Integrand size = 25, antiderivative size = 304 \[ \int (A+B x) (d+e x)^5 \left (a+b x+c x^2\right )^2 \, dx=-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^6}{6 e^6}-\frac {\left (c d^2-b d e+a e^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right ) (d+e x)^7}{7 e^6}-\frac {\left (B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) (d+e x)^8}{8 e^6}-\frac {\left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )\right ) (d+e x)^9}{9 e^6}-\frac {c (5 B c d-2 b B e-A c e) (d+e x)^{10}}{10 e^6}+\frac {B c^2 (d+e x)^{11}}{11 e^6} \]
-1/6*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^6/e^6-1/7*(a*e^2-b*d*e+c*d^2 )*(2*A*e*(-b*e+2*c*d)-B*(5*c*d^2-e*(-a*e+3*b*d)))*(e*x+d)^7/e^6-1/8*(B*(10 *c^2*d^3+b*e^2*(-2*a*e+3*b*d)-6*c*d*e*(-a*e+2*b*d))-A*e*(6*c^2*d^2+b^2*e^2 -2*c*e*(-a*e+3*b*d)))*(e*x+d)^8/e^6-1/9*(2*A*c*e*(-b*e+2*c*d)-B*(10*c^2*d^ 2+b^2*e^2-2*c*e*(-a*e+4*b*d)))*(e*x+d)^9/e^6-1/10*c*(-A*c*e-2*B*b*e+5*B*c* d)*(e*x+d)^10/e^6+1/11*B*c^2*(e*x+d)^11/e^6
Leaf count is larger than twice the leaf count of optimal. \(665\) vs. \(2(304)=608\).
Time = 0.24 (sec) , antiderivative size = 665, normalized size of antiderivative = 2.19 \[ \int (A+B x) (d+e x)^5 \left (a+b x+c x^2\right )^2 \, dx=a^2 A d^5 x+\frac {1}{2} a d^4 (2 A b d+a B d+5 a A e) x^2+\frac {1}{3} d^3 \left (a B d (2 b d+5 a e)+A \left (b^2 d^2+10 a b d e+2 a \left (c d^2+5 a e^2\right )\right )\right ) x^3+\frac {1}{4} d^2 \left (b^2 d^2 (B d+5 A e)+2 b d \left (A c d^2+5 a B d e+10 a A e^2\right )+2 a \left (B c d^3+5 A c d^2 e+5 a B d e^2+5 a A e^3\right )\right ) x^4+\frac {1}{5} d \left (5 b^2 d^2 e (B d+2 A e)+10 a B d e \left (c d^2+a e^2\right )+2 b d \left (B c d^3+5 A c d^2 e+10 a B d e^2+10 a A e^3\right )+A \left (c^2 d^4+20 a c d^2 e^2+5 a^2 e^4\right )\right ) x^5+\frac {1}{6} \left (B \left (c^2 d^5+10 c d^3 e (b d+2 a e)+5 d e^2 \left (2 b^2 d^2+4 a b d e+a^2 e^2\right )\right )+A e \left (5 c^2 d^4+20 c d^2 e (b d+a e)+e^2 \left (10 b^2 d^2+10 a b d e+a^2 e^2\right )\right )\right ) x^6+\frac {1}{7} e \left (A e \left (10 c^2 d^3+10 c d e (2 b d+a e)+b e^2 (5 b d+2 a e)\right )+B \left (5 c^2 d^4+20 c d^2 e (b d+a e)+e^2 \left (10 b^2 d^2+10 a b d e+a^2 e^2\right )\right )\right ) x^7+\frac {1}{8} e^2 \left (A e \left (10 c^2 d^2+b^2 e^2+2 c e (5 b d+a e)\right )+B \left (10 c^2 d^3+10 c d e (2 b d+a e)+b e^2 (5 b d+2 a e)\right )\right ) x^8+\frac {1}{9} e^3 \left (A c e (5 c d+2 b e)+B \left (10 c^2 d^2+b^2 e^2+2 c e (5 b d+a e)\right )\right ) x^9+\frac {1}{10} c e^4 (5 B c d+2 b B e+A c e) x^{10}+\frac {1}{11} B c^2 e^5 x^{11} \]
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 + 2*a*(c*d^2 + 5*a*e^2)))*x^3)/3 + ( d^2*(b^2*d^2*(B*d + 5*A*e) + 2*b*d*(A*c*d^2 + 5*a*B*d*e + 10*a*A*e^2) + 2* a*(B*c*d^3 + 5*A*c*d^2*e + 5*a*B*d*e^2 + 5*a*A*e^3))*x^4)/4 + (d*(5*b^2*d^ 2*e*(B*d + 2*A*e) + 10*a*B*d*e*(c*d^2 + a*e^2) + 2*b*d*(B*c*d^3 + 5*A*c*d^ 2*e + 10*a*B*d*e^2 + 10*a*A*e^3) + A*(c^2*d^4 + 20*a*c*d^2*e^2 + 5*a^2*e^4 ))*x^5)/5 + ((B*(c^2*d^5 + 10*c*d^3*e*(b*d + 2*a*e) + 5*d*e^2*(2*b^2*d^2 + 4*a*b*d*e + a^2*e^2)) + A*e*(5*c^2*d^4 + 20*c*d^2*e*(b*d + a*e) + e^2*(10 *b^2*d^2 + 10*a*b*d*e + a^2*e^2)))*x^6)/6 + (e*(A*e*(10*c^2*d^3 + 10*c*d*e *(2*b*d + a*e) + b*e^2*(5*b*d + 2*a*e)) + B*(5*c^2*d^4 + 20*c*d^2*e*(b*d + a*e) + e^2*(10*b^2*d^2 + 10*a*b*d*e + a^2*e^2)))*x^7)/7 + (e^2*(A*e*(10*c ^2*d^2 + b^2*e^2 + 2*c*e*(5*b*d + a*e)) + B*(10*c^2*d^3 + 10*c*d*e*(2*b*d + a*e) + b*e^2*(5*b*d + 2*a*e)))*x^8)/8 + (e^3*(A*c*e*(5*c*d + 2*b*e) + B* (10*c^2*d^2 + b^2*e^2 + 2*c*e*(5*b*d + a*e)))*x^9)/9 + (c*e^4*(5*B*c*d + 2 *b*B*e + A*c*e)*x^10)/10 + (B*c^2*e^5*x^11)/11
Time = 0.97 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {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.
