Integrand size = 25, antiderivative size = 304 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^5}{5 e^6}-\frac {\left (c d^2-b d e+a e^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right ) (d+e x)^6}{6 e^6}-\frac {\left (B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) (d+e x)^7}{7 e^6}-\frac {\left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )\right ) (d+e x)^8}{8 e^6}-\frac {c (5 B c d-2 b B e-A c e) (d+e x)^9}{9 e^6}+\frac {B c^2 (d+e x)^{10}}{10 e^6} \]
-1/5*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^5/e^6-1/6*(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)^6/e^6-1/7*(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)^7/e^6-1/8*(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)^8/e^6-1/9*c*(-A*c*e-2*B*b*e+5*B*c*d )*(e*x+d)^9/e^6+1/10*B*c^2*(e*x+d)^10/e^6
Time = 0.17 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.81 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=a^2 A d^4 x+\frac {1}{2} a d^3 (2 A b d+a B d+4 a A e) x^2+\frac {1}{3} d^2 \left (2 a B d (b d+2 a e)+A \left (b^2 d^2+8 a b d e+2 a \left (c d^2+3 a e^2\right )\right )\right ) x^3+\frac {1}{4} d \left (b^2 d^2 (B d+4 A e)+2 b d \left (A c d^2+4 a B d e+6 a A e^2\right )+2 a \left (B c d^3+4 A c d^2 e+3 a B d e^2+2 a A e^3\right )\right ) x^4+\frac {1}{5} \left (2 b^2 d^2 e (2 B d+3 A e)+4 a B d e \left (2 c d^2+a e^2\right )+2 b d \left (B c d^3+4 A c d^2 e+6 a B d e^2+4 a A e^3\right )+A \left (c^2 d^4+12 a c d^2 e^2+a^2 e^4\right )\right ) x^5+\frac {1}{6} \left (2 A e \left (2 c^2 d^3+b e^2 (2 b d+a e)+2 c d e (3 b d+2 a e)\right )+B \left (c^2 d^4+4 c d^2 e (2 b d+3 a e)+e^2 \left (6 b^2 d^2+8 a b d e+a^2 e^2\right )\right )\right ) x^6+\frac {1}{7} e \left (A e \left (6 c^2 d^2+b^2 e^2+2 c e (4 b d+a e)\right )+2 B \left (2 c^2 d^3+b e^2 (2 b d+a e)+2 c d e (3 b d+2 a e)\right )\right ) x^7+\frac {1}{8} e^2 \left (2 A c e (2 c d+b e)+B \left (6 c^2 d^2+b^2 e^2+2 c e (4 b d+a e)\right )\right ) x^8+\frac {1}{9} c e^3 (4 B c d+2 b B e+A c e) x^9+\frac {1}{10} B c^2 e^4 x^{10} \]
a^2*A*d^4*x + (a*d^3*(2*A*b*d + a*B*d + 4*a*A*e)*x^2)/2 + (d^2*(2*a*B*d*(b *d + 2*a*e) + A*(b^2*d^2 + 8*a*b*d*e + 2*a*(c*d^2 + 3*a*e^2)))*x^3)/3 + (d *(b^2*d^2*(B*d + 4*A*e) + 2*b*d*(A*c*d^2 + 4*a*B*d*e + 6*a*A*e^2) + 2*a*(B *c*d^3 + 4*A*c*d^2*e + 3*a*B*d*e^2 + 2*a*A*e^3))*x^4)/4 + ((2*b^2*d^2*e*(2 *B*d + 3*A*e) + 4*a*B*d*e*(2*c*d^2 + a*e^2) + 2*b*d*(B*c*d^3 + 4*A*c*d^2*e + 6*a*B*d*e^2 + 4*a*A*e^3) + A*(c^2*d^4 + 12*a*c*d^2*e^2 + a^2*e^4))*x^5) /5 + ((2*A*e*(2*c^2*d^3 + b*e^2*(2*b*d + a*e) + 2*c*d*e*(3*b*d + 2*a*e)) + B*(c^2*d^4 + 4*c*d^2*e*(2*b*d + 3*a*e) + e^2*(6*b^2*d^2 + 8*a*b*d*e + a^2 *e^2)))*x^6)/6 + (e*(A*e*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(4*b*d + a*e)) + 2*B *(2*c^2*d^3 + b*e^2*(2*b*d + a*e) + 2*c*d*e*(3*b*d + 2*a*e)))*x^7)/7 + (e^ 2*(2*A*c*e*(2*c*d + b*e) + B*(6*c^2*d^2 + b^2*e^2 + 2*c*e*(4*b*d + a*e)))* x^8)/8 + (c*e^3*(4*B*c*d + 2*b*B*e + A*c*e)*x^9)/9 + (B*c^2*e^4*x^10)/10
Time = 0.82 (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.
