Integrand size = 43, antiderivative size = 387 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {B (b c-a d)^5 g^2 i^3 n x}{60 b^3 d^2}-\frac {B (b c-a d)^4 g^2 i^3 n (c+d x)^2}{120 b^2 d^3}-\frac {B (b c-a d)^3 g^2 i^3 n (c+d x)^3}{180 b d^3}+\frac {7 B (b c-a d)^2 g^2 i^3 n (c+d x)^4}{120 d^3}-\frac {b B (b c-a d) g^2 i^3 n (c+d x)^5}{30 d^3}+\frac {(b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^3}-\frac {2 b (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac {b^2 g^2 i^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^3}-\frac {B (b c-a d)^6 g^2 i^3 n \log \left (\frac {a+b x}{c+d x}\right )}{60 b^4 d^3}-\frac {B (b c-a d)^6 g^2 i^3 n \log (c+d x)}{60 b^4 d^3} \]
-1/60*B*(-a*d+b*c)^5*g^2*i^3*n*x/b^3/d^2-1/120*B*(-a*d+b*c)^4*g^2*i^3*n*(d *x+c)^2/b^2/d^3-1/180*B*(-a*d+b*c)^3*g^2*i^3*n*(d*x+c)^3/b/d^3+7/120*B*(-a *d+b*c)^2*g^2*i^3*n*(d*x+c)^4/d^3-1/30*b*B*(-a*d+b*c)*g^2*i^3*n*(d*x+c)^5/ d^3+1/4*(-a*d+b*c)^2*g^2*i^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3 -2/5*b*(-a*d+b*c)*g^2*i^3*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3+1/ 6*b^2*g^2*i^3*(d*x+c)^6*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3-1/60*B*(-a*d+b *c)^6*g^2*i^3*n*ln((b*x+a)/(d*x+c))/b^4/d^3-1/60*B*(-a*d+b*c)^6*g^2*i^3*n* ln(d*x+c)/b^4/d^3
Time = 0.22 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.14 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^2 i^3 \left (-15 B (b c-a d)^3 n \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )+12 B (b c-a d)^2 n \left (12 b d (b c-a d)^3 x+6 b^2 (b c-a d)^2 (c+d x)^2+4 b^3 (b c-a d) (c+d x)^3+3 b^4 (c+d x)^4+12 (b c-a d)^4 \log (a+b x)\right )-B (b c-a d) n \left (60 b d (b c-a d)^4 x+30 b^2 (b c-a d)^3 (c+d x)^2+20 b^3 (b c-a d)^2 (c+d x)^3+15 b^4 (b c-a d) (c+d x)^4+12 b^5 (c+d x)^5+60 (b c-a d)^5 \log (a+b x)\right )+90 b^4 (b c-a d)^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-144 b^5 (b c-a d) (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+60 b^6 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{360 b^4 d^3} \]
(g^2*i^3*(-15*B*(b*c - a*d)^3*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]) + 12*B*(b *c - a*d)^2*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*(b*c - a*d)*n*(60*b*d*(b*c - a*d)^4*x + 30*b^2*(b*c - a*d)^3* (c + d*x)^2 + 20*b^3*(b*c - a*d)^2*(c + d*x)^3 + 15*b^4*(b*c - a*d)*(c + d *x)^4 + 12*b^5*(c + d*x)^5 + 60*(b*c - a*d)^5*Log[a + b*x]) + 90*b^4*(b*c - a*d)^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 144*b^5*(b*c - a*d)*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 60*b^6*(c + d *x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))/(360*b^4*d^3)
Time = 0.54 (sec) , antiderivative size = 337, 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, 1195, 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.
