Integrand size = 43, antiderivative size = 477 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {B (b c-a d)^6 g^3 i^3 n x}{140 b^3 d^3}+\frac {B (b c-a d)^5 g^3 i^3 n (c+d x)^2}{280 b^2 d^4}+\frac {B (b c-a d)^4 g^3 i^3 n (c+d x)^3}{420 b d^4}-\frac {17 B (b c-a d)^3 g^3 i^3 n (c+d x)^4}{280 d^4}+\frac {b B (b c-a d)^2 g^3 i^3 n (c+d x)^5}{14 d^4}-\frac {b^2 B (b c-a d) g^3 i^3 n (c+d x)^6}{42 d^4}-\frac {(b c-a d)^3 g^3 i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^4}+\frac {3 b (b c-a d)^2 g^3 i^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^4}-\frac {b^2 (b c-a d) g^3 i^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d^4}+\frac {b^3 g^3 i^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}+\frac {B (b c-a d)^7 g^3 i^3 n \log \left (\frac {a+b x}{c+d x}\right )}{140 b^4 d^4}+\frac {B (b c-a d)^7 g^3 i^3 n \log (c+d x)}{140 b^4 d^4} \]
1/140*B*(-a*d+b*c)^6*g^3*i^3*n*x/b^3/d^3+1/280*B*(-a*d+b*c)^5*g^3*i^3*n*(d *x+c)^2/b^2/d^4+1/420*B*(-a*d+b*c)^4*g^3*i^3*n*(d*x+c)^3/b/d^4-17/280*B*(- a*d+b*c)^3*g^3*i^3*n*(d*x+c)^4/d^4+1/14*b*B*(-a*d+b*c)^2*g^3*i^3*n*(d*x+c) ^5/d^4-1/42*b^2*B*(-a*d+b*c)*g^3*i^3*n*(d*x+c)^6/d^4-1/4*(-a*d+b*c)^3*g^3* i^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^4+3/5*b*(-a*d+b*c)^2*g^3*i ^3*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^4-1/2*b^2*(-a*d+b*c)*g^3*i^ 3*(d*x+c)^6*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^4+1/7*b^3*g^3*i^3*(d*x+c)^7* (A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^4+1/140*B*(-a*d+b*c)^7*g^3*i^3*n*ln((b*x +a)/(d*x+c))/b^4/d^4+1/140*B*(-a*d+b*c)^7*g^3*i^3*n*ln(d*x+c)/b^4/d^4
Time = 0.37 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.32 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^3 i^3 \left (210 d^4 (b c-a d)^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+504 d^5 (b c-a d)^2 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+420 d^6 (b c-a d) (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+120 d^7 (a+b x)^7 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-35 B (b c-a d)^4 n \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )+42 B (b c-a d)^3 n \left (12 b d (b c-a d)^3 x-6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (b c-a d) (a+b x)^3-3 d^4 (a+b x)^4-12 (b c-a d)^4 \log (c+d x)\right )-7 B (b c-a d)^2 n \left (60 b d (b c-a d)^4 x+30 d^2 (-b c+a d)^3 (a+b x)^2+20 d^3 (b c-a d)^2 (a+b x)^3+15 d^4 (-b c+a d) (a+b x)^4+12 d^5 (a+b x)^5-60 (b c-a d)^5 \log (c+d x)\right )+2 B (b c-a d) n \left (60 b d (b c-a d)^5 x-30 d^2 (b c-a d)^4 (a+b x)^2+20 d^3 (b c-a d)^3 (a+b x)^3-15 d^4 (b c-a d)^2 (a+b x)^4+12 d^5 (b c-a d) (a+b x)^5-10 d^6 (a+b x)^6-60 (b c-a d)^6 \log (c+d x)\right )\right )}{840 b^4 d^4} \]
(g^3*i^3*(210*d^4*(b*c - a*d)^3*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d *x))^n]) + 504*d^5*(b*c - a*d)^2*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 420*d^6*(b*c - a*d)*(a + b*x)^6*(A + B*Log[e*((a + b*x)/(c + d *x))^n]) + 120*d^7*(a + b*x)^7*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 35 *B*(b*c - a*d)^4*n*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x) ^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]) + 42*B*(b*c - a*d)^ 3*n*(12*b*d*(b*c - a*d)^3*x - 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(b*c - a*d)*(a + b*x)^3 - 3*d^4*(a + b*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x]) - 7*B*(b*c - a*d)^2*n*(60*b*d*(b*c - a*d)^4*x + 30*d^2*(-(b*c) + a*d)^3*(a + b*x)^2 + 20*d^3*(b*c - a*d)^2*(a + b*x)^3 + 15*d^4*(-(b*c) + a*d)*(a + b *x)^4 + 12*d^5*(a + b*x)^5 - 60*(b*c - a*d)^5*Log[c + d*x]) + 2*B*(b*c - a *d)*n*(60*b*d*(b*c - a*d)^5*x - 30*d^2*(b*c - a*d)^4*(a + b*x)^2 + 20*d^3* (b*c - a*d)^3*(a + b*x)^3 - 15*d^4*(b*c - a*d)^2*(a + b*x)^4 + 12*d^5*(b*c - a*d)*(a + b*x)^5 - 10*d^6*(a + b*x)^6 - 60*(b*c - a*d)^6*Log[c + d*x])) )/(840*b^4*d^4)
Time = 0.65 (sec) , antiderivative size = 414, normalized size of antiderivative = 0.87, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2961, 2782, 27, 2123, 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.