| \(\displaystyle \int (A+B x) (d+e x)^5 \left (a+b x+c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^8 \left (B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )-2 A c e (2 c d-b e)\right )}{e^5}+\frac {(d+e x)^7 \left (A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )-B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )\right )}{e^5}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{e^5}+\frac {(d+e x)^5 (A e-B d) \left (a e^2-b d e+c d^2\right )^2}{e^5}+\frac {c (d+e x)^9 (A c e+2 b B e-5 B c d)}{e^5}+\frac {B c^2 (d+e x)^{10}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(d+e x)^9 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{9 e^6}-\frac {(d+e x)^8 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{8 e^6}+\frac {(d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{7 e^6}-\frac {(d+e x)^6 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{6 e^6}-\frac {c (d+e x)^{10} (-A c e-2 b B e+5 B c d)}{10 e^6}+\frac {B c^2 (d+e x)^{11}}{11 e^6}\) |
-1/6*((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^6)/e^6 + ((c*d^2 - b *d*e + a*e^2)*(5*B*c*d^2 - B*e*(3*b*d - a*e) - 2*A*e*(2*c*d - b*e))*(d + e *x)^7)/(7*e^6) - ((B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e*(2*b*d - a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))*(d + e*x)^8)/(8 *e^6) - ((2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e)))*(d + e*x)^9)/(9*e^6) - (c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^10 )/(10*e^6) + (B*c^2*(d + e*x)^11)/(11*e^6)
3.24.18.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(670\) vs. \(2(292)=584\).
Time = 0.37 (sec) , antiderivative size = 671, normalized size of antiderivative = 2.21
| method | result | size |
| default | \(\frac {B \,e^{5} c^{2} x^{11}}{11}+\frac {\left (\left (A \,e^{5}+5 B d \,e^{4}\right ) c^{2}+2 B \,e^{5} b c \right ) x^{10}}{10}+\frac {\left (\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) c^{2}+2 \left (A \,e^{5}+5 B d \,e^{4}\right ) b c +B \,e^{5} \left (2 a c +b^{2}\right )\right ) x^{9}}{9}+\frac {\left (\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) c^{2}+2 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) b c +\left (A \,e^{5}+5 B d \,e^{4}\right ) \left (2 a c +b^{2}\right )+2 B a b \,e^{5}\right ) x^{8}}{8}+\frac {\left (\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) c^{2}+2 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) b c +\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) \left (2 a c +b^{2}\right )+2 \left (A \,e^{5}+5 B d \,e^{4}\right ) b a +B \,e^{5} a^{2}\right ) x^{7}}{7}+\frac {\left (\left (5 A \,d^{4} e +B \,d^{5}\right ) c^{2}+2 \left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) b c +\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) \left (2 a c +b^{2}\right )+2 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) b a +\left (A \,e^{5}+5 B d \,e^{4}\right ) a^{2}\right ) x^{6}}{6}+\frac {\left (d^{5} A \,c^{2}+2 \left (5 A \,d^{4} e +B \,d^{5}\right ) b c +\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) \left (2 a c +b^{2}\right )+2 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) b a +\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a^{2}\right ) x^{5}}{5}+\frac {\left (2 A \,d^{5} b c +\left (5 A \,d^{4} e +B \,d^{5}\right ) \left (2 a c +b^{2}\right )+2 \left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) b a +\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a^{2}\right ) x^{4}}{4}+\frac {\left (A \,d^{5} \left (2 a c +b^{2}\right )+2 \left (5 A \,d^{4} e +B \,d^{5}\right ) b a +\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a^{2}\right ) x^{3}}{3}+\frac {\left (2 A a b \,d^{5}+\left (5 A \,d^{4} e +B \,d^{5}\right ) a^{2}\right ) x^{2}}{2}+d^{5} A \,a^{2} x\) | \(671\) |
