2} b^{6} c \,d^{6} n +6 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{6} c^{2} d^{5}+9 B x \,a^{2} b^{4} d^{7} n^{2}-3 B x \,b^{6} c^{2} d^{5} n^{2}+12 A x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} c^{2} d^{5}-6 A x \,a^{2} b^{4} d^{7} n +18 A x \,b^{6} c^{2} d^{5} n +6 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a \,b^{5} c^{2} d^{5}+2 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{3} b^{3} d^{7} n +4 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} c^{3} d^{4} n +12 A \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} c^{2} d^{5}-B \,a^{3} b^{3} d^{7} n^{2}+4 B \,b^{6} c^{3} d^{4} n^{2}+2 A \,a^{3} b^{3} d^{7} n +4 A \,b^{6} c^{3} d^{4} n +12 B \,a^{2} b^{4} c \,d^{6} n^{2}-15 B a \,b^{5} c^{2} d^{5} n^{2}-12 A \,a^{2} b^{4} c \,d^{6} n +6 A a \,b^{5} c^{2} d^{5} n}{4 i^{3} g^{2} \left (d x +c \right )^{2} \left (b x +a \right ) \left (a d -c b \right )^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{3} d^{4} n}\) | \(962\) |
-1/4*(-24*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c*d^6*n+6*B*x^3*ln(e*((b*x+a )/(d*x+c))^n)^2*b^6*d^7+12*A*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^6*d^7-18*B*x^ 2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*d^7*n+12*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2 *a*b^5*c*d^6-6*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*d^7*n+12*B*x*ln(e*((b *x+a)/(d*x+c))^n)*b^6*c^2*d^5*n-6*B*x*a*b^5*c*d^6*n^2+24*A*x*ln(e*((b*x+a) /(d*x+c))^n)*a*b^5*c*d^6-12*A*x*a*b^5*c*d^6*n-12*B*ln(e*((b*x+a)/(d*x+c))^ n)*a^2*b^4*c*d^6*n-6*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^6*d^7*n+6*B*x^2*ln( e*((b*x+a)/(d*x+c))^n)^2*a*b^5*d^7+12*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*b^ 6*c*d^6+6*B*x^2*a*b^5*d^7*n^2-6*B*x^2*b^6*c*d^6*n^2+12*A*x^2*ln(e*((b*x+a) /(d*x+c))^n)*a*b^5*d^7+24*A*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c*d^6-12*A*x ^2*a*b^5*d^7*n+12*A*x^2*b^6*c*d^6*n+6*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*b^6* c^2*d^5+9*B*x*a^2*b^4*d^7*n^2-3*B*x*b^6*c^2*d^5*n^2+12*A*x*ln(e*((b*x+a)/( d*x+c))^n)*b^6*c^2*d^5-6*A*x*a^2*b^4*d^7*n+18*A*x*b^6*c^2*d^5*n+6*B*ln(e*( (b*x+a)/(d*x+c))^n)^2*a*b^5*c^2*d^5+2*B*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^3* d^7*n+4*B*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^3*d^4*n+12*A*ln(e*((b*x+a)/(d*x+ c))^n)*a*b^5*c^2*d^5-B*a^3*b^3*d^7*n^2+4*B*b^6*c^3*d^4*n^2+2*A*a^3*b^3*d^7 *n+4*A*b^6*c^3*d^4*n+12*B*a^2*b^4*c*d^6*n^2-15*B*a*b^5*c^2*d^5*n^2-12*A*a^ 2*b^4*c*d^6*n+6*A*a*b^5*c^2*d^5*n)/i^3/g^2/(d*x+c)^2/(b*x+a)/(a*d-b*c)^2/( a^2*d^2-2*a*b*c*d+b^2*c^2)/b^3/d^4/n
Leaf count of result is larger than twice the leaf count of optimal. 949 vs. \(2 (375) = 750\).
