Integrand size = 29, antiderivative size = 120 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=-\frac {(b d-a e)^2 (B d-A e) (d+e x)^6}{6 e^4}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^7}{7 e^4}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^8}{8 e^4}+\frac {b^2 B (d+e x)^9}{9 e^4} \]
-1/6*(-a*e+b*d)^2*(-A*e+B*d)*(e*x+d)^6/e^4+1/7*(-a*e+b*d)*(-2*A*b*e-B*a*e+ 3*B*b*d)*(e*x+d)^7/e^4-1/8*b*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^8/e^4+1/9*b^ 2*B*(e*x+d)^9/e^4
Leaf count is larger than twice the leaf count of optimal. \(330\) vs. \(2(120)=240\).
Time = 0.08 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.75 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=a^2 A d^5 x+\frac {1}{2} a d^4 (2 A b d+a B d+5 a A e) x^2+\frac {1}{3} d^3 \left (a B d (2 b d+5 a e)+A \left (b^2 d^2+10 a b d e+10 a^2 e^2\right )\right ) x^3+\frac {1}{4} d^2 \left (10 a^2 e^2 (B d+A e)+10 a b d e (B d+2 A e)+b^2 d^2 (B d+5 A e)\right ) x^4+d e \left (4 a b d e (B d+A e)+a^2 e^2 (2 B d+A e)+b^2 d^2 (B d+2 A e)\right ) x^5+\frac {1}{6} e^2 \left (10 b^2 d^2 (B d+A e)+10 a b d e (2 B d+A e)+a^2 e^2 (5 B d+A e)\right ) x^6+\frac {1}{7} e^3 \left (a^2 B e^2+5 b^2 d (2 B d+A e)+2 a b e (5 B d+A e)\right ) x^7+\frac {1}{8} b e^4 (5 b B d+A b e+2 a B e) x^8+\frac {1}{9} b^2 B e^5 x^9 \]
a^2*A*d^5*x + (a*d^4*(2*A*b*d + a*B*d + 5*a*A*e)*x^2)/2 + (d^3*(a*B*d*(2*b *d + 5*a*e) + A*(b^2*d^2 + 10*a*b*d*e + 10*a^2*e^2))*x^3)/3 + (d^2*(10*a^2 *e^2*(B*d + A*e) + 10*a*b*d*e*(B*d + 2*A*e) + b^2*d^2*(B*d + 5*A*e))*x^4)/ 4 + d*e*(4*a*b*d*e*(B*d + A*e) + a^2*e^2*(2*B*d + A*e) + b^2*d^2*(B*d + 2* A*e))*x^5 + (e^2*(10*b^2*d^2*(B*d + A*e) + 10*a*b*d*e*(2*B*d + A*e) + a^2* e^2*(5*B*d + A*e))*x^6)/6 + (e^3*(a^2*B*e^2 + 5*b^2*d*(2*B*d + A*e) + 2*a* b*e*(5*B*d + A*e))*x^7)/7 + (b*e^4*(5*b*B*d + A*b*e + 2*a*B*e)*x^8)/8 + (b ^2*B*e^5*x^9)/9
Time = 0.48 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1184, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right ) (A+B x) (d+e x)^5 \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^2 (a+b x)^2 (A+B x) (d+e x)^5dx}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^2 (A+B x) (d+e x)^5dx\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \int \left (\frac {b (d+e x)^7 (2 a B e+A b e-3 b B d)}{e^3}+\frac {(d+e x)^6 (a e-b d) (a B e+2 A b e-3 b B d)}{e^3}+\frac {(d+e x)^5 (a e-b d)^2 (A e-B d)}{e^3}+\frac {b^2 B (d+e x)^8}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b (d+e x)^8 (-2 a B e-A b e+3 b B d)}{8 e^4}+\frac {(d+e x)^7 (b d-a e) (-a B e-2 A b e+3 b B d)}{7 e^4}-\frac {(d+e x)^6 (b d-a e)^2 (B d-A e)}{6 e^4}+\frac {b^2 B (d+e x)^9}{9 e^4}\) |
-1/6*((b*d - a*e)^2*(B*d - A*e)*(d + e*x)^6)/e^4 + ((b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^7)/(7*e^4) - (b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^8)/(8*e^4) + (b^2*B*(d + e*x)^9)/(9*e^4)
3.17.59.3.1 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[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(393\) vs. \(2(112)=224\).
