Integrand size = 41, antiderivative size = 181 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=\frac {B d i n (c+d x)^2}{4 (b c-a d)^2 g^4 (a+b x)^2}-\frac {b B i n (c+d x)^3}{9 (b c-a d)^2 g^4 (a+b x)^3}+\frac {d i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^2 g^4 (a+b x)^2}-\frac {b i (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^2 g^4 (a+b x)^3} \]
1/4*B*d*i*n*(d*x+c)^2/(-a*d+b*c)^2/g^4/(b*x+a)^2-1/9*b*B*i*n*(d*x+c)^3/(-a *d+b*c)^2/g^4/(b*x+a)^3+1/2*d*i*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/ (-a*d+b*c)^2/g^4/(b*x+a)^2-1/3*b*i*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n ))/(-a*d+b*c)^2/g^4/(b*x+a)^3
Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.08 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=-\frac {i \left (\frac {12 A b c}{(a+b x)^3}-\frac {12 a A d}{(a+b x)^3}+\frac {4 b B c n}{(a+b x)^3}-\frac {4 a B d n}{(a+b x)^3}+\frac {18 A d}{(a+b x)^2}+\frac {3 B d n}{(a+b x)^2}-\frac {6 B d^2 n}{(b c-a d) (a+b x)}-\frac {6 B d^3 n \log (a+b x)}{(b c-a d)^2}+\frac {6 B (2 b c+a d+3 b d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3}+\frac {6 B d^3 n \log (c+d x)}{(b c-a d)^2}\right )}{36 b^2 g^4} \]
-1/36*(i*((12*A*b*c)/(a + b*x)^3 - (12*a*A*d)/(a + b*x)^3 + (4*b*B*c*n)/(a + b*x)^3 - (4*a*B*d*n)/(a + b*x)^3 + (18*A*d)/(a + b*x)^2 + (3*B*d*n)/(a + b*x)^2 - (6*B*d^2*n)/((b*c - a*d)*(a + b*x)) - (6*B*d^3*n*Log[a + b*x])/ (b*c - a*d)^2 + (6*B*(2*b*c + a*d + 3*b*d*x)*Log[e*((a + b*x)/(c + d*x))^n ])/(a + b*x)^3 + (6*B*d^3*n*Log[c + d*x])/(b*c - a*d)^2))/(b^2*g^4)
Time = 0.33 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.79, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2961, 2772, 27, 53, 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.
| \(\displaystyle \int \frac {(c i+d i x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(a g+b g x)^4} \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle \frac {i \int \frac {(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^4}d\frac {a+b x}{c+d x}}{g^4 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {i \left (-B n \int -\frac {(c+d x)^4 \left (2 b-\frac {3 d (a+b x)}{c+d x}\right )}{6 (a+b x)^4}d\frac {a+b x}{c+d x}-\frac {b (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}\right )}{g^4 (b c-a d)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {i \left (\frac {1}{6} B n \int \frac {(c+d x)^4 \left (2 b-\frac {3 d (a+b x)}{c+d x}\right )}{(a+b x)^4}d\frac {a+b x}{c+d x}-\frac {b (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}\right )}{g^4 (b c-a d)^2}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {i \left (\frac {1}{6} B n \int \left (\frac {2 b (c+d x)^4}{(a+b x)^4}-\frac {3 d (c+d x)^3}{(a+b x)^3}\right )d\frac {a+b x}{c+d x}-\frac {b (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}\right )}{g^4 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i \left (-\frac {b (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}+\frac {1}{6} B n \left (\frac {3 d (c+d x)^2}{2 (a+b x)^2}-\frac {2 b (c+d x)^3}{3 (a+b x)^3}\right )\right )}{g^4 (b c-a d)^2}\) |
(i*((B*n*((3*d*(c + d*x)^2)/(2*(a + b*x)^2) - (2*b*(c + d*x)^3)/(3*(a + b* x)^3)))/6 + (d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a + b*x)^2) - (b*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(a + b*x)^3)))/((b*c - a*d)^2*g^4)
3.2.15.3.1 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[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
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. \(454\) vs. \(2(173)=346\).