| \(\displaystyle \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^7 \left (B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )-2 A c e (2 c d-b e)\right )}{e^5}+\frac {(d+e x)^6 \left (A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )-B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )\right )}{e^5}+\frac {(d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{e^5}+\frac {(d+e x)^4 (A e-B d) \left (a e^2-b d e+c d^2\right )^2}{e^5}+\frac {c (d+e x)^8 (A c e+2 b B e-5 B c d)}{e^5}+\frac {B c^2 (d+e x)^9}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(d+e x)^8 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{8 e^6}-\frac {(d+e x)^7 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{7 e^6}+\frac {(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{6 e^6}-\frac {(d+e x)^5 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac {c (d+e x)^9 (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac {B c^2 (d+e x)^{10}}{10 e^6}\) |
-1/5*((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^5)/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)^6)/(6*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)^7)/(7 *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)^8)/(8*e^6) - (c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^9) /(9*e^6) + (B*c^2*(d + e*x)^10)/(10*e^6)
3.24.19.3.1 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]
Time = 0.36 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.79
| method | result | size |
| default | \(\frac {B \,e^{4} c^{2} x^{10}}{10}+\frac {\left (\left (A \,e^{4}+4 B \,e^{3} d \right ) c^{2}+2 B \,e^{4} b c \right ) x^{9}}{9}+\frac {\left (\left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) c^{2}+2 \left (A \,e^{4}+4 B \,e^{3} d \right ) b c +B \,e^{4} \left (2 a c +b^{2}\right )\right ) x^{8}}{8}+\frac {\left (\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) c^{2}+2 \left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) b c +\left (A \,e^{4}+4 B \,e^{3} d \right ) \left (2 a c +b^{2}\right )+2 B a b \,e^{4}\right ) x^{7}}{7}+\frac {\left (\left (4 A \,d^{3} e +B \,d^{4}\right ) c^{2}+2 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b c +\left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) \left (2 a c +b^{2}\right )+2 \left (A \,e^{4}+4 B \,e^{3} d \right ) b a +B \,e^{4} a^{2}\right ) x^{6}}{6}+\frac {\left (d^{4} A \,c^{2}+2 \left (4 A \,d^{3} e +B \,d^{4}\right ) b c +\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) \left (2 a c +b^{2}\right )+2 \left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) b a +\left (A \,e^{4}+4 B \,e^{3} d \right ) a^{2}\right ) x^{5}}{5}+\frac {\left (2 A \,d^{4} b c +\left (4 A \,d^{3} e +B \,d^{4}\right ) \left (2 a c +b^{2}\right )+2 \left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) b a +\left (4 A \,e^{3} d +6 B \,d^{2} e^{2}\right ) a^{2}\right ) x^{4}}{4}+\frac {\left (A \,d^{4} \left (2 a c +b^{2}\right )+2 \left (4 A \,d^{3} e +B \,d^{4}\right ) b a +\left (6 A \,d^{2} e^{2}+4 B \,d^{3} e \right ) a^{2}\right ) x^{3}}{3}+\frac {\left (2 A a b \,d^{4}+\left (4 A \,d^{3} e +B \,d^{4}\right ) a^{2}\right ) x^{2}}{2}+d^{4} A \,a^{2} x\) | \(545\) |
| norman | \(\frac {B \,e^{4} c^{2} x^{10}}{10}+\left (\frac {1}{9} A \,c^{2} e^{4}+\frac {2}{9} B \,e^{4} b c +\frac {4}{9} B \,c^{2} d \,e^{3}\right ) x^{9}+\left (\frac {1}{4} A b c \,e^{4}+\frac {1}{2} A \,c^{2} d \,e^{3}+\frac {1}{4} B \,e^{4} a c +\frac {1}{8} B \,b^{2} e^{4}+B b c d \,e^{3}+\frac {3}{4} B \,c^{2} d^{2} e^{2}\right ) x^{8}+\left (\frac {2}{7} A a c \,e^{4}+\frac {1}{7} A \,b^{2} e^{4}+\frac {8}{7} A b c d \,e^{3}+\frac {6}{7} A \,c^{2} d^{2} e^{2}+\frac {2}{7} B a b \,e^{4}+\frac {8}{7} B a c d \,e^{3}+\frac {4}{7} B \,b^{2} e^{3} d +\frac {12}{7} B b c \,d^{2} e^{2}+\frac {4}{7} B \,c^{2} d^{3} e \right ) x^{7}+\left (\frac {1}{3} A a b \,e^{4}+\frac {4}{3} A a c d \,e^{3}+\frac {2}{3} A \,b^{2} d \,e^{3}+2 A b c \,d^{2} e^{2}+\frac {2}{3} A \,c^{2} d^{3} e +\frac {1}{6} B \,e^{4} a^{2}+\frac {4}{3} B a b d \,e^{3}+2 B a