Time = 0.38 (sec) , antiderivative size = 949, normalized size of antiderivative = 2.49 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=-\frac {4 \, A b^{3} c^{3} + 6 \, A a b^{2} c^{2} d - 12 \, A a^{2} b c d^{2} + 2 \, A a^{3} d^{3} + 6 \, {\left (2 \, A b^{3} c d^{2} - 2 \, A a b^{2} d^{3} - {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n\right )} x^{2} + 6 \, {\left (B b^{3} d^{3} n x^{3} + B a b^{2} c^{2} d n + {\left (2 \, B b^{3} c d^{2} + B a b^{2} d^{3}\right )} n x^{2} + {\left (B b^{3} c^{2} d + 2 \, B a b^{2} c d^{2}\right )} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (4 \, B b^{3} c^{3} - 15 \, B a b^{2} c^{2} d + 12 \, B a^{2} b c d^{2} - B a^{3} d^{3}\right )} n + 3 \, {\left (6 \, A b^{3} c^{2} d - 4 \, A a b^{2} c d^{2} - 2 \, A a^{2} b d^{3} - {\left (B b^{3} c^{2} d + 2 \, B a b^{2} c d^{2} - 3 \, B a^{2} b d^{3}\right )} n\right )} x + 2 \, {\left (2 \, B b^{3} c^{3} + 3 \, B a b^{2} c^{2} d - 6 \, B a^{2} b c d^{2} + B a^{3} d^{3} + 6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (3 \, B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2} - B a^{2} b d^{3}\right )} x + 6 \, {\left (B b^{3} d^{3} x^{3} + B a b^{2} c^{2} d + {\left (2 \, B b^{3} c d^{2} + B a b^{2} d^{3}\right )} x^{2} + {\left (B b^{3} c^{2} d + 2 \, B a b^{2} c d^{2}\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \left (e\right ) + 2 \, {\left (6 \, A a b^{2} c^{2} d - 3 \, {\left (B b^{3} d^{3} n - 2 \, A b^{3} d^{3}\right )} x^{3} - 3 \, {\left (3 \, B a b^{2} d^{3} n - 4 \, A b^{3} c d^{2} - 2 \, A a b^{2} d^{3}\right )} x^{2} + {\left (2 \, B b^{3} c^{3} - 6 \, B a^{2} b c d^{2} + B a^{3} d^{3}\right )} n + 3 \, {\left (2 \, A b^{3} c^{2} d + 4 \, A a b^{2} c d^{2} + {\left (2 \, B b^{3} c^{2} d - 4 \, B a b^{2} c d^{2} - B a^{2} b d^{3}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} g^{2} i^{3} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} g^{2} i^{3} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} g^{2} i^{3} x + {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4}\right )} g^{2} i^{3}\right )}} \]
-1/4*(4*A*b^3*c^3 + 6*A*a*b^2*c^2*d - 12*A*a^2*b*c*d^2 + 2*A*a^3*d^3 + 6*( 2*A*b^3*c*d^2 - 2*A*a*b^2*d^3 - (B*b^3*c*d^2 - B*a*b^2*d^3)*n)*x^2 + 6*(B* b^3*d^3*n*x^3 + B*a*b^2*c^2*d*n + (2*B*b^3*c*d^2 + B*a*b^2*d^3)*n*x^2 + (B *b^3*c^2*d + 2*B*a*b^2*c*d^2)*n*x)*log((b*x + a)/(d*x + c))^2 + (4*B*b^3*c ^3 - 15*B*a*b^2*c^2*d + 12*B*a^2*b*c*d^2 - B*a^3*d^3)*n + 3*(6*A*b^3*c^2*d - 4*A*a*b^2*c*d^2 - 2*A*a^2*b*d^3 - (B*b^3*c^2*d + 2*B*a*b^2*c*d^2 - 3*B* a^2*b*d^3)*n)*x + 2*(2*B*b^3*c^3 + 3*B*a*b^2*c^2*d - 6*B*a^2*b*c*d^2 + B*a ^3*d^3 + 6*(B*b^3*c*d^2 - B*a*b^2*d^3)*x^2 + 3*(3*B*b^3*c^2*d - 2*B*a*b^2* c*d^2 - B*a^2*b*d^3)*x + 6*(B*b^3*d^3*x^3 + B*a*b^2*c^2*d + (2*B*b^3*c*d^2 + B*a*b^2*d^3)*x^2 + (B*b^3*c^2*d + 2*B*a*b^2*c*d^2)*x)*log((b*x + a)/(d* x + c)))*log(e) + 2*(6*A*a*b^2*c^2*d - 3*(B*b^3*d^3*n - 2*A*b^3*d^3)*x^3 - 3*(3*B*a*b^2*d^3*n - 4*A*b^3*c*d^2 - 2*A*a*b^2*d^3)*x^2 + (2*B*b^3*c^3 - 6*B*a^2*b*c*d^2 + B*a^3*d^3)*n + 3*(2*A*b^3*c^2*d + 4*A*a*b^2*c*d^2 + (2*B *b^3*c^2*d - 4*B*a*b^2*c*d^2 - B*a^2*b*d^3)*n)*x)*log((b*x + a)/(d*x + c)) )/((b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 - 4*a^3*b^2*c*d^5 + a^4*b*d^6)*g^2*i^3*x^3 + (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 + 8*a^2*b^3*c^3*d^ 3 - 2*a^3*b^2*c^2*d^4 - 2*a^4*b*c*d^5 + a^5*d^6)*g^2*i^3*x^2 + (b^5*c^6 - 2*a*b^4*c^5*d - 2*a^2*b^3*c^4*d^2 + 8*a^3*b^2*c^3*d^3 - 7*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*g^2*i^3*x + (a*b^4*c^6 - 4*a^2*b^3*c^5*d + 6*a^3*b^2*c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*c^2*d^4)*g^2*i^3)
Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1724 vs. \(2 (375) = 750\).
