Integrand size = 41, antiderivative size = 283 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {B (b c-a d)^4 g i^3 n x}{20 b^3 d}+\frac {B (b c-a d)^3 g i^3 n (c+d x)^2}{40 b^2 d^2}+\frac {B (b c-a d)^2 g i^3 n (c+d x)^3}{60 b d^2}-\frac {B (b c-a d) g i^3 n (c+d x)^4}{20 d^2}-\frac {(b c-a d) g i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac {b g i^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}+\frac {B (b c-a d)^5 g i^3 n \log \left (\frac {a+b x}{c+d x}\right )}{20 b^4 d^2}+\frac {B (b c-a d)^5 g i^3 n \log (c+d x)}{20 b^4 d^2} \]
1/20*B*(-a*d+b*c)^4*g*i^3*n*x/b^3/d+1/40*B*(-a*d+b*c)^3*g*i^3*n*(d*x+c)^2/ b^2/d^2+1/60*B*(-a*d+b*c)^2*g*i^3*n*(d*x+c)^3/b/d^2-1/20*B*(-a*d+b*c)*g*i^ 3*n*(d*x+c)^4/d^2-1/4*(-a*d+b*c)*g*i^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c ))^n))/d^2+1/5*b*g*i^3*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^2+1/20* B*(-a*d+b*c)^5*g*i^3*n*ln((b*x+a)/(d*x+c))/b^4/d^2+1/20*B*(-a*d+b*c)^5*g*i ^3*n*ln(d*x+c)/b^4/d^2
Time = 0.14 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.95 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g i^3 \left (\frac {5 B (b c-a d)^2 n \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )}{b^4}-\frac {2 B (b c-a d) n \left (12 b d (b c-a d)^3 x+6 b^2 (b c-a d)^2 (c+d x)^2+4 b^3 (b c-a d) (c+d x)^3+3 b^4 (c+d x)^4+12 (b c-a d)^4 \log (a+b x)\right )}{b^4}-30 (b c-a d) (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+24 b (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{120 d^2} \]
(g*i^3*((5*B*(b*c - a*d)^2*n*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 + 6*(b*c - a*d)^3*Log[a + b*x]))/b^4 - (2*B* (b*c - a*d)*n*(12*b*d*(b*c - a*d)^3*x + 6*b^2*(b*c - a*d)^2*(c + d*x)^2 + 4*b^3*(b*c - a*d)*(c + d*x)^3 + 3*b^4*(c + d*x)^4 + 12*(b*c - a*d)^4*Log[a + b*x]))/b^4 - 30*(b*c - a*d)*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d* x))^n]) + 24*b*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))/(120*d ^2)
Time = 0.43 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2961, 2782, 27, 86, 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.
| \(\displaystyle \int (a g+b g x) (c i+d i x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right ) \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle g i^3 (b c-a d)^5 \int \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2782 |
| \(\displaystyle g i^3 (b c-a d)^5 \left (-B n \int -\frac {(c+d x) \left (b-\frac {5 d (a+b x)}{c+d x}\right )}{20 d^2 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle g i^3 (b c-a d)^5 \left (\frac {B n \int \frac {(c+d x) \left (b-\frac {5 d (a+b x)}{c+d x}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{20 d^2}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle g i^3 (b c-a d)^5 \left (\frac {B n \int \left (\frac {d}{b^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {4 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {c+d x}{b^4 (a+b x)}\right )d\frac {a+b x}{c+d x}}{20 d^2}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g i^3 (b c-a d)^5 \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^4}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^4}+\frac {1}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {1}{3 b \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{20 d^2}\right )\) |
(b*c - a*d)^5*g*i^3*((b*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^2*(b - (d*(a + b*x))/(c + d*x))^5) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(4* d^2*(b - (d*(a + b*x))/(c + d*x))^4) + (B*n*(-(b - (d*(a + b*x))/(c + d*x) )^(-4) + 1/(3*b*(b - (d*(a + b*x))/(c + d*x))^3) + 1/(2*b^2*(b - (d*(a + b *x))/(c + d*x))^2) + 1/(b^3*(b - (d*(a + b*x))/(c + d*x))) + Log[(a + b*x) /(c + d*x)]/b^4 - Log[b - (d*(a + b*x))/(c + d*x)]/b^4))/(20*d^2))
3.2.29.3.1 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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q _), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x)^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}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]
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. \(1118\) vs. \(2(267)=534\).