| \(\displaystyle \int (a g+b g x)^2 (c i+d i x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right ) \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle g^2 i^3 (b c-a d)^6 \int \frac {(a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2782 |
| \(\displaystyle g^2 i^3 (b c-a d)^6 \left (-B n \int \frac {(c+d x) \left (b^2-\frac {6 d (a+b x) b}{c+d x}+\frac {15 d^2 (a+b x)^2}{(c+d x)^2}\right )}{60 d^3 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}+\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle g^2 i^3 (b c-a d)^6 \left (-\frac {B n \int \frac {(c+d x) \left (b^2-\frac {6 d (a+b x) b}{c+d x}+\frac {15 d^2 (a+b x)^2}{(c+d x)^2}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{60 d^3}+\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle g^2 i^3 (b c-a d)^6 \left (-\frac {B n \int \left (\frac {d}{b^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {14 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {10 b d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {c+d x}{b^4 (a+b x)}\right )d\frac {a+b x}{c+d x}}{60 d^3}+\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g^2 i^3 (b c-a d)^6 \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^4}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^4}+\frac {1}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {2 b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {7}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {1}{3 b \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{60 d^3}\right )\) |
(b*c - a*d)^6*g^2*i^3*((b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*d^3 *(b - (d*(a + b*x))/(c + d*x))^6) - (2*b*(A + B*Log[e*((a + b*x)/(c + d*x) )^n]))/(5*d^3*(b - (d*(a + b*x))/(c + d*x))^5) + (A + B*Log[e*((a + b*x)/( c + d*x))^n])/(4*d^3*(b - (d*(a + b*x))/(c + d*x))^4) - (B*n*((2*b)/(b - ( d*(a + b*x))/(c + d*x))^5 - 7/(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))/(60*d^3))
3.2.28.3.1 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[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
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. \(1720\) vs. \(2(367)=734\).
Time = 25.41 (sec) , antiderivative size = 1721, normalized size of antiderivative = 4.45
1/360*(12*B*x^5*a*b^5*d^6*g^2*i^3*n^2+540*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)* a*b^5*c*d^5*g^2*i^3*n+360*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*c*d^5*g^ 2*i^3*n+720*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c^2*d^4*g^2*i^3*n+540*B* x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*c^2*d^4*g^2*i^3*n+360*B*x^2*ln(e*((b *x+a)/(d*x+c))^n)*a*b^5*c^3*d^3*g^2*i^3*n+360*B*x*ln(e*((b*x+a)/(d*x+c))^n )*a^2*b^4*c^3*d^3*g^2*i^3*n+33*B*a^5*b*c*d^5*g^2*i^3*n^2-72*B*a^4*b^2*c^2* d^4*g^2*i^3*n^2-150*B*a^3*b^3*c^3*d^3*g^2*i^3*n^2+168*B*a^2*b^4*c^4*d^2*g^ 2*i^3*n^2+33*B*a*b^5*c^5*d*g^2*i^3*n^2-900*A*a^3*b^3*c^3*d^3*g^2*i^3*n-720 *A*a^2*b^4*c^4*d^2*g^2*i^3*n+90*B*x^2*a^2*b^4*c^2*d^4*g^2*i^3*n^2-102*B*x^ 2*a*b^5*c^3*d^3*g^2*i^3*n^2+540*A*x^2*a^2*b^4*c^2*d^4*g^2*i^3*n+360*A*x^2* a*b^5*c^3*d^3*g^2*i^3*n-36*B*x*a^4*b^2*c*d^5*g^2*i^3*n^2+90*B*x*a^3*b^3*c^ 2*d^4*g^2*i^3*n^2-30*B*x*a^2*b^4*c^3*d^3*g^2*i^3*n^2-36*B*x*a*b^5*c^4*d^2* g^2*i^3*n^2+360*A*x*a^2*b^4*c^3*d^3*g^2*i^3*n+90*B*ln(e*((b*x+a)/(d*x+c))^ n)*a^2*b^4*c^4*d^2*g^2*i^3*n+144*B*x^5*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*d^6 *g^2*i^3*n+216*B*x^5*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c*d^5*g^2*i^3*n+90*B*x^ 4*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*d^6*g^2*i^3*n+270*B*x^4*ln(e*((b*x+a)/ (d*x+c))^n)*b^6*c^2*d^4*g^2*i^3*n+18*B*x^4*a*b^5*c*d^5*g^2*i^3*n^2+540*A*x ^4*a*b^5*c*d^5*g^2*i^3*n+120*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^3*d^3*g ^2*i^3*n+78*B*x^3*a^2*b^4*c*d^5*g^2*i^3*n^2-42*B*x^3*a*b^5*c^2*d^4*g^2*i^3 *n^2+360*A*x^3*a^2*b^4*c*d^5*g^2*i^3*n+720*A*x^3*a*b^5*c^2*d^4*g^2*i^3*...