| norman | \(\frac {B \,e^{5} c^{2} x^{11}}{11}+\left (\frac {1}{10} A \,c^{2} e^{5}+\frac {1}{5} B \,e^{5} b c +\frac {1}{2} B \,c^{2} d \,e^{4}\right ) x^{10}+\left (\frac {2}{9} A b c \,e^{5}+\frac {5}{9} A \,c^{2} d \,e^{4}+\frac {2}{9} B \,e^{5} a c +\frac {1}{9} B \,b^{2} e^{5}+\frac {10}{9} B b c d \,e^{4}+\frac {10}{9} B \,c^{2} d^{2} e^{3}\right ) x^{9}+\left (\frac {1}{4} A a c \,e^{5}+\frac {1}{8} A \,b^{2} e^{5}+\frac {5}{4} A b c d \,e^{4}+\frac {5}{4} A \,c^{2} d^{2} e^{3}+\frac {1}{4} B a b \,e^{5}+\frac {5}{4} B a c d \,e^{4}+\frac {5}{8} B \,b^{2} d \,e^{4}+\frac {5}{2} B b c \,d^{2} e^{3}+\frac {5}{4} B \,c^{2} d^{3} e^{2}\right ) x^{8}+\left (\frac {2}{7} A a b \,e^{5}+\frac {10}{7} A a c d \,e^{4}+\frac {5}{7} A \,b^{2} d \,e^{4}+\frac {20}{7} A b c \,d^{2} e^{3}+\frac {10}{7} A \,c^{2} d^{3} e^{2}+\frac {1}{7} B \,e^{5} a^{2}+\frac {10}{7} B a b d \,e^{4}+\frac {20}{7} B a c \,d^{2} e^{3}+\frac {10}{7} B \,b^{2} d^{2} e^{3}+\frac {20}{7} B b c \,d^{3} e^{2}+\frac {5}{7} B \,c^{2} d^{4} e \right ) x^{7}+\left (\frac {1}{6} A \,a^{2} e^{5}+\frac {5}{3} A a b d \,e^{4}+\frac {10}{3} A a c \,d^{2} e^{3}+\frac {5}{3} A \,b^{2} d^{2} e^{3}+\frac {10}{3} A b c \,d^{3} e^{2}+\frac {5}{6} A \,c^{2} d^{4} e +\frac {5}{6} B \,a^{2} d \,e^{4}+\frac {10}{3} B a b \,d^{2} e^{3}+\frac {10}{3} B a c \,d^{3} e^{2}+\frac {5}{3} B \,b^{2} d^{3} e^{2}+\frac {5}{3} B b c \,d^{4} e +\frac {1}{6} B \,c^{2} d^{5}\right ) x^{6}+\left (A \,a^{2} d \,e^{4}+4 A a b \,d^{2} e^{3}+4 A a c \,d^{3} e^{2}+2 A \,b^{2} d^{3} e^{2}+2 A b c \,d^{4} e +\frac {1}{5} d^{5} A \,c^{2}+2 B \,a^{2} d^{2} e^{3}+4 B a b \,d^{3} e^{2}+2 B a c \,d^{4} e +B \,b^{2} d^{4} e +\frac {2}{5} B b c \,d^{5}\right ) x^{5}+\left (\frac {5}{2} A \,a^{2} d^{2} e^{3}+5 A a b \,d^{3} e^{2}+\frac {5}{2} A a c \,d^{4} e +\frac {5}{4} A \,b^{2} d^{4} e +\frac {1}{2} A \,d^{5} b c +\frac {5}{2} B \,a^{2} d^{3} e^{2}+\frac {5}{2} B a b \,d^{4} e +\frac {1}{2} B a c \,d^{5}+\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 {2}{3} d^{5} A a c +\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\) | \(755\) |
| gosper | \(\frac {5}{2} x^{4} A a c \,d^{4} e +4 x^{5} A a c \,d^{3} e^{2}+2 x^{5} A b c \,d^{4} e +2 x^{5} B a c \,d^{4} e +\frac {5}{3} x^{6} B b c \,d^{4} e +\frac {10}{3} x^{6} B a c \,d^{3} e^{2}+\frac {10}{3} x^{6} A b c \,d^{3} e^{2}+\frac {10}{3} x^{6} A a c \,d^{2} e^{3}+\frac {20}{7} x^{7} B b c \,d^{3} e^{2}+\frac {20}{7} x^{7} B a c \,d^{2} e^{3}+\frac {5}{4} x^{8} A b c d \,e^{4}+\frac {5}{4} x^{8} B a c d \,e^{4}+\frac {5}{2} x^{8} B b c \,d^{2} e^{3}+\frac {10}{7} x^{7} A a c d \,e^{4}+\frac {20}{7} x^{7} A b c \,d^{2} e^{3}+\frac {10}{9} x^{9} B b c d \,e^{4}+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}{5} x^{5} d^{5} A \,c^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{5}+\frac {1}{10} x^{10} A \,c^{2} e^{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}+\frac {1}{11} B \,e^{5} c^{2} x^{11}+d^{5} A \,a^{2} x +\frac {1}{5} x^{10} B \,e^{5} b c +\frac {1}{2} x^{10} B \,c^{2} d \,e^{4}+\frac {2}{9} x^{9} A b c \,e^{5}+\frac {5}{9} x^{9} A \,c^{2} d \,e^{4}+\frac {2}{9} x^{9} B \,e^{5} a c +\frac {10}{9} x^{9} B \,c^{2} d^{2} e^{3}+\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}+\frac {2}{5} x^{5} B b c \,d^{5}+\frac {1}{2} x^{4} A \,d^{5} b c +\frac {1}{2} x^{4} B a c \,d^{5}+\frac {2}{3} x^{3} d^{5} A a c +\frac {1}{4} x^{8} A a c \,e^{5}+\frac {5}{4} x^{8} A \,c^{2} d^{2} e^{3}+\frac {5}{4} x^{8} B \,c^{2} d^{3} e^{2}+\frac {10}{7} x^{7} A \,c^{2} d^{3} e^{2}+\frac {5}{7} x^{7} B \,c^{2} d^{4} e +\frac {5}{6} x^{6} A \,c^{2} d^{4} e +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\) | \(920\) |