Time = 10.20 (sec) , antiderivative size = 455, normalized size of antiderivative = 2.51
| method | result | size |
| parallelrisch | \(-\frac {18 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} c^{2} d^{2} i n -18 B x a \,b^{5} c \,d^{3} i \,n^{2}-36 A x a \,b^{5} c \,d^{3} i n -18 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} c^{2} d^{2} i n -18 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} d^{4} i n -36 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} c \,d^{3} i n +5 B \,a^{3} b^{3} d^{4} i \,n^{2}+4 B \,b^{6} c^{3} d i \,n^{2}+6 A \,a^{3} b^{3} d^{4} i n +12 A \,b^{6} c^{3} d i n -6 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} d^{4} i n +6 B \,x^{2} a \,b^{5} d^{4} i \,n^{2}-6 B \,x^{2} b^{6} c \,d^{3} i \,n^{2}+15 B x \,a^{2} b^{4} d^{4} i \,n^{2}+3 B x \,b^{6} c^{2} d^{2} i \,n^{2}+18 A x \,a^{2} b^{4} d^{4} i n +18 A x \,b^{6} c^{2} d^{2} i n +12 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} c^{3} d i n -9 B a \,b^{5} c^{2} d^{2} i \,n^{2}-18 A a \,b^{5} c^{2} d^{2} i n}{36 g^{4} \left (b x +a \right )^{3} n \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{5} d}\) | \(455\) |
-1/36*(18*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^2*d^2*i*n-18*B*x*a*b^5*c*d^3 *i*n^2-36*A*x*a*b^5*c*d^3*i*n-18*B*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c^2*d^2 *i*n-18*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*d^4*i*n-36*B*x*ln(e*((b*x+a) /(d*x+c))^n)*a*b^5*c*d^3*i*n+5*B*a^3*b^3*d^4*i*n^2+4*B*b^6*c^3*d*i*n^2+6*A *a^3*b^3*d^4*i*n+12*A*b^6*c^3*d*i*n-6*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^6* d^4*i*n+6*B*x^2*a*b^5*d^4*i*n^2-6*B*x^2*b^6*c*d^3*i*n^2+15*B*x*a^2*b^4*d^4 *i*n^2+3*B*x*b^6*c^2*d^2*i*n^2+18*A*x*a^2*b^4*d^4*i*n+18*A*x*b^6*c^2*d^2*i *n+12*B*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^3*d*i*n-9*B*a*b^5*c^2*d^2*i*n^2-18 *A*a*b^5*c^2*d^2*i*n)/g^4/(b*x+a)^3/n/(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^5/d
Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (173) = 346\).
Time = 0.38 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.64 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=\frac {6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} i n x^{2} - {\left (4 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 5 \, B a^{3} d^{3}\right )} i n - 6 \, {\left (2 \, A b^{3} c^{3} - 3 \, A a b^{2} c^{2} d + A a^{3} d^{3}\right )} i - 3 \, {\left ({\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} i n + 6 \, {\left (A b^{3} c^{2} d - 2 \, A a b^{2} c d^{2} + A a^{2} b d^{3}\right )} i\right )} x - 6 \, {\left (3 \, {\left (B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2} + B a^{2} b d^{3}\right )} i x + {\left (2 \, B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + B a^{3} d^{3}\right )} i\right )} \log \left (e\right ) + 6 \, {\left (B b^{3} d^{3} i n x^{3} + 3 \, B a b^{2} d^{3} i n x^{2} - 3 \, {\left (B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2}\right )} i n x - {\left (2 \, B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d\right )} i n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{36 \, {\left ({\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}\right )}} \]
1/36*(6*(B*b^3*c*d^2 - B*a*b^2*d^3)*i*n*x^2 - (4*B*b^3*c^3 - 9*B*a*b^2*c^2 *d + 5*B*a^3*d^3)*i*n - 6*(2*A*b^3*c^3 - 3*A*a*b^2*c^2*d + A*a^3*d^3)*i - 3*((B*b^3*c^2*d - 6*B*a*b^2*c*d^2 + 5*B*a^2*b*d^3)*i*n + 6*(A*b^3*c^2*d - 2*A*a*b^2*c*d^2 + A*a^2*b*d^3)*i)*x - 6*(3*(B*b^3*c^2*d - 2*B*a*b^2*c*d^2 + B*a^2*b*d^3)*i*x + (2*B*b^3*c^3 - 3*B*a*b^2*c^2*d + B*a^3*d^3)*i)*log(e) + 6*(B*b^3*d^3*i*n*x^3 + 3*B*a*b^2*d^3*i*n*x^2 - 3*(B*b^3*c^2*d - 2*B*a*b ^2*c*d^2)*i*n*x - (2*B*b^3*c^3 - 3*B*a*b^2*c^2*d)*i*n)*log((b*x + a)/(d*x + c)))/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a ^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b ^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4)
Timed out. \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 945 vs. \(2 (173) = 346\).
