Integrand size = 43, antiderivative size = 266 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {B n (c+d x)^2 \left (b-\frac {4 d (a+b x)}{c+d x}\right )^2}{4 (b c-a d)^3 g^3 i (a+b x)^2}+\frac {2 b d (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^3 i (a+b x)}-\frac {b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g^3 i (a+b x)^2}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^3 i}-\frac {B d^2 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^3 g^3 i} \]
-1/4*B*n*(d*x+c)^2*(b-4*d*(b*x+a)/(d*x+c))^2/(-a*d+b*c)^3/g^3/i/(b*x+a)^2+ 2*b*d*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^3/i/(b*x+a)-1 /2*b^2*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^3/i/(b*x+a )^2+d^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((b*x+a)/(d*x+c))/(-a*d+b*c)^3/g ^3/i-1/2*B*d^2*n*ln((b*x+a)/(d*x+c))^2/(-a*d+b*c)^3/g^3/i
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.22 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.63 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {-2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+4 d (b c-a d) (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-4 d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+4 B d n (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B n \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )-2 B d^2 n (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+2 B d^2 n (a+b x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{4 (b c-a d)^3 g^3 i (a+b x)^2} \]
(-2*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 4*d*(b*c - a*d) *(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 4*d^2*(a + b*x)^2*Log[ a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 4*d^2*(a + b*x)^2*(A + B *Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 4*B*d*n*(a + b*x)*(b*c - a *d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) - B*n*((b*c - a* d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^2*(a + b*x)^2*Log[a + b*x] + 2*d ^2*(a + b*x)^2*Log[c + d*x]) - 2*B*d^2*n*(a + b*x)^2*(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)]) + 2*B*d^2*n*(a + b*x)^2*((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)^3*g^3*i*(a + b*x)^2)
Time = 0.45 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.77, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2961, 2772, 27, 2010, 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.
| \(\displaystyle \int \frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{(a g+b g x)^3 (c i+d i x)} \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle \frac {\int \frac {(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^3}d\frac {a+b x}{c+d x}}{g^3 i (b c-a d)^3}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {-B n \int -\frac {(c+d x)^3 \left (b^2-\frac {4 d (a+b x) b}{c+d x}-\frac {2 d^2 (a+b x)^2 \log \left (\frac {a+b x}{c+d x}\right )}{(c+d x)^2}\right )}{2 (a+b x)^3}d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+\frac {2 b d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}}{g^3 i (b c-a d)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} B n \int \frac {(c+d x)^3 \left (b^2-\frac {4 d (a+b x) b}{c+d x}-\frac {2 d^2 (a+b x)^2 \log \left (\frac {a+b x}{c+d x}\right )}{(c+d x)^2}\right )}{(a+b x)^3}d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+\frac {2 b d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}}{g^3 i (b c-a d)^3}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {\frac {1}{2} B n \int \left (\frac {b (c+d x)^3 \left (b-\frac {4 d (a+b x)}{c+d x}\right )}{(a+b x)^3}-\frac {2 d^2 (c+d x) \log \left (\frac {a+b x}{c+d x}\right )}{a+b x}\right )d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+\frac {2 b d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}}{g^3 i (b c-a d)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+\frac {2 b d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+\frac {1}{2} B n \left (-d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )-\frac {(c+d x)^2 \left (b-\frac {4 d (a+b x)}{c+d x}\right )^2}{2 (a+b x)^2}\right )}{g^3 i (b c-a d)^3}\) |
((2*b*d*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) - (b^2 *(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a + b*x)^2) + d^2 *(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)] + (B*n*(- 1/2*((c + d*x)^2*(b - (4*d*(a + b*x))/(c + d*x))^2)/(a + b*x)^2 - d^2*Log[ (a + b*x)/(c + d*x)]^2))/2)/((b*c - a*d)^3*g^3*i)
3.2.41.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
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. \(627\) vs. \(2(260)=520\).