| risch | \(\frac {5}{2} x^{4} A a c \,d^{4} e +4 x^{5} A a c \,d^{3} e^{2}+2 x^{5} A b c \,d^{4} e +2 x^{5} B a c \,d^{4} e +\frac {5}{3} x^{6} B b c \,d^{4} e +\frac {10}{3} x^{6} B a c \,d^{3} e^{2}+\frac {10}{3} x^{6} A b c \,d^{3} e^{2}+\frac {10}{3} x^{6} A a c \,d^{2} e^{3}+\frac {20}{7} x^{7} B b c \,d^{3} e^{2}+\frac {20}{7} x^{7} B a c \,d^{2} e^{3}+\frac {5}{4} x^{8} A b c d \,e^{4}+\frac {5}{4} x^{8} B a c d \,e^{4}+\frac {5}{2} x^{8} B b c \,d^{2} e^{3}+\frac {10}{7} x^{7} A a c d \,e^{4}+\frac {20}{7} x^{7} A b c \,d^{2} e^{3}+\frac {10}{9} x^{9} B b c d \,e^{4}+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}{5} x^{5} d^{5} A \,c^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{5}+\frac {1}{10} x^{10} A \,c^{2} e^{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}+\frac {1}{11} B \,e^{5} c^{2} x^{11}+d^{5} A \,a^{2} x +\frac {1}{5} x^{10} B \,e^{5} b c +\frac {1}{2} x^{10} B \,c^{2} d \,e^{4}+\frac {2}{9} x^{9} A b c \,e^{5}+\frac {5}{9} x^{9} A \,c^{2} d \,e^{4}+\frac {2}{9} x^{9} B \,e^{5} a c +\frac {10}{9} x^{9} B \,c^{2} d^{2} e^{3}+\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}+\frac {2}{5} x^{5} B b c \,d^{5}+\frac {1}{2} x^{4} A \,d^{5} b c +\frac {1}{2} x^{4} B a c \,d^{5}+\frac {2}{3} x^{3} d^{5} A a c +\frac {1}{4} x^{8} A a c \,e^{5}+\frac {5}{4} x^{8} A \,c^{2} d^{2} e^{3}+\frac {5}{4} x^{8} B \,c^{2} d^{3} e^{2}+\frac {10}{7} x^{7} A \,c^{2} d^{3} e^{2}+\frac {5}{7} x^{7} B \,c^{2} d^{4} e +\frac {5}{6} x^{6} A \,c^{2} d^{4} e +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\) | \(920\) |
| parallelrisch | \(\frac {5}{2} x^{4} A a c \,d^{4} e +4 x^{5} A a c \,d^{3} e^{2}+2 x^{5} A b c \,d^{4} e +2 x^{5} B a c \,d^{4} e +\frac {5}{3} x^{6} B b c \,d^{4} e +\frac {10}{3} x^{6} B a c \,d^{3} e^{2}+\frac {10}{3} x^{6} A b c \,d^{3} e^{2}+\frac {10}{3} x^{6} A a c \,d^{2} e^{3}+\frac {20}{7} x^{7} B b c \,d^{3} e^{2}+\frac {20}{7} x^{7} B a c \,d^{2} e^{3}+\frac {5}{4} x^{8} A b c d \,e^{4}+\frac {5}{4} x^{8} B a c d \,e^{4}+\frac {5}{2} x^{8} B b c \,d^{2} e^{3}+\frac {10}{7} x^{7} A a c d \,e^{4}+\frac {20}{7} x^{7} A b c \,d^{2} e^{3}+\frac {10}{9} x^{9} B b c d \,e^{4}+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}{5} x^{5} d^{5} A \,c^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{5}+\frac {1}{10} x^{10} A \,c^{2} e^{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}+\frac {1}{11} B \,e^{5} c^{2} x^{11}+d^{5} A \,a^{2} x +\frac {1}{5} x^{10} B \,e^{5} b c +\frac {1}{2} x^{10} B \,c^{2} d \,e^{4}+\frac {2}{9} x^{9} A b c \,e^{5}+\frac {5}{9} x^{9} A \,c^{2} d \,e^{4}+\frac {2}{9} x^{9} B \,e^{5} a c +\frac {10}{9} x^{9} B \,c^{2} d^{2} e^{3}+\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}+\frac {2}{5} x^{5} B b c \,d^{5}+\frac {1}{2} x^{4} A \,d^{5} b c +\frac {1}{2} x^{4} B a c \,d^{5}+\frac {2}{3} x^{3} d^{5} A a c +\frac {1}{4} x^{8} A a c \,e^{5}+\frac {5}{4} x^{8} A \,c^{2} d^{2} e^{3}+\frac {5}{4} x^{8} B \,c^{2} d^{3} e^{2}+\frac {10}{7} x^{7} A \,c^{2} d^{3} e^{2}+\frac {5}{7} x^{7} B \,c^{2} d^{4} e +\frac {5}{6} x^{6} A \,c^{2} d^{4} e +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\) | \(920\) |
1/11*B*e^5*c^2*x^11+1/10*((A*e^5+5*B*d*e^4)*c^2+2*B*e^5*b*c)*x^10+1/9*((5* A*d*e^4+10*B*d^2*e^3)*c^2+2*(A*e^5+5*B*d*e^4)*b*c+B*e^5*(2*a*c+b^2))*x^9+1 /8*((10*A*d^2*e^3+10*B*d^3*e^2)*c^2+2*(5*A*d*e^4+10*B*d^2*e^3)*b*c+(A*e^5+ 5*B*d*e^4)*(2*a*c+b^2)+2*B*a*b*e^5)*x^8+1/7*((10*A*d^3*e^2+5*B*d^4*e)*c^2+ 2*(10*A*d^2*e^3+10*B*d^3*e^2)*b*c+(5*A*d*e^4+10*B*d^2*e^3)*(2*a*c+b^2)+2*( A*e^5+5*B*d*e^4)*b*a+B*e^5*a^2)*x^7+1/6*((5*A*d^4*e+B*d^5)*c^2+2*(10*A*d^3 *e^2+5*B*d^4*e)*b*c+(10*A*d^2*e^3+10*B*d^3*e^2)*(2*a*c+b^2)+2*(5*A*d*e^4+1 0*B*d^2*e^3)*b*a+(A*e^5+5*B*d*e^4)*a^2)*x^6+1/5*(d^5*A*c^2+2*(5*A*d^4*e+B* d^5)*b*c+(10*A*d^3*e^2+5*B*d^4*e)*(2*a*c+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*(2*A*d^5*b*c+(5*A*d^4*e+B*d^5) *(2*a*c+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*d^5*(2*a*c+b^2)+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. 648 vs. \(2 (292) = 584\).