c \,d^{2} e^{2}+B \,b^{2} d^{2} e^{2}+\frac {4}{3} B b c \,d^{3} e +\frac {1}{6} B \,c^{2} d^{4}\right ) x^{6}+\left (\frac {1}{5} A \,a^{2} e^{4}+\frac {8}{5} A a b d \,e^{3}+\frac {12}{5} A a c \,d^{2} e^{2}+\frac {6}{5} A \,b^{2} d^{2} e^{2}+\frac {8}{5} A b c \,d^{3} e +\frac {1}{5} d^{4} A \,c^{2}+\frac {4}{5} B \,a^{2} d \,e^{3}+\frac {12}{5} B a b \,d^{2} e^{2}+\frac {8}{5} B a c \,d^{3} e +\frac {4}{5} B \,b^{2} d^{3} e +\frac {2}{5} B b c \,d^{4}\right ) x^{5}+\left (A \,a^{2} d \,e^{3}+3 A a b \,d^{2} e^{2}+2 A a c \,d^{3} e +A \,b^{2} d^{3} e +\frac {1}{2} A \,d^{4} b c +\frac {3}{2} B \,a^{2} d^{2} e^{2}+2 B a b \,d^{3} e +\frac {1}{2} B a c \,d^{4}+\frac {1}{4} B \,b^{2} d^{4}\right ) x^{4}+\left (2 A \,a^{2} d^{2} e^{2}+\frac {8}{3} A a b \,d^{3} e +\frac {2}{3} d^{4} A a c +\frac {1}{3} A \,b^{2} d^{4}+\frac {4}{3} B \,a^{2} d^{3} e +\frac {2}{3} B a b \,d^{4}\right ) x^{3}+\left (2 A \,a^{2} d^{3} e +A a b \,d^{4}+\frac {1}{2} B \,a^{2} d^{4}\right ) x^{2}+d^{4} A \,a^{2} x\) | \(610\) |
| gosper | \(2 x^{4} A a c \,d^{3} e +\frac {8}{5} x^{5} B a c \,d^{3} e +\frac {12}{5} x^{5} A a c \,d^{2} e^{2}+\frac {8}{5} x^{5} A b c \,d^{3} e +2 x^{6} A b c \,d^{2} e^{2}+2 x^{6} B a c \,d^{2} e^{2}+\frac {4}{3} x^{6} B b c \,d^{3} e +\frac {4}{3} x^{6} A a c d \,e^{3}+\frac {12}{7} x^{7} B b c \,d^{2} e^{2}+\frac {8}{7} x^{7} B a c d \,e^{3}+\frac {8}{5} x^{5} A a b d \,e^{3}+\frac {12}{5} x^{5} B a b \,d^{2} e^{2}+3 x^{4} A a b \,d^{2} e^{2}+2 x^{4} B a b \,d^{3} e +\frac {8}{3} x^{3} A a b \,d^{3} e +\frac {4}{3} x^{6} B a b d \,e^{3}+\frac {8}{7} x^{7} A b c d \,e^{3}+x^{8} B b c d \,e^{3}+\frac {1}{3} x^{3} A \,b^{2} d^{4}+\frac {1}{7} x^{7} A \,b^{2} e^{4}+\frac {1}{4} x^{4} B \,b^{2} d^{4}+\frac {1}{6} x^{6} B \,c^{2} d^{4}+\frac {1}{5} x^{5} d^{4} A \,c^{2}+\frac {1}{9} x^{9} A \,c^{2} e^{4}+\frac {1}{8} B \,b^{2} e^{4} x^{8}+\frac {1}{5} x^{5} A \,a^{2} e^{4}+\frac {1}{2} x^{2} B \,a^{2} d^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{2}+\frac {1}{10} B \,e^{4} c^{2} x^{10}+d^{4} A \,a^{2} x +\frac {4}{9} x^{9} B \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} A b c \,e^{4}+\frac {1}{2} x^{8} A \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} B \,e^{4} a c +\frac {3}{4} x^{8} B \,c^{2} d^{2} e^{2}+\frac {2}{7} x^{7} A a c \,e^{4}+\frac {6}{7} x^{7} A \,c^{2} d^{2} e^{2}+\frac {2}{5} x^{5} B b c \,d^{4}+\frac {1}{2} x^{4} A \,d^{4} b c +\frac {1}{2} x^{4} B a c \,d^{4}+\frac {2}{3} x^{3} d^{4} A a c +\frac {2}{9} x^{9} B \,e^{4} b c +\frac {4}{7} x^{7} B \,c^{2} d^{3} e +\frac {2}{3} x^{6} A \,c^{2} d^{3} e +\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+x^{4} A \,a^{2} d \,e^{3}+\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+2 x^{3} A \,a^{2} d^{2} e^{2}+\frac {4}{3} x^{3} B \,a^{2} d^{3} e +2 x^{2} A \,a^{2} d^{3} e +x^{4} A \,b^{2} d^{3} e +\frac {2}{3} x^{3} B a b \,d^{4}+x^{2} A a b \,d^{4}+\frac {2}{7} x^{7} B a b \,e^{4}+\frac {4}{7} x^{7} B \,b^{2} e^{3} d +\frac {1}{3} x^{6} A a b \,e^{4}+\frac {2}{3} x^{6} A \,b^{2} d \,e^{3}+x^{6} B \,b^{2} d^{2} e^{2}+\frac {6}{5} x^{5} A \,b^{2} d^{2} e^{2}+\frac {4}{5} x^{5} B \,b^{2} d^{3} e\) | \(744\) |
| risch | \(2 x^{4} A a c \,d^{3} e +\frac {8}{5} x^{5} B a c \,d^{3} e +\frac {12}{5} x^{5} A a c \,d^{2} e^{2}+\frac {8}{5} x^{5} A b c \,d^{3} e +2 x^{6} A b c \,d^{2} e^{2}+2 x^{6} B a c \,d^{2} e^{2}+\frac {4}{3} x^{6} B b c \,d^{3} e +\frac {4}{3} x^{6} A a c d \,e^{3}+\frac {12}{7} x^{7} B b c \,d^{2} e^{2}+\frac {8}{7} x^{7} B a c d \,e^{3}+\frac {8}{5} x^{5} A a b d \,e^{3}+\frac {12}{5} x^{5} B a b \,d^{2} e^{2}+3 x^{4} A a b \,d^{2} e^{2}+2 x^{4} B a b \,d^{3} e +\frac {8}{3} x^{3} A a b \,d^{3} e +\frac {4}{3} x^{6} B a b d \,e^{3}+\frac {8}{7} x^{7} A b c d \,e^{3}+x^{8} B b c d \,e^{3}+\frac {1}{3} x^{3} A \,b^{2} d^{4}+\frac {1}{7} x^{7} A \,b^{2} e^{4}+\frac {1}{4} x^{4} B \,b^{2} d^{4}+\frac {1}{6} x^{6} B \,c^{2} d^{4}+\frac {1}{5} x^{5} d^{4} A \,c^{2}+\frac {1}{9} x^{9} A \,c^{2} e^{4}+\frac {1}{8} B \,b^{2} e^{4} x^{8}+\frac {1}{5} x^{5} A \,a^{2} e^{4}+\frac {1}{2} x^{2} B \,a^{2} d^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{2}+\frac {1}{10} B \,e^{4} c^{2} x^{10}+d^{4} A \,a^{2} x +\frac {4}{9} x^{9} B \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} A b c \,e^{4}+\frac {1}{2} x^{8} A \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} B \,e^{4} a c +\frac {3}{4} x^{8} B \,c^{2} d^{2} e^{2}+\frac {2}{7} x^{7} A a c \,e^{4}+\frac {6}{7} x^{7} A \,c^{2} d^{2} e^{2}+\frac {2}{5} x^{5} B b c \,d^{4}+\frac {1}{2} x^{4} A \,d^{4} b c +\frac {1}{2} x^{4} B a c \,d^{4}+\frac {2}{3} x^{3} d^{4} A a c +\frac {2}{9} x^{9} B \,e^{4} b c +\frac {4}{7} x^{7} B \,c^{2} d^{3} e +\frac {2}{3} x^{6} A \,c^{2} d^{3} e +\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+x^{4} A \,a^{2} d \,e^{3}+\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+2 x^{3} A \,a^{2} d^{2} e^{2}+\frac {4}{3} x^{3} B \,a^{2} d^{3} e +2 x^{2} A \,a^{2} d^{3} e +x^{4} A \,b^{2} d^{3} e +\frac {2}{3} x^{3} B a b \,d^{4}+x^{2} A a b \,d^{4}+\frac {2}{7} x^{7} B a b \,e^{4}+\frac {4}{7} x^{7} B \,b^{2} e^{3} d +\frac {1}{3} x^{6} A a b \,e^{4}+\frac {2}{3} x^{6} A \,b^{2} d \,e^{3}+x^{6} B \,b^{2} d^{2} e^{2}+\frac {6}{5} x^{5} A \,b^{2} d^{2} e^{2}+\frac {4}{5} x^{5} B \,b^{2} d^{3} e\) | \(744\) |
| parallelrisch | \(2 x^{4} A a c \,d^{3} e +\frac {8}{5} x^{5} B a c \,d^{3} e +\frac {12}{5} x^{5} A a c \,d^{2} e^{2}+\frac {8}{5} x^{5} A b c \,d^{3} e +2 x^{6} A b c \,d^{2} e^{2}+2 x^{6} B a c \,d^{2} e^{2}+\frac {4}{3} x^{6} B b c \,d^{3} e +\frac {4}{3} x^{6} A a c d \,e^{3}+\frac {12}{7} x^{7} B b c \,d^{2} e^{2}+\frac {8}{7} x^{7} B a c d \,e^{3}+\frac {8}{5} x^{5} A a b d \,e^{3}+\frac {12}{5} x^{5} B a b \,d^{2} e^{2}+3 x^{4} A a b \,d^{2} e^{2}+2 x^{4} B a b \,d^{3} e +\frac {8}{3} x^{3} A a b \,d^{3} e +\frac {4}{3} x^{6} B a b d \,e^{3}+\frac {8}{7} x^{7} A b c d \,e^{3}+x^{8} B b c d \,e^{3}+\frac {1}{3} x^{3} A \,b^{2} d^{4}+\frac {1}{7} x^{7} A \,b^{2} e^{4}+\frac {1}{4} x^{4} B \,b^{2} d^{4}+\frac {1}{6} x^{6} B \,c^{2} d^{4}+\frac {1}{5} x^{5} d^{4} A \,c^{2}+\frac {1}{9} x^{9} A \,c^{2} e^{4}+\frac {1}{8} B \,b^{2} e^{4} x^{8}+\frac {1}{5} x^{5} A \,a^{2} e^{4}+\frac {1}{2} x^{2} B \,a^{2} d^{4}+\frac {1}{6} x^{6} B \,e^{4} a^{2}+\frac {1}{10} B \,e^{4} c^{2} x^{10}+d^{4} A \,a^{2} x +\frac {4}{9} x^{9} B \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} A b c \,e^{4}+\frac {1}{2} x^{8} A \,c^{2} d \,e^{3}+\frac {1}{4} x^{8} B \,e^{4} a c +\frac {3}{4} x^{8} B \,c^{2} d^{2} e^{2}+\frac {2}{7} x^{7} A a c \,e^{4}+\frac {6}{7} x^{7} A \,c^{2} d^{2} e^{2}+\frac {2}{5} x^{5} B b c \,d^{4}+\frac {1}{2} x^{4} A \,d^{4} b c +\frac {1}{2} x^{4} B a c \,d^{4}+\frac {2}{3} x^{3} d^{4} A a c +\frac {2}{9} x^{9} B \,e^{4} b c +\frac {4}{7} x^{7} B \,c^{2} d^{3} e +\frac {2}{3} x^{6} A \,c^{2} d^{3} e +\frac {4}{5} x^{5} B \,a^{2} d \,e^{3}+x^{4} A \,a^{2} d \,e^{3}+\frac {3}{2} x^{4} B \,a^{2} d^{2} e^{2}+2 x^{3} A \,a^{2} d^{2} e^{2}+\frac {4}{3} x^{3} B \,a^{2} d^{3} e +2 x^{2} A \,a^{2} d^{3} e +x^{4} A \,b^{2} d^{3} e +\frac {2}{3} x^{3} B a b \,d^{4}+x^{2} A a b \,d^{4}+\frac {2}{7} x^{7} B a b \,e^{4}+\frac {4}{7} x^{7} B \,b^{2} e^{3} d +\frac {1}{3} x^{6} A a b \,e^{4}+\frac {2}{3} x^{6} A \,b^{2} d \,e^{3}+x^{6} B \,b^{2} d^{2} e^{2}+\frac {6}{5} x^{5} A \,b^{2} d^{2} e^{2}+\frac {4}{5} x^{5} B \,b^{2} d^{3} e\) | \(744\) |
1/10*B*e^4*c^2*x^10+1/9*((A*e^4+4*B*d*e^3)*c^2+2*B*e^4*b*c)*x^9+1/8*((4*A* d*e^3+6*B*d^2*e^2)*c^2+2*(A*e^4+4*B*d*e^3)*b*c+B*e^4*(2*a*c+b^2))*x^8+1/7* ((6*A*d^2*e^2+4*B*d^3*e)*c^2+2*(4*A*d*e^3+6*B*d^2*e^2)*b*c+(A*e^4+4*B*d*e^ 3)*(2*a*c+b^2)+2*B*a*b*e^4)*x^7+1/6*((4*A*d^3*e+B*d^4)*c^2+2*(6*A*d^2*e^2+ 4*B*d^3*e)*b*c+(4*A*d*e^3+6*B*d^2*e^2)*(2*a*c+b^2)+2*(A*e^4+4*B*d*e^3)*b*a +B*e^4*a^2)*x^6+1/5*(d^4*A*c^2+2*(4*A*d^3*e+B*d^4)*b*c+(6*A*d^2*e^2+4*B*d^ 3*e)*(2*a*c+b^2)+2*(4*A*d*e^3+6*B*d^2*e^2)*b*a+(A*e^4+4*B*d*e^3)*a^2)*x^5+ 1/4*(2*A*d^4*b*c+(4*A*d^3*e+B*d^4)*(2*a*c+b^2)+2*(6*A*d^2*e^2+4*B*d^3*e)*b *a+(4*A*d*e^3+6*B*d^2*e^2)*a^2)*x^4+1/3*(A*d^4*(2*a*c+b^2)+2*(4*A*d^3*e+B* d^4)*b*a+(6*A*d^2*e^2+4*B*d^3*e)*a^2)*x^3+1/2*(2*A*a*b*d^4+(4*A*d^3*e+B*d^ 4)*a^2)*x^2+d^4*A*a^2*x