Time = 0.22 (sec) , antiderivative size = 945, normalized size of antiderivative = 5.22 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=-\frac {1}{18} \, B c i n {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x + {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} - \frac {1}{36} \, B d i n {\left (\frac {5 \, a b^{2} c^{2} - 22 \, a^{2} b c d + 5 \, a^{3} d^{2} - 6 \, {\left (3 \, b^{3} c d - a b^{2} d^{2}\right )} x^{2} + 3 \, {\left (3 \, b^{3} c^{2} - 16 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x}{{\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}} - \frac {6 \, {\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (b x + a\right )}{{\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^{4}} + \frac {6 \, {\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (d x + c\right )}{{\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^{4}}\right )} - \frac {{\left (3 \, b x + a\right )} B d i \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{6 \, {\left (b^{5} g^{4} x^{3} + 3 \, a b^{4} g^{4} x^{2} + 3 \, a^{2} b^{3} g^{4} x + a^{3} b^{2} g^{4}\right )}} - \frac {{\left (3 \, b x + a\right )} A d i}{6 \, {\left (b^{5} g^{4} x^{3} + 3 \, a b^{4} g^{4} x^{2} + 3 \, a^{2} b^{3} g^{4} x + a^{3} b^{2} g^{4}\right )}} - \frac {B c i \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} - \frac {A c i}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} \]
-1/18*B*c*i*n*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^ 2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*( a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3* b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g ^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3 *b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c* d^2 - a^3*b*d^3)*g^4)) - 1/36*B*d*i*n*((5*a*b^2*c^2 - 22*a^2*b*c*d + 5*a^3 *d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16*a*b^2*c*d + 5*a^2 *b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a ^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4) - 6*( 3*b*c*d^2 - a*d^3)*log(b*x + a)/((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^4) + 6*(3*b*c*d^2 - a*d^3)*log(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^4)) - 1/6*(3*b*x + a)*B*d*i* log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3* a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/6*(3*b*x + a)*A*d*i/(b^5*g^4*x^3 + 3*a*b^ 4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/3*B*c*i*log(e*(b*x/(d*x + c ) + a/(d*x + c))^n)/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3 *b*g^4) - 1/3*A*c*i/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3 *b*g^4)
Time = 1.08 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.29 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=-\frac {1}{36} \, {\left (\frac {6 \, {\left (2 \, B b i n - \frac {3 \, {\left (b x + a\right )} B d i n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {4 \, B b i n - \frac {9 \, {\left (b x + a\right )} B d i n}{d x + c} + 12 \, B b i \log \left (e\right ) - \frac {18 \, {\left (b x + a\right )} B d i \log \left (e\right )}{d x + c} + 12 \, A b i - \frac {18 \, {\left (b x + a\right )} A d i}{d x + c}}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
-1/36*(6*(2*B*b*i*n - 3*(b*x + a)*B*d*i*n/(d*x + c))*log((b*x + a)/(d*x + c))/((b*x + a)^3*b*c*g^4/(d*x + c)^3 - (b*x + a)^3*a*d*g^4/(d*x + c)^3) + (4*B*b*i*n - 9*(b*x + a)*B*d*i*n/(d*x + c) + 12*B*b*i*log(e) - 18*(b*x + a )*B*d*i*log(e)/(d*x + c) + 12*A*b*i - 18*(b*x + a)*A*d*i/(d*x + c))/((b*x + a)^3*b*c*g^4/(d*x + c)^3 - (b*x + a)^3*a*d*g^4/(d*x + c)^3))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)
Time = 2.12 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.07 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=-\frac {\frac {6\,A\,a^2\,d^2\,i-12\,A\,b^2\,c^2\,i+5\,B\,a^2\,d^2\,i\,n-4\,B\,b^2\,c^2\,i\,n+6\,A\,a\,b\,c\,d\,i+5\,B\,a\,b\,c\,d\,i\,n}{6\,\left (a\,d-b\,c\right )}+\frac {x\,\left (6\,A\,a\,b\,d^2\,i-6\,A\,b^2\,c\,d\,i-B\,b^2\,c\,d\,i\,n+5\,B\,a\,b\,d^2\,i\,n\right )}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b^2\,d^2\,i\,n\,x^2}{a\,d-b\,c}}{6\,a^3\,b^2\,g^4+18\,a^2\,b^3\,g^4\,x+18\,a\,b^4\,g^4\,x^2+6\,b^5\,g^4\,x^3}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B\,c\,i}{3\,b}+\frac {B\,a\,d\,i}{6\,b^2}+\frac {B\,d\,i\,x}{2\,b}\right )}{a^3\,g^4+3\,a^2\,b\,g^4\,x+3\,a\,b^2\,g^4\,x^2+b^3\,g^4\,x^3}-\frac {B\,d^3\,i\,n\,\mathrm {atanh}\left (\frac {6\,b^4\,c^2\,g^4-6\,a^2\,b^2\,d^2\,g^4}{6\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{3\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2} \]
- ((6*A*a^2*d^2*i - 12*A*b^2*c^2*i + 5*B*a^2*d^2*i*n - 4*B*b^2*c^2*i*n + 6 *A*a*b*c*d*i + 5*B*a*b*c*d*i*n)/(6*(a*d - b*c)) + (x*(6*A*a*b*d^2*i - 6*A* b^2*c*d*i - B*b^2*c*d*i*n + 5*B*a*b*d^2*i*n))/(2*(a*d - b*c)) + (B*b^2*d^2 *i*n*x^2)/(a*d - b*c))/(6*a^3*b^2*g^4 + 6*b^5*g^4*x^3 + 18*a^2*b^3*g^4*x + 18*a*b^4*g^4*x^2) - (log(e*((a + b*x)/(c + d*x))^n)*((B*c*i)/(3*b) + (B*a *d*i)/(6*b^2) + (B*d*i*x)/(2*b)))/(a^3*g^4 + b^3*g^4*x^3 + 3*a*b^2*g^4*x^2 + 3*a^2*b*g^4*x) - (B*d^3*i*n*atanh((6*b^4*c^2*g^4 - 6*a^2*b^2*d^2*g^4)/( 6*b^2*g^4*(a*d - b*c)^2) - (2*b*d*x)/(a*d - b*c)))/(3*b^2*g^4*(a*d - b*c)^ 2)