Time = 11.92 (sec) , antiderivative size = 628, normalized size of antiderivative = 2.36
| method | result | size |
| parallelrisch | \(-\frac {6 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{4} b^{2} c^{2} d^{2} n +8 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{5} b \,c^{2} d^{2} n +4 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{4} b^{2} c^{3} d n -2 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{4} b^{2} c^{4} n +B \,x^{2} a^{2} b^{4} c^{4} n^{2}+2 A \,x^{2} a^{2} b^{4} c^{4} n +2 B x \,a^{3} b^{3} c^{4} n^{2}+4 A x \,a^{3} b^{3} c^{4} n +2 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a^{6} c^{2} d^{2}+4 A \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{6} c^{2} d^{2}+2 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a^{4} b^{2} c^{2} d^{2}+7 B \,x^{2} a^{4} b^{2} c^{2} d^{2} n^{2}-8 B \,x^{2} a^{3} b^{3} c^{3} d \,n^{2}+4 A \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{4} b^{2} c^{2} d^{2}+6 A \,x^{2} a^{4} b^{2} c^{2} d^{2} n -8 A \,x^{2} a^{3} b^{3} c^{3} d n +4 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a^{5} b \,c^{2} d^{2}+8 B x \,a^{5} b \,c^{2} d^{2} n^{2}-10 B x \,a^{4} b^{2} c^{3} d \,n^{2}+8 A x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{5} b \,c^{2} d^{2}+8 A x \,a^{5} b \,c^{2} d^{2} n -12 A x \,a^{4} b^{2} c^{3} d n +8 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{5} b \,c^{3} d n}{4 i \,g^{3} \left (b x +a \right )^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -c b \right ) a^{4} c^{2} n}\) | \(628\) |
-1/4*(6*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b^2*c^2*d^2*n+8*B*x*ln(e*((b*x +a)/(d*x+c))^n)*a^5*b*c^2*d^2*n+4*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b^2*c^ 3*d*n-2*B*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b^2*c^4*n+B*x^2*a^2*b^4*c^4*n^2+2* A*x^2*a^2*b^4*c^4*n+2*B*x*a^3*b^3*c^4*n^2+4*A*x*a^3*b^3*c^4*n+2*B*ln(e*((b *x+a)/(d*x+c))^n)^2*a^6*c^2*d^2+4*A*ln(e*((b*x+a)/(d*x+c))^n)*a^6*c^2*d^2+ 2*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^4*b^2*c^2*d^2+7*B*x^2*a^4*b^2*c^2*d^ 2*n^2-8*B*x^2*a^3*b^3*c^3*d*n^2+4*A*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^4*b^2* c^2*d^2+6*A*x^2*a^4*b^2*c^2*d^2*n-8*A*x^2*a^3*b^3*c^3*d*n+4*B*x*ln(e*((b*x +a)/(d*x+c))^n)^2*a^5*b*c^2*d^2+8*B*x*a^5*b*c^2*d^2*n^2-10*B*x*a^4*b^2*c^3 *d*n^2+8*A*x*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b*c^2*d^2+8*A*x*a^5*b*c^2*d^2*n -12*A*x*a^4*b^2*c^3*d*n+8*B*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b*c^3*d*n)/i/g^3 /(b*x+a)^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)/a^4/c^2/n
Time = 0.31 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.82 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {2 \, A b^{2} c^{2} - 8 \, A a b c d + 6 \, A a^{2} d^{2} - 2 \, {\left (B b^{2} d^{2} n x^{2} + 2 \, B a b d^{2} n x + B a^{2} d^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (B b^{2} c^{2} - 8 \, B a b c d + 7 \, B a^{2} d^{2}\right )} n - 2 \, {\left (2 \, A b^{2} c d - 2 \, A a b d^{2} + 3 \, {\left (B b^{2} c d - B a b d^{2}\right )} n\right )} x + 2 \, {\left (B b^{2} c^{2} - 4 \, B a b c d + 3 \, B a^{2} d^{2} - 2 \, {\left (B b^{2} c d - B a b d^{2}\right )} x - 2 \, {\left (B b^{2} d^{2} x^{2} + 2 \, B a b d^{2} x + B a^{2} d^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \left (e\right ) - 2 \, {\left (2 \, A a^{2} d^{2} + {\left (3 \, B b^{2} d^{2} n + 2 \, A b^{2} d^{2}\right )} x^{2} - {\left (B b^{2} c^{2} - 4 \, B a b c d\right )} n + 2 \, {\left (2 \, A a b d^{2} + {\left (B b^{2} c d + 2 \, B a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} g^{3} i x + {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} g^{3} i\right )}} \]
-1/4*(2*A*b^2*c^2 - 8*A*a*b*c*d + 6*A*a^2*d^2 - 2*(B*b^2*d^2*n*x^2 + 2*B*a *b*d^2*n*x + B*a^2*d^2*n)*log((b*x + a)/(d*x + c))^2 + (B*b^2*c^2 - 8*B*a* b*c*d + 7*B*a^2*d^2)*n - 2*(2*A*b^2*c*d - 2*A*a*b*d^2 + 3*(B*b^2*c*d - B*a *b*d^2)*n)*x + 2*(B*b^2*c^2 - 4*B*a*b*c*d + 3*B*a^2*d^2 - 2*(B*b^2*c*d - B *a*b*d^2)*x - 2*(B*b^2*d^2*x^2 + 2*B*a*b*d^2*x + B*a^2*d^2)*log((b*x + a)/ (d*x + c)))*log(e) - 2*(2*A*a^2*d^2 + (3*B*b^2*d^2*n + 2*A*b^2*d^2)*x^2 - (B*b^2*c^2 - 4*B*a*b*c*d)*n + 2*(2*A*a*b*d^2 + (B*b^2*c*d + 2*B*a*b*d^2)*n )*x)*log((b*x + a)/(d*x + c)))/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^3*i*x^2 + 2*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d ^2 - a^4*b*d^3)*g^3*i*x + (a^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3)*g^3*i)
Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 888 vs. \(2 (260) = 520\).