Time = 0.30 (sec) , antiderivative size = 648, normalized size of antiderivative = 2.13 \[ \int (A+B x) (d+e x)^5 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{11} \, B c^{2} e^{5} x^{11} + \frac {1}{10} \, {\left (5 \, B c^{2} d e^{4} + {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{10} + \frac {1}{9} \, {\left (10 \, B c^{2} d^{2} e^{3} + 5 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{9} + A a^{2} d^{5} x + \frac {1}{8} \, {\left (10 \, B c^{2} d^{3} e^{2} + 10 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 5 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (5 \, B c^{2} d^{4} e + 10 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 10 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{5} + A a^{2} e^{5} + 5 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 10 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + 10 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, A a^{2} d e^{4} + {\left (2 \, B b c + A c^{2}\right )} d^{5} + 5 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{4} e + 10 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{3} e^{2} + 10 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (10 \, A a^{2} d^{2} e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{5} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\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} + 2 \, A a c\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} \]
1/11*B*c^2*e^5*x^11 + 1/10*(5*B*c^2*d*e^4 + (2*B*b*c + A*c^2)*e^5)*x^10 + 1/9*(10*B*c^2*d^2*e^3 + 5*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2 + 2*(B*a + A*b) *c)*e^5)*x^9 + A*a^2*d^5*x + 1/8*(10*B*c^2*d^3*e^2 + 10*(2*B*b*c + A*c^2)* d^2*e^3 + 5*(B*b^2 + 2*(B*a + A*b)*c)*d*e^4 + (2*B*a*b + A*b^2 + 2*A*a*c)* e^5)*x^8 + 1/7*(5*B*c^2*d^4*e + 10*(2*B*b*c + A*c^2)*d^3*e^2 + 10*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 + 5*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 + (B*a^2 + 2*A*a*b)*e^5)*x^7 + 1/6*(B*c^2*d^5 + A*a^2*e^5 + 5*(2*B*b*c + A*c^2)*d^4* e + 10*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 + 10*(2*B*a*b + A*b^2 + 2*A*a*c)* d^2*e^3 + 5*(B*a^2 + 2*A*a*b)*d*e^4)*x^6 + 1/5*(5*A*a^2*d*e^4 + (2*B*b*c + A*c^2)*d^5 + 5*(B*b^2 + 2*(B*a + A*b)*c)*d^4*e + 10*(2*B*a*b + A*b^2 + 2* A*a*c)*d^3*e^2 + 10*(B*a^2 + 2*A*a*b)*d^2*e^3)*x^5 + 1/4*(10*A*a^2*d^2*e^3 + (B*b^2 + 2*(B*a + A*b)*c)*d^5 + 5*(2*B*a*b + A*b^2 + 2*A*a*c)*d^4*e + 1 0*(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 + 2*A*a*c)*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. 957 vs. \(2 (294) = 588\).
Time = 0.07 (sec) , antiderivative size = 957, normalized size of antiderivative = 3.15 \[ \int (A+B x) (d+e x)^5 \left (a+b x+c x^2\right )^2 \, dx=A a^{2} d^{5} x + \frac {B c^{2} e^{5} x^{11}}{11} + x^{10} \left (\frac {A c^{2} e^{5}}{10} + \frac {B b c e^{5}}{5} + \frac {B c^{2} d e^{4}}{2}\right ) + x^{9} \cdot \left (\frac {2 A b c e^{5}}{9} + \frac {5 A c^{2} d e^{4}}{9} + \frac {2 B a c e^{5}}{9} + \frac {B b^{2} e^{5}}{9} + \frac {10 B b c d e^{4}}{9} + \frac {10 B c^{2} d^{2} e^{3}}{9}\right ) + x^{8} \left (\frac {A a c e^{5}}{4} + \frac {A b^{2} e^{5}}{8} + \frac {5 A b c d e^{4}}{4} + \frac {5 A c^{2} d^{2} e^{3}}{4} + \frac {B a b e^{5}}{4} + \frac {5 B a c d e^{4}}{4} + \frac {5 B b^{2} d e^{4}}{8} + \frac {5 B b c d^{2} e^{3}}{2} + \frac {5 B c^{2} d^{3} e^{2}}{4}\right ) + x^{7} \cdot \left (\frac {2 A a b e^{5}}{7} + \frac {10 A a c d e^{4}}{7} + \frac {5 A b^{2} d e^{4}}{7} + \frac {20 A b c d^{2} e^{3}}{7} + \frac {10 A c^{2} d^{3} e^{2}}{7} + \frac {B a^{2} e^{5}}{7} + \frac {10 B a b d e^{4}}{7} + \frac {20 B a c d^{2} e^{3}}{7} + \frac {10 B b^{2} d^{2} e^{3}}{7} + \frac {20 B b c d^{3} e^{2}}{7} + \frac {5 B c^{2} d^{4} e}{7}\right ) + x^{6} \left (\frac {A a^{2} e^{5}}{6} + \frac {5 A a b d e^{4}}{3} + \frac {10 A a c d^{2} e^{3}}{3} + \frac {5 A b^{2} d^{2} e^{3}}{3} + \frac {10 A b c d^{3} e^{2}}{3} + \frac {5 A c^{2} d^{4} e}{6} + \frac {5 B a^{2} d e^{4}}{6} + \frac {10 B a b d^{2} e^{3}}{3} + \frac {10 B a c d^{3} e^{2}}{3} + \frac {5 B b^{2} d^{3} e^{2}}{3} + \frac {5 B b c d^{4} e}{3} + \frac {B c^{2} d^{5}}{6}\right ) + x^{5} \left (A a^{2} d e^{4} + 4 A a b d^{2} e^{3} + 4 A a c d^{3} e^{2} + 2 A b^{2} d^{3} e^{2} + 2 A b c d^{4} e + \frac {A c^{2} d^{5}}{5} + 2 B a^{2} d^{2} e^{3} + 4 B a b d^{3} e^{2} + 2 B a c d^{4} e + B b^{2} d^{4} e + \frac {2 B b c d^{5}}{5}\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 a c d^{4} e}{2} + \frac {5 A b^{2} d^{4} e}{4} + \frac {A b c d^{5}}{2} + \frac {5 B a^{2} d^{3} e^{2}}{2} + \frac {5 B a b d^{4} e}{2} + \frac {B a c d^{5}}{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 {2 A a c d^{5}}{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*c**2*e**5*x**11/11 + x**10*(A*c**2*e**5/10 + B*b*c*e**5/ 5 + B*c**2*d*e**4/2) + x**9*(2*A*b*c*e**5/9 + 5*A*c**2*d*e**4/9 + 2*B*a*c* e**5/9 + B*b**2*e**5/9 + 10*B*b*c*d*e**4/9 + 10*B*c**2*d**2*e**3/9) + x**8 *(A*a*c*e**5/4 + A*b**2*e**5/8 + 5*A*b*c*d*e**4/4 + 5*A*c**2*d**2*e**3/4 + B*a*b*e**5/4 + 5*B*a*c*d*e**4/4 + 5*B*b**2*d*e**4/8 + 5*B*b*c*d**2*e**3/2 + 5*B*c**2*d**3*e**2/4) + x**7*(2*A*a*b*e**5/7 + 10*A*a*c*d*e**4/7 + 5*A* b**2*d*e**4/7 + 20*A*b*c*d**2*e**3/7 + 10*A*c**2*d**3*e**2/7 + B*a**2*e**5 /7 + 10*B*a*b*d*e**4/7 + 20*B*a*c*d**2*e**3/7 + 10*B*b**2*d**2*e**3/7 + 20 *B*b*c*d**3*e**2/7 + 5*B*c**2*d**4*e/7) + x**6*(A*a**2*e**5/6 + 5*A*a*b*d* e**4/3 + 10*A*a*c*d**2*e**3/3 + 5*A*b**2*d**2*e**3/3 + 10*A*b*c*d**3*e**2/ 3 + 5*A*c**2*d**4*e/6 + 5*B*a**2*d*e**4/6 + 10*B*a*b*d**2*e**3/3 + 10*B*a* c*d**3*e**2/3 + 5*B*b**2*d**3*e**2/3 + 5*B*b*c*d**4*e/3 + B*c**2*d**5/6) + x**5*(A*a**2*d*e**4 + 4*A*a*b*d**2*e**3 + 4*A*a*c*d**3*e**2 + 2*A*b**2*d* *3*e**2 + 2*A*b*c*d**4*e + A*c**2*d**5/5 + 2*B*a**2*d**2*e**3 + 4*B*a*b*d* *3*e**2 + 2*B*a*c*d**4*e + B*b**2*d**4*e + 2*B*b*c*d**5/5) + x**4*(5*A*a** 2*d**2*e**3/2 + 5*A*a*b*d**3*e**2 + 5*A*a*c*d**4*e/2 + 5*A*b**2*d**4*e/4 + A*b*c*d**5/2 + 5*B*a**2*d**3*e**2/2 + 5*B*a*b*d**4*e/2 + B*a*c*d**5/2 + 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 + 2*A*a*c* d**5/3 + 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. 648 vs. \(2 (292) = 584\).
Time = 0.19 (sec) , antiderivative size = 648, normalized size of antiderivative = 2.13 \[ \int (A+B x) (d+e x)^5 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{11} \, B c^{2} e^{5} x^{11} + \frac {1}{10} \, {\left (5 \, B c^{2} d e^{4} + {\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{10} + \frac {1}{9} \, {\left (10 \, B c^{2} d^{2} e^{3} + 5 \, {\left (2 \, B b c + A c^{2}\right )} d e^{4} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{5}\right )} x^{9} + A a^{2} d^{5} x + \frac {1}{8} \, {\left (10 \, B c^{2} d^{3} e^{2} + 10 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 5 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{4} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (5 \, B c^{2} d^{4} e + 10 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 10 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{3} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + {\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{5} + A a^{2} e^{5} + 5 \, {\left (2 \, B b c + A c^{2}\right )} d^{4} e + 10 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e^{2} + 10 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (5 \, A a^{2} d e^{4} + {\left (2 \, B b c + A c^{2}\right )} d^{5} + 5 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{4} e + 10 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{3} e^{2} + 10 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (10 \, A a^{2} d^{2} e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{5} + 5 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\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} + 2 \, A a c\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} \]
1/11*B*c^2*e^5*x^11 + 1/10*(5*B*c^2*d*e^4 + (2*B*b*c + A*c^2)*e^5)*x^10 + 1/9*(10*B*c^2*d^2*e^3 + 5*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2 + 2*(B*a + A*b) *c)*e^5)*x^9 + A*a^2*d^5*x + 1/8*(10*B*c^2*d^3*e^2 + 10*(2*B*b*c + A*c^2)* d^2*e^3 + 5*(B*b^2 + 2*(B*a + A*b)*c)*d*e^4 + (2*B*a*b + A*b^2 + 2*A*a*c)* e^5)*x^8 + 1/7*(5*B*c^2*d^4*e + 10*(2*B*b*c + A*c^2)*d^3*e^2 + 10*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 + 5*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 + (B*a^2 + 2*A*a*b)*e^5)*x^7 + 1/6*(B*c^2*d^5 + A*a^2*e^5 + 5*(2*B*b*c + A*c^2)*d^4* e + 10*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 + 10*(2*B*a*b + A*b^2 + 2*A*a*c)* d^2*e^3 + 5*(B*a^2 + 2*A*a*b)*d*e^4)*x^6 + 1/5*(5*A*a^2*d*e^4 + (2*B*b*c + A*c^2)*d^5 + 5*(B*b^2 + 2*(B*a + A*b)*c)*d^4*e + 10*(2*B*a*b + A*b^2 + 2* A*a*c)*d^3*e^2 + 10*(B*a^2 + 2*A*a*b)*d^2*e^3)*x^5 + 1/4*(10*A*a^2*d^2*e^3 + (B*b^2 + 2*(B*a + A*b)*c)*d^5 + 5*(2*B*a*b + A*b^2 + 2*A*a*c)*d^4*e + 1 0*(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 + 2*A*a*c)*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. 919 vs. \(2 (292) = 584\).