| \(\displaystyle \int (a g+b g x)^3 (c i+d i x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right ) \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle g^3 i^3 (b c-a d)^7 \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^8}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2782 |
| \(\displaystyle g^3 i^3 (b c-a d)^7 \left (-B n \int -\frac {(c+d x) \left (b^3-\frac {7 d (a+b x) b^2}{c+d x}+\frac {21 d^2 (a+b x)^2 b}{(c+d x)^2}-\frac {35 d^3 (a+b x)^3}{(c+d x)^3}\right )}{140 d^4 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}+\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{7 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {3 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle g^3 i^3 (b c-a d)^7 \left (\frac {B n \int \frac {(c+d x) \left (b^3-\frac {7 d (a+b x) b^2}{c+d x}+\frac {21 d^2 (a+b x)^2 b}{(c+d x)^2}-\frac {35 d^3 (a+b x)^3}{(c+d x)^3}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}}{140 d^4}+\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{7 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {3 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle g^3 i^3 (b c-a d)^7 \left (\frac {B n \int \left (-\frac {20 d b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^7}+\frac {50 d b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {34 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4 b}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3 b^2}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2 b^3}+\frac {c+d x}{(a+b x) b^4}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^4}\right )d\frac {a+b x}{c+d x}}{140 d^4}+\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{7 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {3 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g^3 i^3 (b c-a d)^7 \left (\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{7 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {3 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^4}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^4}+\frac {1}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {10 b^2}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {1}{2 b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {10 b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {17}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {1}{3 b \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{140 d^4}\right )\) |
(b*c - a*d)^7*g^3*i^3*((b^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(7*d^4 *(b - (d*(a + b*x))/(c + d*x))^7) - (b^2*(A + B*Log[e*((a + b*x)/(c + d*x) )^n]))/(2*d^4*(b - (d*(a + b*x))/(c + d*x))^6) + (3*b*(A + B*Log[e*((a + b *x)/(c + d*x))^n]))/(5*d^4*(b - (d*(a + b*x))/(c + d*x))^5) - (A + B*Log[e *((a + b*x)/(c + d*x))^n])/(4*d^4*(b - (d*(a + b*x))/(c + d*x))^4) + (B*n* ((-10*b^2)/(3*(b - (d*(a + b*x))/(c + d*x))^6) + (10*b)/(b - (d*(a + b*x)) /(c + d*x))^5 - 17/(2*(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))/(140*d^4))
3.2.27.3.1 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[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
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. \(2178\) vs. \(2(453)=906\).