Time = 0.25 (sec) , antiderivative size = 888, normalized size of antiderivative = 3.34 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {1}{2} \, B {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} g^{3} i x + {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} g^{3} i} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) - \frac {{\left (b^{2} c^{2} - 8 \, a b c d + 7 \, a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (d x + c\right )^{2} - 6 \, {\left (b^{2} c d - a b d^{2}\right )} x - 6 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, a b d^{2} x + 3 \, a^{2} d^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B n}{4 \, {\left (a^{2} b^{3} c^{3} g^{3} i - 3 \, a^{3} b^{2} c^{2} d g^{3} i + 3 \, a^{4} b c d^{2} g^{3} i - a^{5} d^{3} g^{3} i + {\left (b^{5} c^{3} g^{3} i - 3 \, a b^{4} c^{2} d g^{3} i + 3 \, a^{2} b^{3} c d^{2} g^{3} i - a^{3} b^{2} d^{3} g^{3} i\right )} x^{2} + 2 \, {\left (a b^{4} c^{3} g^{3} i - 3 \, a^{2} b^{3} c^{2} d g^{3} i + 3 \, a^{3} b^{2} c d^{2} g^{3} i - a^{4} b d^{3} g^{3} i\right )} x\right )}} + \frac {1}{2} \, A {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} g^{3} i x + {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} g^{3} i} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i}\right )} \]
1/2*B*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^3* i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*g^3*i*x + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*g^3*i) + 2*d^2*log(b*x + a)/((b^3*c^3 - 3*a*b^2*c^ 2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x + c)/((b^3*c^3 - 3*a *b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i))*log(e*(b*x/(d*x + c) + a/(d* x + c))^n) - 1/4*(b^2*c^2 - 8*a*b*c*d + 7*a^2*d^2 + 2*(b^2*d^2*x^2 + 2*a*b *d^2*x + a^2*d^2)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2) *log(d*x + c)^2 - 6*(b^2*c*d - a*b*d^2)*x - 6*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a) + 2*(3*b^2*d^2*x^2 + 6*a*b*d^2*x + 3*a^2*d^2 - 2*(b ^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a))*log(d*x + c))*B*n/(a^2*b ^3*c^3*g^3*i - 3*a^3*b^2*c^2*d*g^3*i + 3*a^4*b*c*d^2*g^3*i - a^5*d^3*g^3*i + (b^5*c^3*g^3*i - 3*a*b^4*c^2*d*g^3*i + 3*a^2*b^3*c*d^2*g^3*i - a^3*b^2* d^3*g^3*i)*x^2 + 2*(a*b^4*c^3*g^3*i - 3*a^2*b^3*c^2*d*g^3*i + 3*a^3*b^2*c* d^2*g^3*i - a^4*b*d^3*g^3*i)*x) + 1/2*A*((2*b*d*x - b*c + 3*a*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^3*i*x^2 + 2*(a*b^3*c^2 - 2*a^2*b^2*c*d + a ^3*b*d^2)*g^3*i*x + (a^2*b^2*c^2 - 2*a^3*b*c*d + a^4*d^2)*g^3*i) + 2*d^2*l og(b*x + a)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^3*i) - 2*d^2*log(d*x + c)/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*g^ 3*i))