Time = 0.28 (sec) , antiderivative size = 919, normalized size of antiderivative = 3.02 \[ \int (A+B x) (d+e x)^5 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{11} \, B c^{2} e^{5} x^{11} + \frac {1}{2} \, B c^{2} d e^{4} x^{10} + \frac {1}{5} \, B b c e^{5} x^{10} + \frac {1}{10} \, A c^{2} e^{5} x^{10} + \frac {10}{9} \, B c^{2} d^{2} e^{3} x^{9} + \frac {10}{9} \, B b c d e^{4} x^{9} + \frac {5}{9} \, A c^{2} d e^{4} x^{9} + \frac {1}{9} \, B b^{2} e^{5} x^{9} + \frac {2}{9} \, B a c e^{5} x^{9} + \frac {2}{9} \, A b c e^{5} x^{9} + \frac {5}{4} \, B c^{2} d^{3} e^{2} x^{8} + \frac {5}{2} \, B b c d^{2} e^{3} x^{8} + \frac {5}{4} \, A c^{2} d^{2} e^{3} x^{8} + \frac {5}{8} \, B b^{2} d e^{4} x^{8} + \frac {5}{4} \, B a c d e^{4} x^{8} + \frac {5}{4} \, A b c 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 {1}{4} \, A a c e^{5} x^{8} + \frac {5}{7} \, B c^{2} d^{4} e x^{7} + \frac {20}{7} \, B b c d^{3} e^{2} x^{7} + \frac {10}{7} \, A c^{2} d^{3} e^{2} x^{7} + \frac {10}{7} \, B b^{2} d^{2} e^{3} x^{7} + \frac {20}{7} \, B a c d^{2} e^{3} x^{7} + \frac {20}{7} \, A b c 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 {10}{7} \, A a c 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 {1}{6} \, B c^{2} d^{5} x^{6} + \frac {5}{3} \, B b c d^{4} e x^{6} + \frac {5}{6} \, A c^{2} d^{4} e x^{6} + \frac {5}{3} \, B b^{2} d^{3} e^{2} x^{6} + \frac {10}{3} \, B a c d^{3} e^{2} x^{6} + \frac {10}{3} \, A b c 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 {10}{3} \, A a c 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} + \frac {2}{5} \, B b c d^{5} x^{5} + \frac {1}{5} \, A c^{2} d^{5} x^{5} + B b^{2} d^{4} e x^{5} + 2 \, B a c d^{4} e x^{5} + 2 \, A b c 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} + 4 \, A a c 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 {1}{2} \, B a c d^{5} x^{4} + \frac {1}{2} \, A b c 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} \, A a c 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 {2}{3} \, A a c 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 \]
1/11*B*c^2*e^5*x^11 + 1/2*B*c^2*d*e^4*x^10 + 1/5*B*b*c*e^5*x^10 + 1/10*A*c ^2*e^5*x^10 + 10/9*B*c^2*d^2*e^3*x^9 + 10/9*B*b*c*d*e^4*x^9 + 5/9*A*c^2*d* e^4*x^9 + 1/9*B*b^2*e^5*x^9 + 2/9*B*a*c*e^5*x^9 + 2/9*A*b*c*e^5*x^9 + 5/4* B*c^2*d^3*e^2*x^8 + 5/2*B*b*c*d^2*e^3*x^8 + 5/4*A*c^2*d^2*e^3*x^8 + 5/8*B* b^2*d*e^4*x^8 + 5/4*B*a*c*d*e^4*x^8 + 5/4*A*b*c*d*e^4*x^8 + 1/4*B*a*b*e^5* x^8 + 1/8*A*b^2*e^5*x^8 + 1/4*A*a*c*e^5*x^8 + 5/7*B*c^2*d^4*e*x^7 + 20/7*B *b*c*d^3*e^2*x^7 + 10/7*A*c^2*d^3*e^2*x^7 + 10/7*B*b^2*d^2*e^3*x^7 + 20/7* B*a*c*d^2*e^3*x^7 + 20/7*A*b*c*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 + 10/7*A*a*c*d*e^4*x^7 + 1/7*B*a^2*e^5*x^7 + 2/7*A*a*b*e^5*x ^7 + 1/6*B*c^2*d^5*x^6 + 5/3*B*b*c*d^4*e*x^6 + 5/6*A*c^2*d^4*e*x^6 + 5/3*B *b^2*d^3*e^2*x^6 + 10/3*B*a*c*d^3*e^2*x^6 + 10/3*A*b*c*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 + 10/3*A*a*c*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 + 2/5*B*b*c*d^5*x ^5 + 1/5*A*c^2*d^5*x^5 + B*b^2*d^4*e*x^5 + 2*B*a*c*d^4*e*x^5 + 2*A*b*c*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 + 4*A*a*c*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 + 1/2*B*a*c*d^5*x^4 + 1/2*A*b*c*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*A*a*c*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 + 2/3*A*a*c*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/...