Time = 0.21 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.28
| method | result | size |
| default | \(\frac {B \,b^{2} e^{5} x^{9}}{9}+\frac {\left (\left (A \,e^{5}+5 B d \,e^{4}\right ) b^{2}+2 B a b \,e^{5}\right ) x^{8}}{8}+\frac {\left (\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) b^{2}+2 \left (A \,e^{5}+5 B d \,e^{4}\right ) b a +B \,e^{5} a^{2}\right ) x^{7}}{7}+\frac {\left (\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) b^{2}+2 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) b a +\left (A \,e^{5}+5 B d \,e^{4}\right ) a^{2}\right ) x^{6}}{6}+\frac {\left (\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) b^{2}+2 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) b a +\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a^{2}\right ) x^{5}}{5}+\frac {\left (\left (5 A \,d^{4} e +B \,d^{5}\right ) b^{2}+2 \left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) b a +\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a^{2}\right ) x^{4}}{4}+\frac {\left (A \,b^{2} d^{5}+2 \left (5 A \,d^{4} e +B \,d^{5}\right ) b a +\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a^{2}\right ) x^{3}}{3}+\frac {\left (2 A a b \,d^{5}+\left (5 A \,d^{4} e +B \,d^{5}\right ) a^{2}\right ) x^{2}}{2}+d^{5} A \,a^{2} x\) | \(394\) |
| norman | \(\frac {B \,b^{2} e^{5} x^{9}}{9}+\left (\frac {1}{8} A \,b^{2} e^{5}+\frac {1}{4} B a b \,e^{5}+\frac {5}{8} B \,b^{2} d \,e^{4}\right ) x^{8}+\left (\frac {2}{7} A a b \,e^{5}+\frac {5}{7} A \,b^{2} d \,e^{4}+\frac {1}{7} B \,e^{5} a^{2}+\frac {10}{7} B a b d \,e^{4}+\frac {10}{7} B \,b^{2} d^{2} e^{3}\right ) x^{7}+\left (\frac {1}{6} A \,a^{2} e^{5}+\frac {5}{3} A a b d \,e^{4}+\frac {5}{3} A \,b^{2} d^{2} e^{3}+\frac {5}{6} B \,a^{2} d \,e^{4}+\frac {10}{3} B a b \,d^{2} e^{3}+\frac {5}{3} B \,b^{2} d^{3} e^{2}\right ) x^{6}+\left (A \,a^{2} d \,e^{4}+4 A a b \,d^{2} e^{3}+2 A \,b^{2} d^{3} e^{2}+2 B \,a^{2} d^{2} e^{3}+4 B a b \,d^{3} e^{2}+B \,b^{2} d^{4} e \right ) x^{5}+\left (\frac {5}{2} A \,a^{2} d^{2} e^{3}+5 A a b \,d^{3} e^{2}+\frac {5}{4} A \,b^{2} d^{4} e +\frac {5}{2} B \,a^{2} d^{3} e^{2}+\frac {5}{2} B a b \,d^{4} e +\frac {1}{4} B \,b^{2} d^{5}\right ) x^{4}+\left (\frac {10}{3} A \,a^{2} d^{3} e^{2}+\frac {10}{3} A a b \,d^{4} e +\frac {1}{3} A \,b^{2} d^{5}+\frac {5}{3} B \,a^{2} d^{4} e +\frac {2}{3} B a b \,d^{5}\right ) x^{3}+\left (\frac {5}{2} A \,a^{2} d^{4} e +A a b \,d^{5}+\frac {1}{2} B \,a^{2} d^{5}\right ) x^{2}+d^{5} A \,a^{2} x\) | \(397\) |