Time = 0.32 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.75 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{10} \, B c^{2} e^{4} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{2} d e^{3} + {\left (2 \, B b c + A c^{2}\right )} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (6 \, B c^{2} d^{2} e^{2} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{4}\right )} x^{8} + A a^{2} d^{4} x + \frac {1}{7} \, {\left (4 \, B c^{2} d^{3} e + 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{2} + 4 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{3} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{4} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 6 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{2} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{2} e^{4} + {\left (2 \, B b c + A c^{2}\right )} d^{4} + 4 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e + 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{2} d e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{4} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{3} e + 6 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{2} d^{2} e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{4} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{2} d^{3} e + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \]
1/10*B*c^2*e^4*x^10 + 1/9*(4*B*c^2*d*e^3 + (2*B*b*c + A*c^2)*e^4)*x^9 + 1/ 8*(6*B*c^2*d^2*e^2 + 4*(2*B*b*c + A*c^2)*d*e^3 + (B*b^2 + 2*(B*a + A*b)*c) *e^4)*x^8 + A*a^2*d^4*x + 1/7*(4*B*c^2*d^3*e + 6*(2*B*b*c + A*c^2)*d^2*e^2 + 4*(B*b^2 + 2*(B*a + A*b)*c)*d*e^3 + (2*B*a*b + A*b^2 + 2*A*a*c)*e^4)*x^ 7 + 1/6*(B*c^2*d^4 + 4*(2*B*b*c + A*c^2)*d^3*e + 6*(B*b^2 + 2*(B*a + A*b)* c)*d^2*e^2 + 4*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^3 + (B*a^2 + 2*A*a*b)*e^4)* x^6 + 1/5*(A*a^2*e^4 + (2*B*b*c + A*c^2)*d^4 + 4*(B*b^2 + 2*(B*a + A*b)*c) *d^3*e + 6*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^2 + 4*(B*a^2 + 2*A*a*b)*d*e^3 )*x^5 + 1/4*(4*A*a^2*d*e^3 + (B*b^2 + 2*(B*a + A*b)*c)*d^4 + 4*(2*B*a*b + A*b^2 + 2*A*a*c)*d^3*e + 6*(B*a^2 + 2*A*a*b)*d^2*e^2)*x^4 + 1/3*(6*A*a^2*d ^2*e^2 + (2*B*a*b + A*b^2 + 2*A*a*c)*d^4 + 4*(B*a^2 + 2*A*a*b)*d^3*e)*x^3 + 1/2*(4*A*a^2*d^3*e + (B*a^2 + 2*A*a*b)*d^4)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 765 vs. \(2 (292) = 584\).
Time = 0.06 (sec) , antiderivative size = 765, normalized size of antiderivative = 2.52 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=A a^{2} d^{4} x + \frac {B c^{2} e^{4} x^{10}}{10} + x^{9} \left (\frac {A c^{2} e^{4}}{9} + \frac {2 B b c e^{4}}{9} + \frac {4 B c^{2} d e^{3}}{9}\right ) + x^{8} \left (\frac {A b c e^{4}}{4} + \frac {A c^{2} d e^{3}}{2} + \frac {B a c e^{4}}{4} + \frac {B b^{2} e^{4}}{8} + B b c d e^{3} + \frac {3 B c^{2} d^{2} e^{2}}{4}\right ) + x^{7} \cdot \left (\frac {2 A a c e^{4}}{7} + \frac {A b^{2} e^{4}}{7} + \frac {8 A b c d e^{3}}{7} + \frac {6 A c^{2} d^{2} e^{2}}{7} + \frac {2 B a b e^{4}}{7} + \frac {8 B a c d e^{3}}{7} + \frac {4 B b^{2} d e^{3}}{7} + \frac {12 B b c d^{2} e^{2}}{7} + \frac {4 B c^{2} d^{3} e}{7}\right ) + x^{6} \left (\frac {A a b e^{4}}{3} + \frac {4 A a c d e^{3}}{3} + \frac {2 A b^{2} d e^{3}}{3} + 2 A b c d^{2} e^{2} + \frac {2 A c^{2} d^{3} e}{3} + \frac {B a^{2} e^{4}}{6} + \frac {4 B a b d e^{3}}{3} + 2 B a c d^{2} e^{2} + B b^{2} d^{2} e^{2} + \frac {4 B b c d^{3} e}{3} + \frac {B c^{2} d^{4}}{6}\right ) + x^{5} \left (\frac {A a^{2} e^{4}}{5} + \frac {8 A a b d e^{3}}{5} + \frac {12 A a c d^{2} e^{2}}{5} + \frac {6 A b^{2} d^{2} e^{2}}{5} + \frac {8 A b c d^{3} e}{5} + \frac {A c^{2} d^{4}}{5} + \frac {4 B a^{2} d e^{3}}{5} + \frac {12 B a b d^{2} e^{2}}{5} + \frac {8 B a c d^{3} e}{5} + \frac {4 B b^{2} d^{3} e}{5} + \frac {2 B b c d^{4}}{5}\right ) + x^{4} \left (A a^{2} d e^{3} + 3 A a b d^{2} e^{2} + 2 A a c d^{3} e + A b^{2} d^{3} e + \frac {A b c d^{4}}{2} + \frac {3 B a^{2} d^{2} e^{2}}{2} + 2 B a b d^{3} e + \frac {B a c d^{4}}{2} + \frac {B b^{2} d^{4}}{4}\right ) + x^{3} \cdot \left (2 A a^{2} d^{2} e^{2} + \frac {8 A a b d^{3} e}{3} + \frac {2 A a c d^{4}}{3} + \frac {A b^{2} d^{4}}{3} + \frac {4 B a^{2} d^{3} e}{3} + \frac {2 B a b d^{4}}{3}\right ) + x^{2} \cdot \left (2 A a^{2} d^{3} e + A a b d^{4} + \frac {B a^{2} d^{4}}{2}\right ) \]