Leaf count of result is larger than twice the leaf count of optimal. 1075 vs. \(2 (367) = 734\).
Time = 0.66 (sec) , antiderivative size = 1075, normalized size of antiderivative = 2.78 \[ \int (a g+b g x)^2 (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 {60 \, A b^{6} d^{6} g^{2} i^{3} x^{6} + 6 \, {\left (20 \, B a^{3} b^{3} c^{3} d^{3} - 15 \, B a^{4} b^{2} c^{2} d^{4} + 6 \, B a^{5} b c d^{5} - B a^{6} d^{6}\right )} g^{2} i^{3} n \log \left (b x + a\right ) - 6 \, {\left (B b^{6} c^{6} - 6 \, B a b^{5} c^{5} d + 15 \, B a^{2} b^{4} c^{4} d^{2}\right )} g^{2} i^{3} n \log \left (d x + c\right ) - 12 \, {\left ({\left (B b^{6} c d^{5} - B a b^{5} d^{6}\right )} g^{2} i^{3} n - 6 \, {\left (3 \, A b^{6} c d^{5} + 2 \, A a b^{5} d^{6}\right )} g^{2} i^{3}\right )} x^{5} - 3 \, {\left ({\left (13 \, B b^{6} c^{2} d^{4} - 6 \, B a b^{5} c d^{5} - 7 \, B a^{2} b^{4} d^{6}\right )} g^{2} i^{3} n - 30 \, {\left (3 \, A b^{6} c^{2} d^{4} + 6 \, A a b^{5} c d^{5} + A a^{2} b^{4} d^{6}\right )} g^{2} i^{3}\right )} x^{4} - 2 \, {\left ({\left (19 \, B b^{6} c^{3} d^{3} + 21 \, B a b^{5} c^{2} d^{4} - 39 \, B a^{2} b^{4} c d^{5} - B a^{3} b^{3} d^{6}\right )} g^{2} i^{3} n - 60 \, {\left (A b^{6} c^{3} d^{3} + 6 \, A a b^{5} c^{2} d^{4} + 3 \, A a^{2} b^{4} c d^{5}\right )} g^{2} i^{3}\right )} x^{3} - 3 \, {\left ({\left (B b^{6} c^{4} d^{2} + 34 \, B a b^{5} c^{3} d^{3} - 30 \, B a^{2} b^{4} c^{2} d^{4} - 6 \, B a^{3} b^{3} c d^{5} + B a^{4} b^{2} d^{6}\right )} g^{2} i^{3} n - 60 \, {\left (2 \, A a b^{5} c^{3} d^{3} + 3 \, A a^{2} b^{4} c^{2} d^{4}\right )} g^{2} i^{3}\right )} x^{2} + 6 \, {\left (60 \, A a^{2} b^{4} c^{3} d^{3} g^{2} i^{3} + {\left (B b^{6} c^{5} d - 6 \, B a b^{5} c^{4} d^{2} - 5 \, B a^{2} b^{4} c^{3} d^{3} + 15 \, B a^{3} b^{3} c^{2} d^{4} - 6 \, B a^{4} b^{2} c d^{5} + B a^{5} b d^{6}\right )} g^{2} i^{3} n\right )} x + 6 \, {\left (10 \, B b^{6} d^{6} g^{2} i^{3} x^{6} + 60 \, B a^{2} b^{4} c^{3} d^{3} g^{2} i^{3} x + 12 \, {\left (3 \, B b^{6} c d^{5} + 2 \, B a b^{5} d^{6}\right )} g^{2} i^{3} x^{5} + 15 \, {\left (3 \, B b^{6} c^{2} d^{4} + 6 \, B a b^{5} c d^{5} + B a^{2} b^{4} d^{6}\right )} g^{2} i^{3} x^{4} + 20 \, {\left (B b^{6} c^{3} d^{3} + 6 \, B a b^{5} c^{2} d^{4} + 3 \, B a^{2} b^{4} c