| gosper | \(\frac {x \left (56 B \,b^{2} e^{5} x^{8}+63 x^{7} A \,b^{2} e^{5}+126 x^{7} B a b \,e^{5}+315 x^{7} B \,b^{2} d \,e^{4}+144 x^{6} A a b \,e^{5}+360 x^{6} A \,b^{2} d \,e^{4}+72 x^{6} B \,e^{5} a^{2}+720 x^{6} B a b d \,e^{4}+720 x^{6} B \,b^{2} d^{2} e^{3}+84 x^{5} A \,a^{2} e^{5}+840 x^{5} A a b d \,e^{4}+840 x^{5} A \,b^{2} d^{2} e^{3}+420 x^{5} B \,a^{2} d \,e^{4}+1680 x^{5} B a b \,d^{2} e^{3}+840 x^{5} B \,b^{2} d^{3} e^{2}+504 A \,a^{2} d \,e^{4} x^{4}+2016 A a b \,d^{2} e^{3} x^{4}+1008 A \,b^{2} d^{3} e^{2} x^{4}+1008 B \,a^{2} d^{2} e^{3} x^{4}+2016 B a b \,d^{3} e^{2} x^{4}+504 B \,b^{2} d^{4} e \,x^{4}+1260 x^{3} A \,a^{2} d^{2} e^{3}+2520 x^{3} A a b \,d^{3} e^{2}+630 x^{3} A \,b^{2} d^{4} e +1260 x^{3} B \,a^{2} d^{3} e^{2}+1260 x^{3} B a b \,d^{4} e +126 x^{3} B \,b^{2} d^{5}+1680 x^{2} A \,a^{2} d^{3} e^{2}+1680 x^{2} A a b \,d^{4} e +168 x^{2} A \,b^{2} d^{5}+840 x^{2} B \,a^{2} d^{4} e +336 x^{2} B a b \,d^{5}+1260 x A \,a^{2} d^{4} e +504 x A a b \,d^{5}+252 x B \,a^{2} d^{5}+504 A \,a^{2} d^{5}\right )}{504}\) | \(464\) |
| risch | \(4 B a b \,d^{3} e^{2} x^{5}+\frac {10}{3} x^{3} A a b \,d^{4} e +4 A a b \,d^{2} e^{3} x^{5}+5 x^{4} A a b \,d^{3} e^{2}+\frac {5}{2} x^{4} B a b \,d^{4} e +\frac {10}{3} x^{6} B a b \,d^{2} e^{3}+\frac {10}{7} x^{7} B a b d \,e^{4}+\frac {5}{3} x^{6} A a b d \,e^{4}+\frac {1}{3} x^{3} A \,b^{2} d^{5}+\frac {1}{8} x^{8} A \,b^{2} e^{5}+\frac {1}{4} x^{4} B \,b^{2} d^{5}+\frac {1}{9} B \,b^{2} e^{5} x^{9}+\frac {1}{2} x^{2} B \,a^{2} d^{5}+\frac {1}{7} x^{7} B \,e^{5} a^{2}+\frac {1}{6} x^{6} A \,a^{2} e^{5}+d^{5} A \,a^{2} x +\frac {5}{4} x^{4} A \,b^{2} d^{4} e +\frac {2}{3} x^{3} B a b \,d^{5}+x^{2} A a b \,d^{5}+B \,b^{2} d^{4} e \,x^{5}+2 A \,b^{2} d^{3} e^{2} x^{5}+\frac {1}{4} x^{8} B a b \,e^{5}+\frac {5}{8} x^{8} B \,b^{2} d \,e^{4}+\frac {2}{7} x^{7} A a b \,e^{5}+\frac {5}{7} x^{7} A \,b^{2} d \,e^{4}+\frac {10}{7} x^{7} B \,b^{2} d^{2} e^{3}+\frac {5}{3} x^{6} A \,b^{2} d^{2} e^{3}+\frac {5}{3} x^{6} B \,b^{2} d^{3} e^{2}+\frac {5}{6} x^{6} B \,a^{2} d \,e^{4}+x^{5} A \,a^{2} d \,e^{4}+2 x^{5} B \,a^{2} d^{2} e^{3}+\frac {5}{2} x^{4} A \,a^{2} d^{2} e^{3}+\frac {5}{2} x^{4} B \,a^{2} d^{3} e^{2}+\frac {10}{3} x^{3} A \,a^{2} d^{3} e^{2}+\frac {5}{3} x^{3} B \,a^{2} d^{4} e +\frac {5}{2} x^{2} A \,a^{2} d^{4} e\) | \(464\) |
| parallelrisch | \(4 B a b \,d^{3} e^{2} x^{5}+\frac {10}{3} x^{3} A a b \,d^{4} e +4 A a b \,d^{2} e^{3} x^{5}+5 x^{4} A a b \,d^{3} e^{2}+\frac {5}{2} x^{4} B a b \,d^{4} e +\frac {10}{3} x^{6} B a b \,d^{2} e^{3}+\frac {10}{7} x^{7} B a b d \,e^{4}+\frac {5}{3} x^{6} A a b d \,e^{4}+\frac {1}{3} x^{3} A \,b^{2} d^{5}+\frac {1}{8} x^{8} A \,b^{2} e^{5}+\frac {1}{4} x^{4} B \,b^{2} d^{5}+\frac {1}{9} B \,b^{2} e^{5} x^{9}+\frac {1}{2} x^{2} B \,a^{2} d^{5}+\frac {1}{7} x^{7} B \,e^{5} a^{2}+\frac {1}{6} x^{6} A \,a^{2} e^{5}+d^{5} A \,a^{2} x +\frac {5}{4} x^{4} A \,b^{2} d^{4} e +\frac {2}{3} x^{3} B a b \,d^{5}+x^{2} A a b \,d^{5}+B \,b^{2} d^{4} e \,x^{5}+2 A \,b^{2} d^{3} e^{2} x^{5}+\frac {1}{4} x^{8} B a b \,e^{5}+\frac {5}{8} x^{8} B \,b^{2} d \,e^{4}+\frac {2}{7} x^{7} A a b \,e^{5}+\frac {5}{7} x^{7} A \,b^{2} d \,e^{4}+\frac {10}{7} x^{7} B \,b^{2} d^{2} e^{3}+\frac {5}{3} x^{6} A \,b^{2} d^{2} e^{3}+\frac {5}{3} x^{6} B \,b^{2} d^{3} e^{2}+\frac {5}{6} x^{6} B \,a^{2} d \,e^{4}+x^{5} A \,a^{2} d \,e^{4}+2 x^{5} B \,a^{2} d^{2} e^{3}+\frac {5}{2} x^{4} A \,a^{2} d^{2} e^{3}+\frac {5}{2} x^{4} B \,a^{2} d^{3} e^{2}+\frac {10}{3} x^{3} A \,a^{2} d^{3} e^{2}+\frac {5}{3} x^{3} B \,a^{2} d^{4} e +\frac {5}{2} x^{2} A \,a^{2} d^{4} e\) | \(464\) |
1/9*B*b^2*e^5*x^9+1/8*((A*e^5+5*B*d*e^4)*b^2+2*B*a*b*e^5)*x^8+1/7*((5*A*d* e^4+10*B*d^2*e^3)*b^2+2*(A*e^5+5*B*d*e^4)*b*a+B*e^5*a^2)*x^7+1/6*((10*A*d^ 2*e^3+10*B*d^3*e^2)*b^2+2*(5*A*d*e^4+10*B*d^2*e^3)*b*a+(A*e^5+5*B*d*e^4)*a ^2)*x^6+1/5*((10*A*d^3*e^2+5*B*d^4*e)*b^2+2*(10*A*d^2*e^3+10*B*d^3*e^2)*b* a+(5*A*d*e^4+10*B*d^2*e^3)*a^2)*x^5+1/4*((5*A*d^4*e+B*d^5)*b^2+2*(10*A*d^3 *e^2+5*B*d^4*e)*b*a+(10*A*d^2*e^3+10*B*d^3*e^2)*a^2)*x^4+1/3*(A*b^2*d^5+2* (5*A*d^4*e+B*d^5)*b*a+(10*A*d^3*e^2+5*B*d^4*e)*a^2)*x^3+1/2*(2*A*a*b*d^5+( 5*A*d^4*e+B*d^5)*a^2)*x^2+d^5*A*a^2*x
Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (112) = 224\).