Time = 104.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.39 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {1}{4} \, {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} {\left (\frac {2 \, {\left (d x + c\right )}^{2} B n \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )}^{2} g^{3} i} + \frac {{\left (B n + 2 \, B \log \left (e\right ) + 2 \, A\right )} {\left (d x + c\right )}^{2}}{{\left (b x + a\right )}^{2} g^{3} i}\right )} \]
-1/4*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)^2*(2*(d*x + c)^2*B*n*log((b*x + a)/(d*x + c))/((b*x + a)^2*g^3*i) + (B*n + 2*B*log(e) + 2*A)*(d*x + c)^ 2/((b*x + a)^2*g^3*i))
Time = 3.08 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.15 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {B\,d^2\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (2\,a\,d-b\,c\right )}{2\,d^2}+\frac {a\,g^3\,i\,n\,\left (a\,d-b\,c\right )}{2\,d}+\frac {b\,g^3\,i\,n\,x\,\left (a\,d-b\,c\right )}{d}\right )}{g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (i\,a^2\,g^3+2\,i\,a\,b\,g^3\,x+i\,b^2\,g^3\,x^2\right )}-\frac {B\,d^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{2\,g^3\,i\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {\frac {6\,A\,a\,d-2\,A\,b\,c+7\,B\,a\,d\,n-B\,b\,c\,n}{2\,\left (a\,d-b\,c\right )}+\frac {d\,x\,\left (2\,A\,b+3\,B\,b\,n\right )}{a\,d-b\,c}}{x^2\,\left (2\,b^3\,c\,g^3\,i-2\,a\,b^2\,d\,g^3\,i\right )+x\,\left (4\,a\,b^2\,c\,g^3\,i-4\,a^2\,b\,d\,g^3\,i\right )-2\,a^3\,d\,g^3\,i+2\,a^2\,b\,c\,g^3\,i}+\frac {d^2\,\mathrm {atan}\left (\frac {d^2\,\left (A+\frac {3\,B\,n}{2}\right )\,\left (2\,i\,a^3\,d^3\,g^3-2\,i\,a^2\,b\,c\,d^2\,g^3-2\,i\,a\,b^2\,c^2\,d\,g^3+2\,i\,b^3\,c^3\,g^3\right )\,1{}\mathrm {i}}{g^3\,i\,\left (2\,A\,d^2+3\,B\,d^2\,n\right )\,{\left (a\,d-b\,c\right )}^3}+\frac {b\,d^3\,x\,\left (A+\frac {3\,B\,n}{2}\right )\,\left (i\,a^2\,d^2\,g^3-2\,i\,a\,b\,c\,d\,g^3+i\,b^2\,c^2\,g^3\right )\,4{}\mathrm {i}}{g^3\,i\,\left (2\,A\,d^2+3\,B\,d^2\,n\right )\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (A+\frac {3\,B\,n}{2}\right )\,2{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3} \]
(d^2*atan((d^2*(A + (3*B*n)/2)*(2*a^3*d^3*g^3*i + 2*b^3*c^3*g^3*i - 2*a*b^ 2*c^2*d*g^3*i - 2*a^2*b*c*d^2*g^3*i)*1i)/(g^3*i*(2*A*d^2 + 3*B*d^2*n)*(a*d - b*c)^3) + (b*d^3*x*(A + (3*B*n)/2)*(a^2*d^2*g^3*i + b^2*c^2*g^3*i - 2*a *b*c*d*g^3*i)*4i)/(g^3*i*(2*A*d^2 + 3*B*d^2*n)*(a*d - b*c)^3))*(A + (3*B*n )/2)*2i)/(g^3*i*(a*d - b*c)^3) - ((6*A*a*d - 2*A*b*c + 7*B*a*d*n - B*b*c*n )/(2*(a*d - b*c)) + (d*x*(2*A*b + 3*B*b*n))/(a*d - b*c))/(x^2*(2*b^3*c*g^3 *i - 2*a*b^2*d*g^3*i) + x*(4*a*b^2*c*g^3*i - 4*a^2*b*d*g^3*i) - 2*a^3*d*g^ 3*i + 2*a^2*b*c*g^3*i) - (B*d^2*log(e*((a + b*x)/(c + d*x))^n)^2)/(2*g^3*i *n*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (B*d^2*log(e*((a + b*x)/ (c + d*x))^n)*((g^3*i*n*(a*d - b*c)*(2*a*d - b*c))/(2*d^2) + (a*g^3*i*n*(a *d - b*c))/(2*d) + (b*g^3*i*n*x*(a*d - b*c))/d))/(g^3*i*n*(a*d - b*c)*(a^2 *d^2 + b^2*c^2 - 2*a*b*c*d)*(a^2*g^3*i + b^2*g^3*i*x^2 + 2*a*b*g^3*i*x))