A*a**2*d**4*x + B*c**2*e**4*x**10/10 + x**9*(A*c**2*e**4/9 + 2*B*b*c*e**4/ 9 + 4*B*c**2*d*e**3/9) + x**8*(A*b*c*e**4/4 + A*c**2*d*e**3/2 + B*a*c*e**4 /4 + B*b**2*e**4/8 + B*b*c*d*e**3 + 3*B*c**2*d**2*e**2/4) + x**7*(2*A*a*c* e**4/7 + A*b**2*e**4/7 + 8*A*b*c*d*e**3/7 + 6*A*c**2*d**2*e**2/7 + 2*B*a*b *e**4/7 + 8*B*a*c*d*e**3/7 + 4*B*b**2*d*e**3/7 + 12*B*b*c*d**2*e**2/7 + 4* B*c**2*d**3*e/7) + x**6*(A*a*b*e**4/3 + 4*A*a*c*d*e**3/3 + 2*A*b**2*d*e**3 /3 + 2*A*b*c*d**2*e**2 + 2*A*c**2*d**3*e/3 + B*a**2*e**4/6 + 4*B*a*b*d*e** 3/3 + 2*B*a*c*d**2*e**2 + B*b**2*d**2*e**2 + 4*B*b*c*d**3*e/3 + B*c**2*d** 4/6) + x**5*(A*a**2*e**4/5 + 8*A*a*b*d*e**3/5 + 12*A*a*c*d**2*e**2/5 + 6*A *b**2*d**2*e**2/5 + 8*A*b*c*d**3*e/5 + A*c**2*d**4/5 + 4*B*a**2*d*e**3/5 + 12*B*a*b*d**2*e**2/5 + 8*B*a*c*d**3*e/5 + 4*B*b**2*d**3*e/5 + 2*B*b*c*d** 4/5) + x**4*(A*a**2*d*e**3 + 3*A*a*b*d**2*e**2 + 2*A*a*c*d**3*e + A*b**2*d **3*e + A*b*c*d**4/2 + 3*B*a**2*d**2*e**2/2 + 2*B*a*b*d**3*e + B*a*c*d**4/ 2 + B*b**2*d**4/4) + x**3*(2*A*a**2*d**2*e**2 + 8*A*a*b*d**3*e/3 + 2*A*a*c *d**4/3 + A*b**2*d**4/3 + 4*B*a**2*d**3*e/3 + 2*B*a*b*d**4/3) + x**2*(2*A* a**2*d**3*e + A*a*b*d**4 + B*a**2*d**4/2)
Time = 0.19 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.75 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{10} \, B c^{2} e^{4} x^{10} + \frac {1}{9} \, {\left (4 \, B c^{2} d e^{3} + {\left (2 \, B b c + A c^{2}\right )} e^{4}\right )} x^{9} + \frac {1}{8} \, {\left (6 \, B c^{2} d^{2} e^{2} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{4}\right )} x^{8} + A a^{2} d^{4} x + \frac {1}{7} \, {\left (4 \, B c^{2} d^{3} e + 6 \, {\left (2 \, B b c + A c^{2}\right )} d^{2} e^{2} + 4 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e^{3} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{4}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{4} + 4 \, {\left (2 \, B b c + A c^{2}\right )} d^{3} e + 6 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} e^{2} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{3} + {\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac {1}{5} \, {\left (A a^{2} e^{4} + {\left (2 \, B b c + A c^{2}\right )} d^{4} + 4 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{3} e + 6 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{2} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (4 \, A a^{2} d e^{3} + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{4} + 4 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{3} e + 6 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, A a^{2} d^{2} e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{4} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (4 \, A a^{2} d^{3} e + {\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \]
1/10*B*c^2*e^4*x^10 + 1/9*(4*B*c^2*d*e^3 + (2*B*b*c + A*c^2)*e^4)*x^9 + 1/ 8*(6*B*c^2*d^2*e^2 + 4*(2*B*b*c + A*c^2)*d*e^3 + (B*b^2 + 2*(B*a + A*b)*c) *e^4)*x^8 + A*a^2*d^4*x + 1/7*(4*B*c^2*d^3*e + 6*(2*B*b*c + A*c^2)*d^2*e^2 + 4*(B*b^2 + 2*(B*a + A*b)*c)*d*e^3 + (2*B*a*b + A*b^2 + 2*A*a*c)*e^4)*x^ 7 + 1/6*(B*c^2*d^4 + 4*(2*B*b*c + A*c^2)*d^3*e + 6*(B*b^2 + 2*(B*a + A*b)* c)*d^2*e^2 + 4*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^3 + (B*a^2 + 2*A*a*b)*e^4)* x^6 + 1/5*(A*a^2*e^4 + (2*B*b*c + A*c^2)*d^4 + 4*(B*b^2 + 2*(B*a + A*b)*c) *d^3*e + 6*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^2 + 4*(B*a^2 + 2*A*a*b)*d*e^3 )*x^5 + 1/4*(4*A*a^2*d*e^3 + (B*b^2 + 2*(B*a + A*b)*c)*d^4 + 4*(2*B*a*b + A*b^2 + 2*A*a*c)*d^3*e + 6*(B*a^2 + 2*A*a*b)*d^2*e^2)*x^4 + 1/3*(6*A*a^2*d ^2*e^2 + (2*B*a*b + A*b^2 + 2*A*a*c)*d^4 + 4*(B*a^2 + 2*A*a*b)*d^3*e)*x^3 + 1/2*(4*A*a^2*d^3*e + (B*a^2 + 2*A*a*b)*d^4)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (292) = 584\).