Time = 11.25 (sec) , antiderivative size = 739, normalized size of antiderivative = 2.43 \[ \int (A+B x) (d+e x)^5 \left (a+b x+c x^2\right )^2 \, dx=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 {2\,A\,c\,a\,d^5}{3}+\frac {A\,b^2\,d^5}{3}\right )+x^9\,\left (\frac {B\,b^2\,e^5}{9}+\frac {10\,B\,b\,c\,d\,e^4}{9}+\frac {2\,A\,b\,c\,e^5}{9}+\frac {10\,B\,c^2\,d^2\,e^3}{9}+\frac {5\,A\,c^2\,d\,e^4}{9}+\frac {2\,B\,a\,c\,e^5}{9}\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\,c\,a\,d^5}{2}+\frac {5\,A\,c\,a\,d^4\,e}{2}+\frac {B\,b^2\,d^5}{4}+\frac {5\,A\,b^2\,d^4\,e}{4}+\frac {A\,c\,b\,d^5}{2}\right )+x^8\,\left (\frac {5\,B\,b^2\,d\,e^4}{8}+\frac {A\,b^2\,e^5}{8}+\frac {5\,B\,b\,c\,d^2\,e^3}{2}+\frac {5\,A\,b\,c\,d\,e^4}{4}+\frac {B\,a\,b\,e^5}{4}+\frac {5\,B\,c^2\,d^3\,e^2}{4}+\frac {5\,A\,c^2\,d^2\,e^3}{4}+\frac {5\,B\,a\,c\,d\,e^4}{4}+\frac {A\,a\,c\,e^5}{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 {10\,B\,a\,c\,d^3\,e^2}{3}+\frac {10\,A\,a\,c\,d^2\,e^3}{3}+\frac {5\,B\,b^2\,d^3\,e^2}{3}+\frac {5\,A\,b^2\,d^2\,e^3}{3}+\frac {5\,B\,b\,c\,d^4\,e}{3}+\frac {10\,A\,b\,c\,d^3\,e^2}{3}+\frac {B\,c^2\,d^5}{6}+\frac {5\,A\,c^2\,d^4\,e}{6}\right )+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+2\,B\,a\,c\,d^4\,e+4\,A\,a\,c\,d^3\,e^2+B\,b^2\,d^4\,e+2\,A\,b^2\,d^3\,e^2+\frac {2\,B\,b\,c\,d^5}{5}+2\,A\,b\,c\,d^4\,e+\frac {A\,c^2\,d^5}{5}\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 {20\,B\,a\,c\,d^2\,e^3}{7}+\frac {10\,A\,a\,c\,d\,e^4}{7}+\frac {10\,B\,b^2\,d^2\,e^3}{7}+\frac {5\,A\,b^2\,d\,e^4}{7}+\frac {20\,B\,b\,c\,d^3\,e^2}{7}+\frac {20\,A\,b\,c\,d^2\,e^3}{7}+\frac {5\,B\,c^2\,d^4\,e}{7}+\frac {10\,A\,c^2\,d^3\,e^2}{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 {c\,e^4\,x^{10}\,\left (A\,c\,e+2\,B\,b\,e+5\,B\,c\,d\right )}{10}+\frac {B\,c^2\,e^5\,x^{11}}{11} \]
x^3*((A*b^2*d^5)/3 + (2*A*a*c*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^9*((B*b^2*e^5)/9 + (2*A* b*c*e^5)/9 + (2*B*a*c*e^5)/9 + (5*A*c^2*d*e^4)/9 + (10*B*c^2*d^2*e^3)/9 + (10*B*b*c*d*e^4)/9) + x^4*((B*b^2*d^5)/4 + (A*b*c*d^5)/2 + (B*a*c*d^5)/2 + (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*A*a*c* d^4*e)/2 + (5*B*a*b*d^4*e)/2 + 5*A*a*b*d^3*e^2) + x^8*((A*b^2*e^5)/8 + (A* a*c*e^5)/4 + (B*a*b*e^5)/4 + (5*B*b^2*d*e^4)/8 + (5*A*c^2*d^2*e^3)/4 + (5* B*c^2*d^3*e^2)/4 + (5*A*b*c*d*e^4)/4 + (5*B*a*c*d*e^4)/4 + (5*B*b*c*d^2*e^ 3)/2) + x^6*((A*a^2*e^5)/6 + (B*c^2*d^5)/6 + (5*B*a^2*d*e^4)/6 + (5*A*c^2* d^4*e)/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 + (5*B*b*c*d^4*e)/3 + (10*A*a*c*d^2*e^3)/3 + (10*B*a*b*d^2*e^3)/3 + (10*A*b *c*d^3*e^2)/3 + (10*B*a*c*d^3*e^2)/3) + x^5*((A*c^2*d^5)/5 + (2*B*b*c*d^5) /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 + 2*A*b *c*d^4*e + 2*B*a*c*d^4*e + 4*A*a*b*d^2*e^3 + 4*A*a*c*d^3*e^2 + 4*B*a*b*d^3 *e^2) + x^7*((B*a^2*e^5)/7 + (2*A*a*b*e^5)/7 + (5*A*b^2*d*e^4)/7 + (5*B*c^ 2*d^4*e)/7 + (10*A*c^2*d^3*e^2)/7 + (10*B*b^2*d^2*e^3)/7 + (10*A*a*c*d*e^4 )/7 + (10*B*a*b*d*e^4)/7 + (20*A*b*c*d^2*e^3)/7 + (20*B*a*c*d^2*e^3)/7 + ( 20*B*b*c*d^3*e^2)/7) + A*a^2*d^5*x + (a*d^4*x^2*(5*A*a*e + 2*A*b*d + B*a*d ))/2 + (c*e^4*x^10*(A*c*e + 2*B*b*e + 5*B*c*d))/10 + (B*c^2*e^5*x^11)/11