Time = 11.62 (sec) , antiderivative size = 1119, normalized size of antiderivative = 3.95
1/120*(6*B*ln(b*x+a)*b^5*c^5*g*i^3*n^2-6*B*ln(b*x+a)*a^5*d^5*g*i^3*n^2-6*B *ln(e*((b*x+a)/(d*x+c))^n)*b^5*c^5*g*i^3*n+24*A*x^5*b^5*d^5*g*i^3*n+24*B*x ^5*ln(e*((b*x+a)/(d*x+c))^n)*b^5*d^5*g*i^3*n+120*B*x^3*ln(e*((b*x+a)/(d*x+ c))^n)*a*b^4*c*d^4*g*i^3*n+180*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^4*c^2*d ^3*g*i^3*n+120*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b^4*c^3*d^2*g*i^3*n+30*B*ln (e*((b*x+a)/(d*x+c))^n)*a*b^4*c^4*d*g*i^3*n+30*B*x^4*ln(e*((b*x+a)/(d*x+c) )^n)*a*b^4*d^5*g*i^3*n+90*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*b^5*c*d^4*g*i^3* n+120*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^5*c^2*d^3*g*i^3*n-27*B*x^2*b^5*c^3 *d^2*g*i^3*n^2+60*A*x^2*b^5*c^3*d^2*g*i^3*n+6*B*x*a^4*b*d^5*g*i^3*n^2-6*B* x*b^5*c^4*d*g*i^3*n^2+6*B*b^5*c^5*g*i^3*n^2-6*B*a^5*d^5*g*i^3*n^2+20*B*x^3 *a*b^4*c*d^4*g*i^3*n^2+120*A*x^3*a*b^4*c*d^4*g*i^3*n+60*B*x^2*ln(e*((b*x+a )/(d*x+c))^n)*b^5*c^3*d^2*g*i^3*n+15*B*x^2*a^2*b^3*c*d^4*g*i^3*n^2+15*B*x^ 2*a*b^4*c^2*d^3*g*i^3*n^2+180*A*x^2*a*b^4*c^2*d^3*g*i^3*n-30*B*x*a^3*b^2*c *d^4*g*i^3*n^2+60*B*x*a^2*b^3*c^2*d^3*g*i^3*n^2-30*B*x*a*b^4*c^3*d^2*g*i^3 *n^2+120*A*x*a*b^4*c^3*d^2*g*i^3*n+30*B*ln(b*x+a)*a^4*b*c*d^4*g*i^3*n^2-60 *B*ln(b*x+a)*a^3*b^2*c^2*d^3*g*i^3*n^2+60*B*ln(b*x+a)*a^2*b^3*c^3*d^2*g*i^ 3*n^2-30*B*ln(b*x+a)*a*b^4*c^4*d*g*i^3*n^2+27*B*a^4*b*c*d^4*g*i^3*n^2-45*B *a^3*b^2*c^2*d^3*g*i^3*n^2-45*B*a^2*b^3*c^3*d^2*g*i^3*n^2+63*B*a*b^4*c^4*d *g*i^3*n^2-300*A*a^2*b^3*c^3*d^2*g*i^3*n-180*A*a*b^4*c^4*d*g*i^3*n+6*B*x^4 *a*b^4*d^5*g*i^3*n^2-6*B*x^4*b^5*c*d^4*g*i^3*n^2+30*A*x^4*a*b^4*d^5*g*i...
Leaf count of result is larger than twice the leaf count of optimal. 721 vs. \(2 (267) = 534\).