d^{5}\right )} g^{2} i^{3} x^{3} + 30 \, {\left (2 \, B a b^{5} c^{3} d^{3} + 3 \, B a^{2} b^{4} c^{2} d^{4}\right )} g^{2} i^{3} x^{2}\right )} \log \left (e\right ) + 6 \, {\left (10 \, B b^{6} d^{6} g^{2} i^{3} n x^{6} + 60 \, B a^{2} b^{4} c^{3} d^{3} g^{2} i^{3} n x + 12 \, {\left (3 \, B b^{6} c d^{5} + 2 \, B a b^{5} d^{6}\right )} g^{2} i^{3} n x^{5} + 15 \, {\left (3 \, B b^{6} c^{2} d^{4} + 6 \, B a b^{5} c d^{5} + B a^{2} b^{4} d^{6}\right )} g^{2} i^{3} n x^{4} + 20 \, {\left (B b^{6} c^{3} d^{3} + 6 \, B a b^{5} c^{2} d^{4} + 3 \, B a^{2} b^{4} c d^{5}\right )} g^{2} i^{3} n x^{3} + 30 \, {\left (2 \, B a b^{5} c^{3} d^{3} + 3 \, B a^{2} b^{4} c^{2} d^{4}\right )} g^{2} i^{3} n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{360 \, b^{4} d^{3}} \]
1/360*(60*A*b^6*d^6*g^2*i^3*x^6 + 6*(20*B*a^3*b^3*c^3*d^3 - 15*B*a^4*b^2*c ^2*d^4 + 6*B*a^5*b*c*d^5 - B*a^6*d^6)*g^2*i^3*n*log(b*x + a) - 6*(B*b^6*c^ 6 - 6*B*a*b^5*c^5*d + 15*B*a^2*b^4*c^4*d^2)*g^2*i^3*n*log(d*x + c) - 12*(( B*b^6*c*d^5 - B*a*b^5*d^6)*g^2*i^3*n - 6*(3*A*b^6*c*d^5 + 2*A*a*b^5*d^6)*g ^2*i^3)*x^5 - 3*((13*B*b^6*c^2*d^4 - 6*B*a*b^5*c*d^5 - 7*B*a^2*b^4*d^6)*g^ 2*i^3*n - 30*(3*A*b^6*c^2*d^4 + 6*A*a*b^5*c*d^5 + A*a^2*b^4*d^6)*g^2*i^3)* x^4 - 2*((19*B*b^6*c^3*d^3 + 21*B*a*b^5*c^2*d^4 - 39*B*a^2*b^4*c*d^5 - B*a ^3*b^3*d^6)*g^2*i^3*n - 60*(A*b^6*c^3*d^3 + 6*A*a*b^5*c^2*d^4 + 3*A*a^2*b^ 4*c*d^5)*g^2*i^3)*x^3 - 3*((B*b^6*c^4*d^2 + 34*B*a*b^5*c^3*d^3 - 30*B*a^2* b^4*c^2*d^4 - 6*B*a^3*b^3*c*d^5 + B*a^4*b^2*d^6)*g^2*i^3*n - 60*(2*A*a*b^5 *c^3*d^3 + 3*A*a^2*b^4*c^2*d^4)*g^2*i^3)*x^2 + 6*(60*A*a^2*b^4*c^3*d^3*g^2 *i^3 + (B*b^6*c^5*d - 6*B*a*b^5*c^4*d^2 - 5*B*a^2*b^4*c^3*d^3 + 15*B*a^3*b ^3*c^2*d^4 - 6*B*a^4*b^2*c*d^5 + B*a^5*b*d^6)*g^2*i^3*n)*x + 6*(10*B*b^6*d ^6*g^2*i^3*x^6 + 60*B*a^2*b^4*c^3*d^3*g^2*i^3*x + 12*(3*B*b^6*c*d^5 + 2*B* a*b^5*d^6)*g^2*i^3*x^5 + 15*(3*B*b^6*c^2*d^4 + 6*B*a*b^5*c*d^5 + B*a^2*b^4 *d^6)*g^2*i^3*x^4 + 20*(B*b^6*c^3*d^3 + 6*B*a*b^5*c^2*d^4 + 3*B*a^2*b^4*c* d^5)*g^2*i^3*x^3 + 30*(2*B*a*b^5*c^3*d^3 + 3*B*a^2*b^4*c^2*d^4)*g^2*i^3*x^ 2)*log(e) + 6*(10*B*b^6*d^6*g^2*i^3*n*x^6 + 60*B*a^2*b^4*c^3*d^3*g^2*i^3*n *x + 12*(3*B*b^6*c*d^5 + 2*B*a*b^5*d^6)*g^2*i^3*n*x^5 + 15*(3*B*b^6*c^2*d^ 4 + 6*B*a*b^5*c*d^5 + B*a^2*b^4*d^6)*g^2*i^3*n*x^4 + 20*(B*b^6*c^3*d^3 ...