Time = 0.32 (sec) , antiderivative size = 369, normalized size of antiderivative = 3.08 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{9} \, B b^{2} e^{5} x^{9} + A a^{2} d^{5} x + \frac {1}{8} \, {\left (5 \, B b^{2} d e^{4} + {\left (2 \, B a b + A b^{2}\right )} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, B b^{2} d^{2} e^{3} + 5 \, {\left (2 \, B a b + A b^{2}\right )} d e^{4} + {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, B b^{2} d^{3} e^{2} + A a^{2} e^{5} + 10 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{3} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{6} + {\left (B b^{2} d^{4} e + A a^{2} d e^{4} + 2 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{5} + 10 \, A a^{2} d^{2} e^{3} + 5 \, {\left (2 \, B a b + A b^{2}\right )} d^{4} e + 10 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, A a^{2} d^{3} e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{5} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} d^{4} e\right )} x^{3} + \frac {1}{2} \, {\left (5 \, A a^{2} d^{4} e + {\left (B a^{2} + 2 \, A a b\right )} d^{5}\right )} x^{2} \]
1/9*B*b^2*e^5*x^9 + A*a^2*d^5*x + 1/8*(5*B*b^2*d*e^4 + (2*B*a*b + A*b^2)*e ^5)*x^8 + 1/7*(10*B*b^2*d^2*e^3 + 5*(2*B*a*b + A*b^2)*d*e^4 + (B*a^2 + 2*A *a*b)*e^5)*x^7 + 1/6*(10*B*b^2*d^3*e^2 + A*a^2*e^5 + 10*(2*B*a*b + A*b^2)* d^2*e^3 + 5*(B*a^2 + 2*A*a*b)*d*e^4)*x^6 + (B*b^2*d^4*e + A*a^2*d*e^4 + 2* (2*B*a*b + A*b^2)*d^3*e^2 + 2*(B*a^2 + 2*A*a*b)*d^2*e^3)*x^5 + 1/4*(B*b^2* d^5 + 10*A*a^2*d^2*e^3 + 5*(2*B*a*b + A*b^2)*d^4*e + 10*(B*a^2 + 2*A*a*b)* d^3*e^2)*x^4 + 1/3*(10*A*a^2*d^3*e^2 + (2*B*a*b + A*b^2)*d^5 + 5*(B*a^2 + 2*A*a*b)*d^4*e)*x^3 + 1/2*(5*A*a^2*d^4*e + (B*a^2 + 2*A*a*b)*d^5)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (114) = 228\).
Time = 0.04 (sec) , antiderivative size = 481, normalized size of antiderivative = 4.01 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=A a^{2} d^{5} x + \frac {B b^{2} e^{5} x^{9}}{9} + x^{8} \left (\frac {A b^{2} e^{5}}{8} + \frac {B a b e^{5}}{4} + \frac {5 B b^{2} d e^{4}}{8}\right ) + x^{7} \cdot \left (\frac {2 A a b e^{5}}{7} + \frac {5 A b^{2} d e^{4}}{7} + \frac {B a^{2} e^{5}}{7} + \frac {10 B a b d e^{4}}{7} + \frac {10 B b^{2} d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac {A a^{2} e^{5}}{6} + \frac {5 A a b d e^{4}}{3} + \frac {5 A b^{2} d^{2} e^{3}}{3} + \frac {5 B a^{2} d e^{4}}{6} + \frac {10 B a b d^{2} e^{3}}{3} + \frac {5 B b^{2} d^{3} e^{2}}{3}\right ) + x^{5} \left (A a^{2} d e^{4} + 4 A a b d^{2} e^{3} + 2 A b^{2} d^{3} e^{2} + 2 B a^{2} d^{2} e^{3} + 4 B a b d^{3} e^{2} + B b^{2} d^{4} e\right ) + x^{4} \cdot \left (\frac {5 A a^{2} d^{2} e^{3}}{2} + 5 A a b d^{3} e^{2} + \frac {5 A b^{2} d^{4} e}{4} + \frac {5 B a^{2} d^{3} e^{2}}{2} + \frac {5 B a b d^{4} e}{2} + \frac {B b^{2} d^{5}}{4}\right ) + x^{3} \cdot \left (\frac {10 A a^{2} d^{3} e^{2}}{3} + \frac {10 A a b d^{4} e}{3} + \frac {A b^{2} d^{5}}{3} + \frac {5 B a^{2} d^{4} e}{3} + \frac {2 B a b d^{5}}{3}\right ) + x^{2} \cdot \left (\frac {5 A a^{2} d^{4} e}{2} + A a b d^{5} + \frac {B a^{2} d^{5}}{2}\right ) \]