Time = 0.45 (sec) , antiderivative size = 721, normalized size of antiderivative = 2.55 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {24 \, A b^{5} d^{5} g i^{3} x^{5} + 6 \, {\left (10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3} + 5 \, B a^{4} b c d^{4} - B a^{5} d^{5}\right )} g i^{3} n \log \left (b x + a\right ) + 6 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d\right )} g i^{3} n \log \left (d x + c\right ) - 6 \, {\left ({\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g i^{3} n - 5 \, {\left (3 \, A b^{5} c d^{4} + A a b^{4} d^{5}\right )} g i^{3}\right )} x^{4} - 2 \, {\left ({\left (11 \, B b^{5} c^{2} d^{3} - 10 \, B a b^{4} c d^{4} - B a^{2} b^{3} d^{5}\right )} g i^{3} n - 60 \, {\left (A b^{5} c^{2} d^{3} + A a b^{4} c d^{4}\right )} g i^{3}\right )} x^{3} - 3 \, {\left ({\left (9 \, B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} - 5 \, B a^{2} b^{3} c d^{4} + B a^{3} b^{2} d^{5}\right )} g i^{3} n - 20 \, {\left (A b^{5} c^{3} d^{2} + 3 \, A a b^{4} c^{2} d^{3}\right )} g i^{3}\right )} x^{2} + 6 \, {\left (20 \, A a b^{4} c^{3} d^{2} g i^{3} - {\left (B b^{5} c^{4} d + 5 \, B a b^{4} c^{3} d^{2} - 10 \, B a^{2} b^{3} c^{2} d^{3} + 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g i^{3} n\right )} x + 6 \, {\left (4 \, B b^{5} d^{5} g i^{3} x^{5} + 20 \, B a b^{4} c^{3} d^{2} g i^{3} x + 5 \, {\left (3 \, B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g i^{3} x^{4} + 20 \, {\left (B b^{5} c^{2} d^{3} + B a b^{4} c d^{4}\right )} g i^{3} x^{3} + 10 \, {\left (B b^{5} c^{3} d^{2} + 3 \, B a b^{4} c^{2} d^{3}\right )} g i^{3} x^{2}\right )} \log \left (e\right ) + 6 \, {\left (4 \, B b^{5} d^{5} g i^{3} n x^{5} + 20 \, B a b^{4} c^{3} d^{2} g i^{3} n x + 5 \, {\left (3 \, B b^{5} c d^{4} + B a b^{4} d^{5}\right )} g i^{3} n x^{4} + 20 \, {\left (B b^{5} c^{2} d^{3} + B a b^{4} c d^{4}\right )} g i^{3} n x^{3} + 10 \, {\left (B b^{5} c^{3} d^{2} + 3 \, B a b^{4} c^{2} d^{3}\right )} g i^{3} n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{120 \, b^{4} d^{2}} \]
1/120*(24*A*b^5*d^5*g*i^3*x^5 + 6*(10*B*a^2*b^3*c^3*d^2 - 10*B*a^3*b^2*c^2 *d^3 + 5*B*a^4*b*c*d^4 - B*a^5*d^5)*g*i^3*n*log(b*x + a) + 6*(B*b^5*c^5 - 5*B*a*b^4*c^4*d)*g*i^3*n*log(d*x + c) - 6*((B*b^5*c*d^4 - B*a*b^4*d^5)*g*i ^3*n - 5*(3*A*b^5*c*d^4 + A*a*b^4*d^5)*g*i^3)*x^4 - 2*((11*B*b^5*c^2*d^3 - 10*B*a*b^4*c*d^4 - B*a^2*b^3*d^5)*g*i^3*n - 60*(A*b^5*c^2*d^3 + A*a*b^4*c *d^4)*g*i^3)*x^3 - 3*((9*B*b^5*c^3*d^2 - 5*B*a*b^4*c^2*d^3 - 5*B*a^2*b^3*c *d^4 + B*a^3*b^2*d^5)*g*i^3*n - 20*(A*b^5*c^3*d^2 + 3*A*a*b^4*c^2*d^3)*g*i ^3)*x^2 + 6*(20*A*a*b^4*c^3*d^2*g*i^3 - (B*b^5*c^4*d + 5*B*a*b^4*c^3*d^2 - 10*B*a^2*b^3*c^2*d^3 + 5*B*a^3*b^2*c*d^4 - B*a^4*b*d^5)*g*i^3*n)*x + 6*(4 *B*b^5*d^5*g*i^3*x^5 + 20*B*a*b^4*c^3*d^2*g*i^3*x + 5*(3*B*b^5*c*d^4 + B*a *b^4*d^5)*g*i^3*x^4 + 20*(B*b^5*c^2*d^3 + B*a*b^4*c*d^4)*g*i^3*x^3 + 10*(B *b^5*c^3*d^2 + 3*B*a*b^4*c^2*d^3)*g*i^3*x^2)*log(e) + 6*(4*B*b^5*d^5*g*i^3 *n*x^5 + 20*B*a*b^4*c^3*d^2*g*i^3*n*x + 5*(3*B*b^5*c*d^4 + B*a*b^4*d^5)*g* i^3*n*x^4 + 20*(B*b^5*c^2*d^3 + B*a*b^4*c*d^4)*g*i^3*n*x^3 + 10*(B*b^5*c^3 *d^2 + 3*B*a*b^4*c^2*d^3)*g*i^3*n*x^2)*log((b*x + a)/(d*x + c)))/(b^4*d^2)
Timed out. \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1118 vs. \(2 (267) = 534\).