Time = 50.10 (sec) , antiderivative size = 2179, normalized size of antiderivative = 4.57
1/840*(-6*B*x*b^7*c^6*d*g^3*i^3*n^2+120*B*x^7*ln(e*((b*x+a)/(d*x+c))^n)*b^ 7*d^7*g^3*i^3*n+20*B*x^6*a*b^6*d^7*g^3*i^3*n^2-20*B*x^6*b^7*c*d^6*g^3*i^3* n^2+420*A*x^6*a*b^6*d^7*g^3*i^3*n+420*A*x^6*b^7*c*d^6*g^3*i^3*n+60*B*x^5*a ^2*b^5*d^7*g^3*i^3*n^2-60*B*x^5*b^7*c^2*d^5*g^3*i^3*n^2+504*A*x^5*a^2*b^5* d^7*g^3*i^3*n+504*A*x^5*b^7*c^2*d^5*g^3*i^3*n+51*B*x^4*a^3*b^4*d^7*g^3*i^3 *n^2-51*B*x^4*b^7*c^3*d^4*g^3*i^3*n^2+1512*B*x^5*ln(e*((b*x+a)/(d*x+c))^n) *a*b^6*c*d^6*g^3*i^3*n+1890*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^5*c*d^6* g^3*i^3*n+1890*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a*b^6*c^2*d^5*g^3*i^3*n+840 *B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^4*c*d^6*g^3*i^3*n+2520*B*x^3*ln(e*( (b*x+a)/(d*x+c))^n)*a^2*b^5*c^2*d^5*g^3*i^3*n+840*B*x^3*ln(e*((b*x+a)/(d*x +c))^n)*a*b^6*c^3*d^4*g^3*i^3*n+1260*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b ^4*c^2*d^5*g^3*i^3*n+1260*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^5*c^3*d^4* g^3*i^3*n+840*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^4*c^3*d^4*g^3*i^3*n-21*B *x^2*a*b^6*c^4*d^3*g^3*i^3*n^2+1260*A*x^2*a^3*b^4*c^2*d^5*g^3*i^3*n+1260*A *x^2*a^2*b^5*c^3*d^4*g^3*i^3*n+420*B*x^6*ln(e*((b*x+a)/(d*x+c))^n)*a*b^6*d ^7*g^3*i^3*n+420*B*x^6*ln(e*((b*x+a)/(d*x+c))^n)*b^7*c*d^6*g^3*i^3*n+504*B *x^5*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^5*d^7*g^3*i^3*n+39*B*a^6*b*c*d^6*g^3* i^3*n^2-105*B*a^5*b^2*c^2*d^5*g^3*i^3*n^2-378*B*a^4*b^3*c^3*d^4*g^3*i^3*n^ 2+378*B*a^3*b^4*c^4*d^3*g^3*i^3*n^2+504*B*x^5*ln(e*((b*x+a)/(d*x+c))^n)*b^ 7*c^2*d^5*g^3*i^3*n+1512*A*x^5*a*b^6*c*d^6*g^3*i^3*n+210*B*x^4*ln(e*((b...
Leaf count of result is larger than twice the leaf count of optimal. 1336 vs. \(2 (453) = 906\).
Time = 0.93 (sec) , antiderivative size = 1336, normalized size of antiderivative = 2.80 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]
1/840*(120*A*b^7*d^7*g^3*i^3*x^7 + 6*(35*B*a^4*b^3*c^3*d^4 - 21*B*a^5*b^2* c^2*d^5 + 7*B*a^6*b*c*d^6 - B*a^7*d^7)*g^3*i^3*n*log(b*x + a) + 6*(B*b^7*c ^7 - 7*B*a*b^6*c^6*d + 21*B*a^2*b^5*c^5*d^2 - 35*B*a^3*b^4*c^4*d^3)*g^3*i^ 3*n*log(d*x + c) - 20*((B*b^7*c*d^6 - B*a*b^6*d^7)*g^3*i^3*n - 21*(A*b^7*c *d^6 + A*a*b^6*d^7)*g^3*i^3)*x^6 - 12*(5*(B*b^7*c^2*d^5 - B*a^2*b^5*d^7)*g ^3*i^3*n - 42*(A*b^7*c^2*d^5 + 3*A*a*b^6*c*d^6 + A*a^2*b^5*d^7)*g^3*i^3)*x ^5 - 3*((17*B*b^7*c^3*d^4 + 49*B*a*b^6*c^2*d^5 - 49*B*a^2*b^5*c*d^6 - 17*B *a^3*b^4*d^7)*g^3*i^3*n - 70*(A*b^7*c^3*d^4 + 9*A*a*b^6*c^2*d^5 + 9*A*a^2* b^5*c*d^6 + A*a^3*b^4*d^7)*g^3*i^3)*x^4 - 2*((B*b^7*c^4*d^3 + 98*B*a*b^6*c ^3*d^4 - 98*B*a^3*b^4*c*d^6 - B*a^4*b^3*d^7)*g^3*i^3*n - 420*(A*a*b^6*c^3* d^4 + 3*A*a^2*b^5*c^2*d^5 + A*a^3*b^4*c*d^6)*g^3*i^3)*x^3 + 3*((B*b^7*c^5* d^2 - 7*B*a*b^6*c^4*d^3 - 84*B*a^2*b^5*c^3*d^4 + 84*B*a^3*b^4*c^2*d^5 + 7* B*a^4*b^3*c*d^6 - B*a^5*b^2*d^7)*g^3*i^3*n + 420*(A*a^2*b^5*c^3*d^4 + A*a^ 3*b^4*c^2*d^5)*g^3*i^3)*x^2 + 6*(140*A*a^3*b^4*c^3*d^4*g^3*i^3 - (B*b^7*c^ 6*d - 7*B*a*b^6*c^5*d^2 + 21*B*a^2*b^5*c^4*d^3 - 21*B*a^4*b^3*c^2*d^5 + 7* B*a^5*b^2*c*d^6 - B*a^6*b*d^7)*g^3*i^3*n)*x + 6*(20*B*b^7*d^7*g^3*i^3*x^7 + 140*B*a^3*b^4*c^3*d^4*g^3*i^3*x + 70*(B*b^7*c*d^6 + B*a*b^6*d^7)*g^3*i^3 *x^6 + 84*(B*b^7*c^2*d^5 + 3*B*a*b^6*c*d^6 + B*a^2*b^5*d^7)*g^3*i^3*x^5 + 35*(B*b^7*c^3*d^4 + 9*B*a*b^6*c^2*d^5 + 9*B*a^2*b^5*c*d^6 + B*a^3*b^4*d^7) *g^3*i^3*x^4 + 140*(B*a*b^6*c^3*d^4 + 3*B*a^2*b^5*c^2*d^5 + B*a^3*b^4*c...
Timed out. \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2901 vs. \(2 (453) = 906\).
Time = 0.28 (sec) , antiderivative size = 2901, normalized size of antiderivative = 6.08 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]
1/7*B*b^3*d^3*g^3*i^3*x^7*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/7*A*b ^3*d^3*g^3*i^3*x^7 + 1/2*B*b^3*c*d^2*g^3*i^3*x^6*log(e*(b*x/(d*x + c) + a/ (d*x + c))^n) + 1/2*B*a*b^2*d^3*g^3*i^3*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*A*b^3*c*d^2*g^3*i^3*x^6 + 1/2*A*a*b^2*d^3*g^3*i^3*x^6 + 3/5 *B*b^3*c^2*d*g^3*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 9/5*B*a* b^2*c*d^2*g^3*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/5*B*a^2*b *d^3*g^3*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/5*A*b^3*c^2*d* g^3*i^3*x^5 + 9/5*A*a*b^2*c*d^2*g^3*i^3*x^5 + 3/5*A*a^2*b*d^3*g^3*i^3*x^5 + 1/4*B*b^3*c^3*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 9/4*B *a*b^2*c^2*d*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 9/4*B*a^ 2*b*c*d^2*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*B*a^3*d ^3*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A*b^3*c^3*g^3* i^3*x^4 + 9/4*A*a*b^2*c^2*d*g^3*i^3*x^4 + 9/4*A*a^2*b*c*d^2*g^3*i^3*x^4 + 1/4*A*a^3*d^3*g^3*i^3*x^4 + B*a*b^2*c^3*g^3*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3*B*a^2*b*c^2*d*g^3*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d *x + c))^n) + B*a^3*c*d^2*g^3*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^ n) + A*a*b^2*c^3*g^3*i^3*x^3 + 3*A*a^2*b*c^2*d*g^3*i^3*x^3 + A*a^3*c*d^2*g ^3*i^3*x^3 + 3/2*B*a^2*b*c^3*g^3*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c ))^n) + 3/2*B*a^3*c^2*d*g^3*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*A*a^2*b*c^3*g^3*i^3*x^2 + 3/2*A*a^3*c^2*d*g^3*i^3*x^2 + 1/420*B*...
Leaf count of result is larger than twice the leaf count of optimal. 5934 vs. \(2 (453) = 906\).