A*a**2*d**5*x + B*b**2*e**5*x**9/9 + x**8*(A*b**2*e**5/8 + B*a*b*e**5/4 + 5*B*b**2*d*e**4/8) + x**7*(2*A*a*b*e**5/7 + 5*A*b**2*d*e**4/7 + B*a**2*e** 5/7 + 10*B*a*b*d*e**4/7 + 10*B*b**2*d**2*e**3/7) + x**6*(A*a**2*e**5/6 + 5 *A*a*b*d*e**4/3 + 5*A*b**2*d**2*e**3/3 + 5*B*a**2*d*e**4/6 + 10*B*a*b*d**2 *e**3/3 + 5*B*b**2*d**3*e**2/3) + x**5*(A*a**2*d*e**4 + 4*A*a*b*d**2*e**3 + 2*A*b**2*d**3*e**2 + 2*B*a**2*d**2*e**3 + 4*B*a*b*d**3*e**2 + B*b**2*d** 4*e) + x**4*(5*A*a**2*d**2*e**3/2 + 5*A*a*b*d**3*e**2 + 5*A*b**2*d**4*e/4 + 5*B*a**2*d**3*e**2/2 + 5*B*a*b*d**4*e/2 + B*b**2*d**5/4) + x**3*(10*A*a* *2*d**3*e**2/3 + 10*A*a*b*d**4*e/3 + A*b**2*d**5/3 + 5*B*a**2*d**4*e/3 + 2 *B*a*b*d**5/3) + x**2*(5*A*a**2*d**4*e/2 + A*a*b*d**5 + B*a**2*d**5/2)
Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (112) = 224\).
Time = 0.20 (sec) , antiderivative size = 369, normalized size of antiderivative = 3.08 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{9} \, B b^{2} e^{5} x^{9} + A a^{2} d^{5} x + \frac {1}{8} \, {\left (5 \, B b^{2} d e^{4} + {\left (2 \, B a b + A b^{2}\right )} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (10 \, B b^{2} d^{2} e^{3} + 5 \, {\left (2 \, B a b + A b^{2}\right )} d e^{4} + {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, B b^{2} d^{3} e^{2} + A a^{2} e^{5} + 10 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{3} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{6} + {\left (B b^{2} d^{4} e + A a^{2} d e^{4} + 2 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{5} + 10 \, A a^{2} d^{2} e^{3} + 5 \, {\left (2 \, B a b + A b^{2}\right )} d^{4} e + 10 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (10 \, A a^{2} d^{3} e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{5} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} d^{4} e\right )} x^{3} + \frac {1}{2} \, {\left (5 \, A a^{2} d^{4} e + {\left (B a^{2} + 2 \, A a b\right )} d^{5}\right )} x^{2} \]
1/9*B*b^2*e^5*x^9 + A*a^2*d^5*x + 1/8*(5*B*b^2*d*e^4 + (2*B*a*b + A*b^2)*e ^5)*x^8 + 1/7*(10*B*b^2*d^2*e^3 + 5*(2*B*a*b + A*b^2)*d*e^4 + (B*a^2 + 2*A *a*b)*e^5)*x^7 + 1/6*(10*B*b^2*d^3*e^2 + A*a^2*e^5 + 10*(2*B*a*b + A*b^2)* d^2*e^3 + 5*(B*a^2 + 2*A*a*b)*d*e^4)*x^6 + (B*b^2*d^4*e + A*a^2*d*e^4 + 2* (2*B*a*b + A*b^2)*d^3*e^2 + 2*(B*a^2 + 2*A*a*b)*d^2*e^3)*x^5 + 1/4*(B*b^2* d^5 + 10*A*a^2*d^2*e^3 + 5*(2*B*a*b + A*b^2)*d^4*e + 10*(B*a^2 + 2*A*a*b)* d^3*e^2)*x^4 + 1/3*(10*A*a^2*d^3*e^2 + (2*B*a*b + A*b^2)*d^5 + 5*(B*a^2 + 2*A*a*b)*d^4*e)*x^3 + 1/2*(5*A*a^2*d^4*e + (B*a^2 + 2*A*a*b)*d^5)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (112) = 224\).