Time = 0.23 (sec) , antiderivative size = 1118, normalized size of antiderivative = 3.95 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]
1/5*B*b*d^3*g*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/5*A*b*d^3 *g*i^3*x^5 + 3/4*B*b*c*d^2*g*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n ) + 1/4*B*a*d^3*g*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/4*A*b *c*d^2*g*i^3*x^4 + 1/4*A*a*d^3*g*i^3*x^4 + B*b*c^2*d*g*i^3*x^3*log(e*(b*x/ (d*x + c) + a/(d*x + c))^n) + B*a*c*d^2*g*i^3*x^3*log(e*(b*x/(d*x + c) + a /(d*x + c))^n) + A*b*c^2*d*g*i^3*x^3 + A*a*c*d^2*g*i^3*x^3 + 1/2*B*b*c^3*g *i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*B*a*c^2*d*g*i^3*x^2* log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*A*b*c^3*g*i^3*x^2 + 3/2*A*a*c ^2*d*g*i^3*x^2 + 1/60*B*b*d^3*g*i^3*n*(12*a^5*log(b*x + a)/b^5 - 12*c^5*lo g(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2 *d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4 *d^4)) - 1/8*B*b*c*d^2*g*i^3*n*(6*a^4*log(b*x + a)/b^4 - 6*c^4*log(d*x + c )/d^4 + (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6 *(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3)) - 1/24*B*a*d^3*g*i^3*n*(6*a^4*log(b*x + a)/b^4 - 6*c^4*log(d*x + c)/d^4 + (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3 *c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3)) + 1/2*B*b*c^ 2*d*g*i^3*n*(2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2)) + 1/2*B*a*c*d^2*g*i^3* n*(2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)* x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2)) - 1/2*B*b*c^3*g*i^3*n*(a^2*lo...
Leaf count of result is larger than twice the leaf count of optimal. 2584 vs. \(2 (267) = 534\).
Time = 1.18 (sec) , antiderivative size = 2584, normalized size of antiderivative = 9.13 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]
-1/120*(6*(B*b^7*c^6*g*i^3*n - 6*B*a*b^6*c^5*d*g*i^3*n - 5*(b*x + a)*B*b^6 *c^6*d*g*i^3*n/(d*x + c) + 15*B*a^2*b^5*c^4*d^2*g*i^3*n + 30*(b*x + a)*B*a *b^5*c^5*d^2*g*i^3*n/(d*x + c) - 20*B*a^3*b^4*c^3*d^3*g*i^3*n - 75*(b*x + a)*B*a^2*b^4*c^4*d^3*g*i^3*n/(d*x + c) + 15*B*a^4*b^3*c^2*d^4*g*i^3*n + 10 0*(b*x + a)*B*a^3*b^3*c^3*d^4*g*i^3*n/(d*x + c) - 6*B*a^5*b^2*c*d^5*g*i^3* n - 75*(b*x + a)*B*a^4*b^2*c^2*d^5*g*i^3*n/(d*x + c) + B*a^6*b*d^6*g*i^3*n + 30*(b*x + a)*B*a^5*b*c*d^6*g*i^3*n/(d*x + c) - 5*(b*x + a)*B*a^6*d^7*g* i^3*n/(d*x + c))*log((b*x + a)/(d*x + c))/(b^5*d^2 - 5*(b*x + a)*b^4*d^3/( d*x + c) + 10*(b*x + a)^2*b^3*d^4/(d*x + c)^2 - 10*(b*x + a)^3*b^2*d^5/(d* x + c)^3 + 5*(b*x + a)^4*b*d^6/(d*x + c)^4 - (b*x + a)^5*d^7/(d*x + c)^5) - (5*B*b^10*c^6*g*i^3*n - 30*B*a*b^9*c^5*d*g*i^3*n - 31*(b*x + a)*B*b^9*c^ 6*d*g*i^3*n/(d*x + c) + 75*B*a^2*b^8*c^4*d^2*g*i^3*n + 186*(b*x + a)*B*a*b ^8*c^5*d^2*g*i^3*n/(d*x + c) + 47*(b*x + a)^2*B*b^8*c^6*d^2*g*i^3*n/(d*x + c)^2 - 100*B*a^3*b^7*c^3*d^3*g*i^3*n - 465*(b*x + a)*B*a^2*b^7*c^4*d^3*g* i^3*n/(d*x + c) - 282*(b*x + a)^2*B*a*b^7*c^5*d^3*g*i^3*n/(d*x + c)^2 - 27 *(b*x + a)^3*B*b^7*c^6*d^3*g*i^3*n/(d*x + c)^3 + 75*B*a^4*b^6*c^2*d^4*g*i^ 3*n + 620*(b*x + a)*B*a^3*b^6*c^3*d^4*g*i^3*n/(d*x + c) + 705*(b*x + a)^2* B*a^2*b^6*c^4*d^4*g*i^3*n/(d*x + c)^2 + 162*(b*x + a)^3*B*a*b^6*c^5*d^4*g* i^3*n/(d*x + c)^3 + 6*(b*x + a)^4*B*b^6*c^6*d^4*g*i^3*n/(d*x + c)^4 - 30*B *a^5*b^5*c*d^5*g*i^3*n - 465*(b*x + a)*B*a^4*b^5*c^2*d^5*g*i^3*n/(d*x +...
Time = 2.33 (sec) , antiderivative size = 1234, normalized size of antiderivative = 4.36 \[ \int (a g+b g x) (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]
x*((a*c*(((20*a*d + 20*b*c)*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B *b*c*n))/5 - (A*d^2*g*i^3*(20*a*d + 20*b*c))/20))/(20*b*d) - (d*g*i^3*(4*A *a^2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 32*A*a*b*c*d + 2*B *a*b*c*d*n))/(4*b) + A*a*c*d^2*g*i^3))/(b*d) - ((20*a*d + 20*b*c)*(((20*a* d + 20*b*c)*(((20*a*d + 20*b*c)*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*a*d + 20*b*c))/20))/(20*b*d) - (d*g*i^3* (4*A*a^2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 32*A*a*b*c*d + 2*B*a*b*c*d*n))/(4*b) + A*a*c*d^2*g*i^3))/(20*b*d) - (a*c*((d^2*g*i^3*(10 *A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*a*d + 20*b*c) )/20))/(b*d) + (c*g*i^3*(4*A*a^2*d^2 + 4*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c ^2*n + 12*A*a*b*c*d))/b))/(20*b*d) + (c^2*g*i^3*(12*A*a^2*d^2 + 2*A*b^2*c^ 2 + 3*B*a^2*d^2*n - B*b^2*c^2*n + 16*A*a*b*c*d - 2*B*a*b*c*d*n))/(2*b*d)) - x^3*(((20*a*d + 20*b*c)*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*b *c*n))/5 - (A*d^2*g*i^3*(20*a*d + 20*b*c))/20))/(60*b*d) - (d*g*i^3*(4*A*a ^2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 32*A*a*b*c*d + 2*B*a *b*c*d*n))/(12*b) + (A*a*c*d^2*g*i^3)/3) + x^2*(((20*a*d + 20*b*c)*(((20*a *d + 20*b*c)*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A *d^2*g*i^3*(20*a*d + 20*b*c))/20))/(20*b*d) - (d*g*i^3*(4*A*a^2*d^2 + 24*A *b^2*c^2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 32*A*a*b*c*d + 2*B*a*b*c*d*n))/(4 *b) + A*a*c*d^2*g*i^3))/(40*b*d) - (a*c*((d^2*g*i^3*(10*A*a*d + 20*A*b*...