Time = 2.50 (sec) , antiderivative size = 5934, normalized size of antiderivative = 12.44 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]
-1/840*(6*(B*b^11*c^8*g^3*i^3*n - 8*B*a*b^10*c^7*d*g^3*i^3*n - 7*(b*x + a) *B*b^10*c^8*d*g^3*i^3*n/(d*x + c) + 28*B*a^2*b^9*c^6*d^2*g^3*i^3*n + 56*(b *x + a)*B*a*b^9*c^7*d^2*g^3*i^3*n/(d*x + c) + 21*(b*x + a)^2*B*b^9*c^8*d^2 *g^3*i^3*n/(d*x + c)^2 - 56*B*a^3*b^8*c^5*d^3*g^3*i^3*n - 196*(b*x + a)*B* a^2*b^8*c^6*d^3*g^3*i^3*n/(d*x + c) - 168*(b*x + a)^2*B*a*b^8*c^7*d^3*g^3* i^3*n/(d*x + c)^2 - 35*(b*x + a)^3*B*b^8*c^8*d^3*g^3*i^3*n/(d*x + c)^3 + 7 0*B*a^4*b^7*c^4*d^4*g^3*i^3*n + 392*(b*x + a)*B*a^3*b^7*c^5*d^4*g^3*i^3*n/ (d*x + c) + 588*(b*x + a)^2*B*a^2*b^7*c^6*d^4*g^3*i^3*n/(d*x + c)^2 + 280* (b*x + a)^3*B*a*b^7*c^7*d^4*g^3*i^3*n/(d*x + c)^3 - 56*B*a^5*b^6*c^3*d^5*g ^3*i^3*n - 490*(b*x + a)*B*a^4*b^6*c^4*d^5*g^3*i^3*n/(d*x + c) - 1176*(b*x + a)^2*B*a^3*b^6*c^5*d^5*g^3*i^3*n/(d*x + c)^2 - 980*(b*x + a)^3*B*a^2*b^ 6*c^6*d^5*g^3*i^3*n/(d*x + c)^3 + 28*B*a^6*b^5*c^2*d^6*g^3*i^3*n + 392*(b* x + a)*B*a^5*b^5*c^3*d^6*g^3*i^3*n/(d*x + c) + 1470*(b*x + a)^2*B*a^4*b^5* c^4*d^6*g^3*i^3*n/(d*x + c)^2 + 1960*(b*x + a)^3*B*a^3*b^5*c^5*d^6*g^3*i^3 *n/(d*x + c)^3 - 8*B*a^7*b^4*c*d^7*g^3*i^3*n - 196*(b*x + a)*B*a^6*b^4*c^2 *d^7*g^3*i^3*n/(d*x + c) - 1176*(b*x + a)^2*B*a^5*b^4*c^3*d^7*g^3*i^3*n/(d *x + c)^2 - 2450*(b*x + a)^3*B*a^4*b^4*c^4*d^7*g^3*i^3*n/(d*x + c)^3 + B*a ^8*b^3*d^8*g^3*i^3*n + 56*(b*x + a)*B*a^7*b^3*c*d^8*g^3*i^3*n/(d*x + c) + 588*(b*x + a)^2*B*a^6*b^3*c^2*d^8*g^3*i^3*n/(d*x + c)^2 + 1960*(b*x + a)^3 *B*a^5*b^3*c^3*d^8*g^3*i^3*n/(d*x + c)^3 - 7*(b*x + a)*B*a^8*b^2*d^9*g^...
Time = 3.06 (sec) , antiderivative size = 4476, normalized size of antiderivative = 9.38 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]
x^4*((g^3*i^3*(20*A*a^3*d^3 + 20*A*b^3*c^3 + 3*B*a^3*d^3*n - 3*B*b^3*c^3*n + 120*A*a*b^2*c^2*d + 120*A*a^2*b*c*d^2 - 6*B*a*b^2*c^2*d*n + 6*B*a^2*b*c *d^2*n))/20 + ((140*a*d + 140*b*c)*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b* c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*( 140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/( 560*b*d) - (a*c*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n ))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140))/(4*b*d)) + x^3*((g^3* i^3*(4*A*a^4*d^4 + 4*A*b^4*c^4 + B*a^4*d^4*n - B*b^4*c^4*n + 144*A*a^2*b^2 *c^2*d^2 + 64*A*a*b^3*c^3*d + 64*A*a^3*b*c*d^3 - 8*B*a*b^3*c^3*d*n + 8*B*a ^3*b*c*d^3*n))/(12*b*d) - ((140*a*d + 140*b*c)*((g^3*i^3*(20*A*a^3*d^3 + 2 0*A*b^3*c^3 + 3*B*a^3*d^3*n - 3*B*b^3*c^3*n + 120*A*a*b^2*c^2*d + 120*A*a^ 2*b*c*d^2 - 6*B*a*b^2*c^2*d*n + 6*B*a^2*b*c*d^2*n))/5 + ((140*a*d + 140*b* c)*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b ^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 32 *A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(140*b*d) - (a*c*((b^2*d^2*g^3*i^ 3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a *d + 140*b*c))/140))/(b*d)))/(420*b*d) + (a*c*((((b^2*d^2*g^3*i^3*(28*A*a* d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 14...