Time = 0.29 (sec) , antiderivative size = 463, normalized size of antiderivative = 3.86 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {1}{9} \, B b^{2} e^{5} x^{9} + \frac {5}{8} \, B b^{2} d e^{4} x^{8} + \frac {1}{4} \, B a b e^{5} x^{8} + \frac {1}{8} \, A b^{2} e^{5} x^{8} + \frac {10}{7} \, B b^{2} d^{2} e^{3} x^{7} + \frac {10}{7} \, B a b d e^{4} x^{7} + \frac {5}{7} \, A b^{2} d e^{4} x^{7} + \frac {1}{7} \, B a^{2} e^{5} x^{7} + \frac {2}{7} \, A a b e^{5} x^{7} + \frac {5}{3} \, B b^{2} d^{3} e^{2} x^{6} + \frac {10}{3} \, B a b d^{2} e^{3} x^{6} + \frac {5}{3} \, A b^{2} d^{2} e^{3} x^{6} + \frac {5}{6} \, B a^{2} d e^{4} x^{6} + \frac {5}{3} \, A a b d e^{4} x^{6} + \frac {1}{6} \, A a^{2} e^{5} x^{6} + B b^{2} d^{4} e x^{5} + 4 \, B a b d^{3} e^{2} x^{5} + 2 \, A b^{2} d^{3} e^{2} x^{5} + 2 \, B a^{2} d^{2} e^{3} x^{5} + 4 \, A a b d^{2} e^{3} x^{5} + A a^{2} d e^{4} x^{5} + \frac {1}{4} \, B b^{2} d^{5} x^{4} + \frac {5}{2} \, B a b d^{4} e x^{4} + \frac {5}{4} \, A b^{2} d^{4} e x^{4} + \frac {5}{2} \, B a^{2} d^{3} e^{2} x^{4} + 5 \, A a b d^{3} e^{2} x^{4} + \frac {5}{2} \, A a^{2} d^{2} e^{3} x^{4} + \frac {2}{3} \, B a b d^{5} x^{3} + \frac {1}{3} \, A b^{2} d^{5} x^{3} + \frac {5}{3} \, B a^{2} d^{4} e x^{3} + \frac {10}{3} \, A a b d^{4} e x^{3} + \frac {10}{3} \, A a^{2} d^{3} e^{2} x^{3} + \frac {1}{2} \, B a^{2} d^{5} x^{2} + A a b d^{5} x^{2} + \frac {5}{2} \, A a^{2} d^{4} e x^{2} + A a^{2} d^{5} x \]
1/9*B*b^2*e^5*x^9 + 5/8*B*b^2*d*e^4*x^8 + 1/4*B*a*b*e^5*x^8 + 1/8*A*b^2*e^ 5*x^8 + 10/7*B*b^2*d^2*e^3*x^7 + 10/7*B*a*b*d*e^4*x^7 + 5/7*A*b^2*d*e^4*x^ 7 + 1/7*B*a^2*e^5*x^7 + 2/7*A*a*b*e^5*x^7 + 5/3*B*b^2*d^3*e^2*x^6 + 10/3*B *a*b*d^2*e^3*x^6 + 5/3*A*b^2*d^2*e^3*x^6 + 5/6*B*a^2*d*e^4*x^6 + 5/3*A*a*b *d*e^4*x^6 + 1/6*A*a^2*e^5*x^6 + B*b^2*d^4*e*x^5 + 4*B*a*b*d^3*e^2*x^5 + 2 *A*b^2*d^3*e^2*x^5 + 2*B*a^2*d^2*e^3*x^5 + 4*A*a*b*d^2*e^3*x^5 + A*a^2*d*e ^4*x^5 + 1/4*B*b^2*d^5*x^4 + 5/2*B*a*b*d^4*e*x^4 + 5/4*A*b^2*d^4*e*x^4 + 5 /2*B*a^2*d^3*e^2*x^4 + 5*A*a*b*d^3*e^2*x^4 + 5/2*A*a^2*d^2*e^3*x^4 + 2/3*B *a*b*d^5*x^3 + 1/3*A*b^2*d^5*x^3 + 5/3*B*a^2*d^4*e*x^3 + 10/3*A*a*b*d^4*e* x^3 + 10/3*A*a^2*d^3*e^2*x^3 + 1/2*B*a^2*d^5*x^2 + A*a*b*d^5*x^2 + 5/2*A*a ^2*d^4*e*x^2 + A*a^2*d^5*x