Time = 0.30 (sec) , antiderivative size = 1724, normalized size of antiderivative = 4.52 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Too large to display} \]
-1/2*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + a*b*d^2)*x)/((b^4*c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5) *g^2*i^3*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*b* c*d^4 - a^4*d^5)*g^2*i^3*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*g^2*i^3*x + (a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3)*g^2*i^3) + 6*b^2*d*log(b*x + a)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3) - 6*b^2*d*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^ 3*b*c*d^3 + a^4*d^4)*g^2*i^3))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) - 1/ 4*(4*b^3*c^3 - 15*a*b^2*c^2*d + 12*a^2*b*c*d^2 - a^3*d^3 - 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)* x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a)^2 - 6*(b^3*d^3*x^3 + a*b ^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)* log(d*x + c)^2 - 3*(b^3*c^2*d + 2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x - 6*(b^3*d^ 3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2 *c*d^2)*x)*log(b*x + a) + 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a* b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x + 2*(b^3*d^3*x^3 + a*b^2*c^2* d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a))*log(d*x + c))*B*n/(a*b^4*c^6*g^2*i^3 - 4*a^2*b^3*c^5*d*g^2*i^3 + 6* a^3*b^2*c^4*d^2*g^2*i^3 - 4*a^4*b*c^3*d^3*g^2*i^3 + a^5*c^2*d^4*g^2*i^3...
\[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\int { \frac {B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}{{\left (b g x + a g\right )}^{2} {\left (d i x + c i\right )}^{3}} \,d x } \]
Time = 4.07 (sec) , antiderivative size = 1018, normalized size of antiderivative = 2.67 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\frac {\frac {4\,A\,b^2\,c^2-2\,A\,a^2\,d^2+B\,a^2\,d^2\,n+4\,B\,b^2\,c^2\,n+10\,A\,a\,b\,c\,d-11\,B\,a\,b\,c\,d\,n}{2\,\left (a\,d-b\,c\right )}+\frac {3\,x^2\,\left (2\,A\,b^2\,d^2-B\,b^2\,d^2\,n\right )}{a\,d-b\,c}+\frac {3\,x\,\left (2\,A\,a\,b\,d^2+6\,A\,b^2\,c\,d-3\,B\,a\,b\,d^2\,n-B\,b^2\,c\,d\,n\right )}{2\,\left (a\,d-b\,c\right )}}{x\,\left (4\,a^3\,c\,d^3\,g^2\,i^3-6\,a^2\,b\,c^2\,d^2\,g^2\,i^3+2\,b^3\,c^4\,g^2\,i^3\right )+x^2\,\left (2\,a^3\,d^4\,g^2\,i^3-6\,a\,b^2\,c^2\,d^2\,g^2\,i^3+4\,b^3\,c^3\,d\,g^2\,i^3\right )+x^3\,\left (2\,a^2\,b\,d^4\,g^2\,i^3-4\,a\,b^2\,c\,d^3\,g^2\,i^3+2\,b^3\,c^2\,d^2\,g^2\,i^3\right )+2\,a^3\,c^2\,d^2\,g^2\,i^3+2\,a\,b^2\,c^4\,g^2\,i^3-4\,a^2\,b\,c^3\,d\,g^2\,i^3}-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {\frac {B\,\left (a\,d+2\,b\,c\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {3\,B\,b\,d\,x}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x\,\left (b\,c^2\,g^2\,i^3+2\,a\,d\,c\,g^2\,i^3\right )+x^2\,\left (a\,d^2\,g^2\,i^3+2\,b\,c\,d\,g^2\,i^3\right )+a\,c^2\,g^2\,i^3+b\,d^2\,g^2\,i^3\,x^3}-\frac {3\,B\,b^2\,d\,\left (d\,g^2\,i^3\,n\,x^2\,\left (a\,d-b\,c\right )+\frac {a\,c\,g^2\,i^3\,n\,\left (a\,d-b\,c\right )}{b}+\frac {g^2\,i^3\,n\,x\,\left (a\,d+b\,c\right )\,\left (a\,d-b\,c\right )}{b}\right )}{g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^4\,\left (x\,\left (b\,c^2\,g^2\,i^3+2\,a\,d\,c\,g^2\,i^3\right )+x^2\,\left (a\,d^2\,g^2\,i^3+2\,b\,c\,d\,g^2\,i^3\right )+a\,c^2\,g^2\,i^3+b\,d^2\,g^2\,i^3\,x^3\right )}\right )-\frac {3\,B\,b^2\,d\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{2\,g^2\,i^3\,n\,{\left (a\,d-b\,c\right )}^4}+\frac {b^2\,d\,\mathrm {atan}\left (\frac {b^2\,d\,\left (2\,A-B\,n\right )\,\left (\frac {a^4\,d^4\,g^2\,i^3-2\,a^3\,b\,c\,d^3\,g^2\,i^3+2\,a\,b^3\,c^3\,d\,g^2\,i^3-b^4\,c^4\,g^2\,i^3}{a^3\,d^3\,g^2\,i^3-3\,a^2\,b\,c\,d^2\,g^2\,i^3+3\,a\,b^2\,c^2\,d\,g^2\,i^3-b^3\,c^3\,g^2\,i^3}+2\,b\,d\,x\right )\,\left (a^3\,d^3\,g^2\,i^3-3\,a^2\,b\,c\,d^2\,g^2\,i^3+3\,a\,b^2\,c^2\,d\,g^2\,i^3-b^3\,c^3\,g^2\,i^3\right )\,3{}\mathrm {i}}{g^2\,i^3\,\left (6\,A\,b^2\,d-3\,B\,b^2\,d\,n\right )\,{\left (a\,d-b\,c\right )}^4}\right )\,\left (2\,A-B\,n\right )\,3{}\mathrm {i}}{g^2\,i^3\,{\left (a\,d-b\,c\right )}^4} \]
((4*A*b^2*c^2 - 2*A*a^2*d^2 + B*a^2*d^2*n + 4*B*b^2*c^2*n + 10*A*a*b*c*d - 11*B*a*b*c*d*n)/(2*(a*d - b*c)) + (3*x^2*(2*A*b^2*d^2 - B*b^2*d^2*n))/(a* d - b*c) + (3*x*(2*A*a*b*d^2 + 6*A*b^2*c*d - 3*B*a*b*d^2*n - B*b^2*c*d*n)) /(2*(a*d - b*c)))/(x*(2*b^3*c^4*g^2*i^3 + 4*a^3*c*d^3*g^2*i^3 - 6*a^2*b*c^ 2*d^2*g^2*i^3) + x^2*(2*a^3*d^4*g^2*i^3 + 4*b^3*c^3*d*g^2*i^3 - 6*a*b^2*c^ 2*d^2*g^2*i^3) + x^3*(2*b^3*c^2*d^2*g^2*i^3 + 2*a^2*b*d^4*g^2*i^3 - 4*a*b^ 2*c*d^3*g^2*i^3) + 2*a^3*c^2*d^2*g^2*i^3 + 2*a*b^2*c^4*g^2*i^3 - 4*a^2*b*c ^3*d*g^2*i^3) - log(e*((a + b*x)/(c + d*x))^n)*(((B*(a*d + 2*b*c))/(2*(a^2 *d^2 + b^2*c^2 - 2*a*b*c*d)) + (3*B*b*d*x)/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c *d)))/(x*(b*c^2*g^2*i^3 + 2*a*c*d*g^2*i^3) + x^2*(a*d^2*g^2*i^3 + 2*b*c*d* g^2*i^3) + a*c^2*g^2*i^3 + b*d^2*g^2*i^3*x^3) - (3*B*b^2*d*(d*g^2*i^3*n*x^ 2*(a*d - b*c) + (a*c*g^2*i^3*n*(a*d - b*c))/b + (g^2*i^3*n*x*(a*d + b*c)*( a*d - b*c))/b))/(g^2*i^3*n*(a*d - b*c)^4*(x*(b*c^2*g^2*i^3 + 2*a*c*d*g^2*i ^3) + x^2*(a*d^2*g^2*i^3 + 2*b*c*d*g^2*i^3) + a*c^2*g^2*i^3 + b*d^2*g^2*i^ 3*x^3))) + (b^2*d*atan((b^2*d*(2*A - B*n)*((a^4*d^4*g^2*i^3 - b^4*c^4*g^2* i^3 + 2*a*b^3*c^3*d*g^2*i^3 - 2*a^3*b*c*d^3*g^2*i^3)/(a^3*d^3*g^2*i^3 - b^ 3*c^3*g^2*i^3 + 3*a*b^2*c^2*d*g^2*i^3 - 3*a^2*b*c*d^2*g^2*i^3) + 2*b*d*x)* (a^3*d^3*g^2*i^3 - b^3*c^3*g^2*i^3 + 3*a*b^2*c^2*d*g^2*i^3 - 3*a^2*b*c*d^2 *g^2*i^3)*3i)/(g^2*i^3*(6*A*b^2*d - 3*B*b^2*d*n)*(a*d - b*c)^4))*(2*A - B* n)*3i)/(g^2*i^3*(a*d - b*c)^4) - (3*B*b^2*d*log(e*((a + b*x)/(c + d*x))...