Timed out. \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1978 vs. \(2 (367) = 734\).
Time = 0.25 (sec) , antiderivative size = 1978, normalized size of antiderivative = 5.11 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]
1/6*B*b^2*d^3*g^2*i^3*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/6*A*b ^2*d^3*g^2*i^3*x^6 + 3/5*B*b^2*c*d^2*g^2*i^3*x^5*log(e*(b*x/(d*x + c) + a/ (d*x + c))^n) + 2/5*B*a*b*d^3*g^2*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/5*A*b^2*c*d^2*g^2*i^3*x^5 + 2/5*A*a*b*d^3*g^2*i^3*x^5 + 3/4*B*b ^2*c^2*d*g^2*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*B*a*b*c* d^2*g^2*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*B*a^2*d^3*g^2 *i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/4*A*b^2*c^2*d*g^2*i^3* x^4 + 3/2*A*a*b*c*d^2*g^2*i^3*x^4 + 1/4*A*a^2*d^3*g^2*i^3*x^4 + 1/3*B*b^2* c^3*g^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*B*a*b*c^2*d*g^2 *i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + B*a^2*c*d^2*g^2*i^3*x^3* log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/3*A*b^2*c^3*g^2*i^3*x^3 + 2*A*a *b*c^2*d*g^2*i^3*x^3 + A*a^2*c*d^2*g^2*i^3*x^3 + B*a*b*c^3*g^2*i^3*x^2*log (e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*B*a^2*c^2*d*g^2*i^3*x^2*log(e*(b *x/(d*x + c) + a/(d*x + c))^n) + A*a*b*c^3*g^2*i^3*x^2 + 3/2*A*a^2*c^2*d*g ^2*i^3*x^2 - 1/360*B*b^2*d^3*g^2*i^3*n*(60*a^6*log(b*x + a)/b^6 - 60*c^6*l og(d*x + c)/d^6 + (12*(b^5*c*d^4 - a*b^4*d^5)*x^5 - 15*(b^5*c^2*d^3 - a^2* b^3*d^5)*x^4 + 20*(b^5*c^3*d^2 - a^3*b^2*d^5)*x^3 - 30*(b^5*c^4*d - a^4*b* d^5)*x^2 + 60*(b^5*c^5 - a^5*d^5)*x)/(b^5*d^5)) + 1/20*B*b^2*c*d^2*g^2*i^3 *n*(12*a^5*log(b*x + a)/b^5 - 12*c^5*log(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...
Leaf count of result is larger than twice the leaf count of optimal. 4300 vs. \(2 (367) = 734\).