Time = 0.17 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.18 \[ \int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right ) \, dx=x^5\,\left (2\,B\,a^2\,d^2\,e^3+A\,a^2\,d\,e^4+4\,B\,a\,b\,d^3\,e^2+4\,A\,a\,b\,d^2\,e^3+B\,b^2\,d^4\,e+2\,A\,b^2\,d^3\,e^2\right )+x^4\,\left (\frac {5\,B\,a^2\,d^3\,e^2}{2}+\frac {5\,A\,a^2\,d^2\,e^3}{2}+\frac {5\,B\,a\,b\,d^4\,e}{2}+5\,A\,a\,b\,d^3\,e^2+\frac {B\,b^2\,d^5}{4}+\frac {5\,A\,b^2\,d^4\,e}{4}\right )+x^6\,\left (\frac {5\,B\,a^2\,d\,e^4}{6}+\frac {A\,a^2\,e^5}{6}+\frac {10\,B\,a\,b\,d^2\,e^3}{3}+\frac {5\,A\,a\,b\,d\,e^4}{3}+\frac {5\,B\,b^2\,d^3\,e^2}{3}+\frac {5\,A\,b^2\,d^2\,e^3}{3}\right )+x^3\,\left (\frac {5\,B\,a^2\,d^4\,e}{3}+\frac {10\,A\,a^2\,d^3\,e^2}{3}+\frac {2\,B\,a\,b\,d^5}{3}+\frac {10\,A\,a\,b\,d^4\,e}{3}+\frac {A\,b^2\,d^5}{3}\right )+x^7\,\left (\frac {B\,a^2\,e^5}{7}+\frac {10\,B\,a\,b\,d\,e^4}{7}+\frac {2\,A\,a\,b\,e^5}{7}+\frac {10\,B\,b^2\,d^2\,e^3}{7}+\frac {5\,A\,b^2\,d\,e^4}{7}\right )+A\,a^2\,d^5\,x+\frac {a\,d^4\,x^2\,\left (5\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b\,e^4\,x^8\,\left (A\,b\,e+2\,B\,a\,e+5\,B\,b\,d\right )}{8}+\frac {B\,b^2\,e^5\,x^9}{9} \]
x^5*(A*a^2*d*e^4 + B*b^2*d^4*e + 2*A*b^2*d^3*e^2 + 2*B*a^2*d^2*e^3 + 4*A*a *b*d^2*e^3 + 4*B*a*b*d^3*e^2) + x^4*((B*b^2*d^5)/4 + (5*A*b^2*d^4*e)/4 + ( 5*A*a^2*d^2*e^3)/2 + (5*B*a^2*d^3*e^2)/2 + (5*B*a*b*d^4*e)/2 + 5*A*a*b*d^3 *e^2) + x^6*((A*a^2*e^5)/6 + (5*B*a^2*d*e^4)/6 + (5*A*b^2*d^2*e^3)/3 + (5* B*b^2*d^3*e^2)/3 + (5*A*a*b*d*e^4)/3 + (10*B*a*b*d^2*e^3)/3) + x^3*((A*b^2 *d^5)/3 + (2*B*a*b*d^5)/3 + (5*B*a^2*d^4*e)/3 + (10*A*a^2*d^3*e^2)/3 + (10 *A*a*b*d^4*e)/3) + x^7*((B*a^2*e^5)/7 + (2*A*a*b*e^5)/7 + (5*A*b^2*d*e^4)/ 7 + (10*B*b^2*d^2*e^3)/7 + (10*B*a*b*d*e^4)/7) + A*a^2*d^5*x + (a*d^4*x^2* (5*A*a*e + 2*A*b*d + B*a*d))/2 + (b*e^4*x^8*(A*b*e + 2*B*a*e + 5*B*b*d))/8 + (B*b^2*e^5*x^9)/9