Time = 1.78 (sec) , antiderivative size = 4300, normalized size of antiderivative = 11.11 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]
1/360*(6*(B*b^9*c^7*g^2*i^3*n - 7*B*a*b^8*c^6*d*g^2*i^3*n - 6*(b*x + a)*B* b^8*c^7*d*g^2*i^3*n/(d*x + c) + 21*B*a^2*b^7*c^5*d^2*g^2*i^3*n + 42*(b*x + a)*B*a*b^7*c^6*d^2*g^2*i^3*n/(d*x + c) + 15*(b*x + a)^2*B*b^7*c^7*d^2*g^2 *i^3*n/(d*x + c)^2 - 35*B*a^3*b^6*c^4*d^3*g^2*i^3*n - 126*(b*x + a)*B*a^2* b^6*c^5*d^3*g^2*i^3*n/(d*x + c) - 105*(b*x + a)^2*B*a*b^6*c^6*d^3*g^2*i^3* n/(d*x + c)^2 + 35*B*a^4*b^5*c^3*d^4*g^2*i^3*n + 210*(b*x + a)*B*a^3*b^5*c ^4*d^4*g^2*i^3*n/(d*x + c) + 315*(b*x + a)^2*B*a^2*b^5*c^5*d^4*g^2*i^3*n/( d*x + c)^2 - 21*B*a^5*b^4*c^2*d^5*g^2*i^3*n - 210*(b*x + a)*B*a^4*b^4*c^3* d^5*g^2*i^3*n/(d*x + c) - 525*(b*x + a)^2*B*a^3*b^4*c^4*d^5*g^2*i^3*n/(d*x + c)^2 + 7*B*a^6*b^3*c*d^6*g^2*i^3*n + 126*(b*x + a)*B*a^5*b^3*c^2*d^6*g^ 2*i^3*n/(d*x + c) + 525*(b*x + a)^2*B*a^4*b^3*c^3*d^6*g^2*i^3*n/(d*x + c)^ 2 - B*a^7*b^2*d^7*g^2*i^3*n - 42*(b*x + a)*B*a^6*b^2*c*d^7*g^2*i^3*n/(d*x + c) - 315*(b*x + a)^2*B*a^5*b^2*c^2*d^7*g^2*i^3*n/(d*x + c)^2 + 6*(b*x + a)*B*a^7*b*d^8*g^2*i^3*n/(d*x + c) + 105*(b*x + a)^2*B*a^6*b*c*d^8*g^2*i^3 *n/(d*x + c)^2 - 15*(b*x + a)^2*B*a^7*d^9*g^2*i^3*n/(d*x + c)^2)*log((b*x + a)/(d*x + c))/(b^6*d^3 - 6*(b*x + a)*b^5*d^4/(d*x + c) + 15*(b*x + a)^2* b^4*d^5/(d*x + c)^2 - 20*(b*x + a)^3*b^3*d^6/(d*x + c)^3 + 15*(b*x + a)^4* b^2*d^7/(d*x + c)^4 - 6*(b*x + a)^5*b*d^8/(d*x + c)^5 + (b*x + a)^6*d^9/(d *x + c)^6) - (2*B*b^12*c^7*g^2*i^3*n - 14*B*a*b^11*c^6*d*g^2*i^3*n - 18*(b *x + a)*B*b^11*c^7*d*g^2*i^3*n/(d*x + c) + 42*B*a^2*b^10*c^5*d^2*g^2*i^...
Time = 2.83 (sec) , antiderivative size = 2547, normalized size of antiderivative = 6.58 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \]
x^2*((a*c*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d* g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 60* A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(2*b*d) - ((60*a*d + 6 0*b*c)*((g^2*i^3*(4*A*a^3*d^3 + 16*A*b^3*c^3 + B*a^3*d^3*n - 3*B*b^3*c^3*n + 72*A*a*b^2*c^2*d + 48*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d*n + 5*B*a^2*b*c*d ^2*n))/(4*b) + ((60*a*d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2 *n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3) )/(60*b*d) - (a*c*((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n ))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60))/(b*d)))/(120*b*d) + (c*g^2 *i^3*(12*A*a^3*d^3 + 3*A*b^3*c^3 + 3*B*a^3*d^3*n - B*b^3*c^3*n + 36*A*a*b^ 2*c^2*d + 54*A*a^2*b*c*d^2 - 5*B*a*b^2*c^2*d*n + 3*B*a^2*b*c*d^2*n))/(6*b* d)) + x^3*((g^2*i^3*(4*A*a^3*d^3 + 16*A*b^3*c^3 + B*a^3*d^3*n - 3*B*b^3*c^ 3*n + 72*A*a*b^2*c^2*d + 48*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d*n + 5*B*a^2*b* c*d^2*n))/(12*b) + ((60*a*d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b *c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a *d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2 *d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g...