Time = 0.28 (sec) , antiderivative size = 743, normalized size of antiderivative = 2.44 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{10} \, B c^{2} e^{4} x^{10} + \frac {4}{9} \, B c^{2} d e^{3} x^{9} + \frac {2}{9} \, B b c e^{4} x^{9} + \frac {1}{9} \, A c^{2} e^{4} x^{9} + \frac {3}{4} \, B c^{2} d^{2} e^{2} x^{8} + B b c d e^{3} x^{8} + \frac {1}{2} \, A c^{2} d e^{3} x^{8} + \frac {1}{8} \, B b^{2} e^{4} x^{8} + \frac {1}{4} \, B a c e^{4} x^{8} + \frac {1}{4} \, A b c e^{4} x^{8} + \frac {4}{7} \, B c^{2} d^{3} e x^{7} + \frac {12}{7} \, B b c d^{2} e^{2} x^{7} + \frac {6}{7} \, A c^{2} d^{2} e^{2} x^{7} + \frac {4}{7} \, B b^{2} d e^{3} x^{7} + \frac {8}{7} \, B a c d e^{3} x^{7} + \frac {8}{7} \, A b c d e^{3} x^{7} + \frac {2}{7} \, B a b e^{4} x^{7} + \frac {1}{7} \, A b^{2} e^{4} x^{7} + \frac {2}{7} \, A a c e^{4} x^{7} + \frac {1}{6} \, B c^{2} d^{4} x^{6} + \frac {4}{3} \, B b c d^{3} e x^{6} + \frac {2}{3} \, A c^{2} d^{3} e x^{6} + B b^{2} d^{2} e^{2} x^{6} + 2 \, B a c d^{2} e^{2} x^{6} + 2 \, A b c d^{2} e^{2} x^{6} + \frac {4}{3} \, B a b d e^{3} x^{6} + \frac {2}{3} \, A b^{2} d e^{3} x^{6} + \frac {4}{3} \, A a c d e^{3} x^{6} + \frac {1}{6} \, B a^{2} e^{4} x^{6} + \frac {1}{3} \, A a b e^{4} x^{6} + \frac {2}{5} \, B b c d^{4} x^{5} + \frac {1}{5} \, A c^{2} d^{4} x^{5} + \frac {4}{5} \, B b^{2} d^{3} e x^{5} + \frac {8}{5} \, B a c d^{3} e x^{5} + \frac {8}{5} \, A b c d^{3} e x^{5} + \frac {12}{5} \, B a b d^{2} e^{2} x^{5} + \frac {6}{5} \, A b^{2} d^{2} e^{2} x^{5} + \frac {12}{5} \, A a c d^{2} e^{2} x^{5} + \frac {4}{5} \, B a^{2} d e^{3} x^{5} + \frac {8}{5} \, A a b d e^{3} x^{5} + \frac {1}{5} \, A a^{2} e^{4} x^{5} + \frac {1}{4} \, B b^{2} d^{4} x^{4} + \frac {1}{2} \, B a c d^{4} x^{4} + \frac {1}{2} \, A b c d^{4} x^{4} + 2 \, B a b d^{3} e x^{4} + A b^{2} d^{3} e x^{4} + 2 \, A a c d^{3} e x^{4} + \frac {3}{2} \, B a^{2} d^{2} e^{2} x^{4} + 3 \, A a b d^{2} e^{2} x^{4} + A a^{2} d e^{3} x^{4} + \frac {2}{3} \, B a b d^{4} x^{3} + \frac {1}{3} \, A b^{2} d^{4} x^{3} + \frac {2}{3} \, A a c d^{4} x^{3} + \frac {4}{3} \, B a^{2} d^{3} e x^{3} + \frac {8}{3} \, A a b d^{3} e x^{3} + 2 \, A a^{2} d^{2} e^{2} x^{3} + \frac {1}{2} \, B a^{2} d^{4} x^{2} + A a b d^{4} x^{2} + 2 \, A a^{2} d^{3} e x^{2} + A a^{2} d^{4} x \]
1/10*B*c^2*e^4*x^10 + 4/9*B*c^2*d*e^3*x^9 + 2/9*B*b*c*e^4*x^9 + 1/9*A*c^2* e^4*x^9 + 3/4*B*c^2*d^2*e^2*x^8 + B*b*c*d*e^3*x^8 + 1/2*A*c^2*d*e^3*x^8 + 1/8*B*b^2*e^4*x^8 + 1/4*B*a*c*e^4*x^8 + 1/4*A*b*c*e^4*x^8 + 4/7*B*c^2*d^3* e*x^7 + 12/7*B*b*c*d^2*e^2*x^7 + 6/7*A*c^2*d^2*e^2*x^7 + 4/7*B*b^2*d*e^3*x ^7 + 8/7*B*a*c*d*e^3*x^7 + 8/7*A*b*c*d*e^3*x^7 + 2/7*B*a*b*e^4*x^7 + 1/7*A *b^2*e^4*x^7 + 2/7*A*a*c*e^4*x^7 + 1/6*B*c^2*d^4*x^6 + 4/3*B*b*c*d^3*e*x^6 + 2/3*A*c^2*d^3*e*x^6 + B*b^2*d^2*e^2*x^6 + 2*B*a*c*d^2*e^2*x^6 + 2*A*b*c *d^2*e^2*x^6 + 4/3*B*a*b*d*e^3*x^6 + 2/3*A*b^2*d*e^3*x^6 + 4/3*A*a*c*d*e^3 *x^6 + 1/6*B*a^2*e^4*x^6 + 1/3*A*a*b*e^4*x^6 + 2/5*B*b*c*d^4*x^5 + 1/5*A*c ^2*d^4*x^5 + 4/5*B*b^2*d^3*e*x^5 + 8/5*B*a*c*d^3*e*x^5 + 8/5*A*b*c*d^3*e*x ^5 + 12/5*B*a*b*d^2*e^2*x^5 + 6/5*A*b^2*d^2*e^2*x^5 + 12/5*A*a*c*d^2*e^2*x ^5 + 4/5*B*a^2*d*e^3*x^5 + 8/5*A*a*b*d*e^3*x^5 + 1/5*A*a^2*e^4*x^5 + 1/4*B *b^2*d^4*x^4 + 1/2*B*a*c*d^4*x^4 + 1/2*A*b*c*d^4*x^4 + 2*B*a*b*d^3*e*x^4 + A*b^2*d^3*e*x^4 + 2*A*a*c*d^3*e*x^4 + 3/2*B*a^2*d^2*e^2*x^4 + 3*A*a*b*d^2 *e^2*x^4 + A*a^2*d*e^3*x^4 + 2/3*B*a*b*d^4*x^3 + 1/3*A*b^2*d^4*x^3 + 2/3*A *a*c*d^4*x^3 + 4/3*B*a^2*d^3*e*x^3 + 8/3*A*a*b*d^3*e*x^3 + 2*A*a^2*d^2*e^2 *x^3 + 1/2*B*a^2*d^4*x^2 + A*a*b*d^4*x^2 + 2*A*a^2*d^3*e*x^2 + A*a^2*d^4*x
Time = 11.28 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.95 \[ \int (A+B x) (d+e x)^4 \left (a+b x+c x^2\right )^2 \, dx=x^3\,\left (\frac {4\,B\,a^2\,d^3\,e}{3}+2\,A\,a^2\,d^2\,e^2+\frac {2\,B\,a\,b\,d^4}{3}+\frac {8\,A\,a\,b\,d^3\,e}{3}+\frac {2\,A\,c\,a\,d^4}{3}+\frac {A\,b^2\,d^4}{3}\right )+x^4\,\left (\frac {3\,B\,a^2\,d^2\,e^2}{2}+A\,a^2\,d\,e^3+2\,B\,a\,b\,d^3\,e+3\,A\,a\,b\,d^2\,e^2+\frac {B\,c\,a\,d^4}{2}+2\,A\,c\,a\,d^3\,e+\frac {B\,b^2\,d^4}{4}+A\,b^2\,d^3\,e+\frac {A\,c\,b\,d^4}{2}\right )+x^8\,\left (\frac {B\,b^2\,e^4}{8}+B\,b\,c\,d\,e^3+\frac {A\,b\,c\,e^4}{4}+\frac {3\,B\,c^2\,d^2\,e^2}{4}+\frac {A\,c^2\,d\,e^3}{2}+\frac {B\,a\,c\,e^4}{4}\right )+x^7\,\left (\frac {4\,B\,b^2\,d\,e^3}{7}+\frac {A\,b^2\,e^4}{7}+\frac {12\,B\,b\,c\,d^2\,e^2}{7}+\frac {8\,A\,b\,c\,d\,e^3}{7}+\frac {2\,B\,a\,b\,e^4}{7}+\frac {4\,B\,c^2\,d^3\,e}{7}+\frac {6\,A\,c^2\,d^2\,e^2}{7}+\frac {8\,B\,a\,c\,d\,e^3}{7}+\frac {2\,A\,a\,c\,e^4}{7}\right )+x^5\,\left (\frac {4\,B\,a^2\,d\,e^3}{5}+\frac {A\,a^2\,e^4}{5}+\frac {12\,B\,a\,b\,d^2\,e^2}{5}+\frac {8\,A\,a\,b\,d\,e^3}{5}+\frac {8\,B\,a\,c\,d^3\,e}{5}+\frac {12\,A\,a\,c\,d^2\,e^2}{5}+\frac {4\,B\,b^2\,d^3\,e}{5}+\frac {6\,A\,b^2\,d^2\,e^2}{5}+\frac {2\,B\,b\,c\,d^4}{5}+\frac {8\,A\,b\,c\,d^3\,e}{5}+\frac {A\,c^2\,d^4}{5}\right )+x^6\,\left (\frac {B\,a^2\,e^4}{6}+\frac {4\,B\,a\,b\,d\,e^3}{3}+\frac {A\,a\,b\,e^4}{3}+2\,B\,a\,c\,d^2\,e^2+\frac {4\,A\,a\,c\,d\,e^3}{3}+B\,b^2\,d^2\,e^2+\frac {2\,A\,b^2\,d\,e^3}{3}+\frac {4\,B\,b\,c\,d^3\,e}{3}+2\,A\,b\,c\,d^2\,e^2+\frac {B\,c^2\,d^4}{6}+\frac {2\,A\,c^2\,d^3\,e}{3}\right )+A\,a^2\,d^4\,x+\frac {a\,d^3\,x^2\,\left (4\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {c\,e^3\,x^9\,\left (A\,c\,e+2\,B\,b\,e+4\,B\,c\,d\right )}{9}+\frac {B\,c^2\,e^4\,x^{10}}{10} \]
x^3*((A*b^2*d^4)/3 + (2*A*a*c*d^4)/3 + (2*B*a*b*d^4)/3 + (4*B*a^2*d^3*e)/3 + 2*A*a^2*d^2*e^2 + (8*A*a*b*d^3*e)/3) + x^4*((B*b^2*d^4)/4 + (A*b*c*d^4) /2 + (B*a*c*d^4)/2 + A*a^2*d*e^3 + A*b^2*d^3*e + (3*B*a^2*d^2*e^2)/2 + 2*A *a*c*d^3*e + 2*B*a*b*d^3*e + 3*A*a*b*d^2*e^2) + x^8*((B*b^2*e^4)/8 + (A*b* c*e^4)/4 + (B*a*c*e^4)/4 + (A*c^2*d*e^3)/2 + (3*B*c^2*d^2*e^2)/4 + B*b*c*d *e^3) + x^7*((A*b^2*e^4)/7 + (2*A*a*c*e^4)/7 + (2*B*a*b*e^4)/7 + (4*B*b^2* d*e^3)/7 + (4*B*c^2*d^3*e)/7 + (6*A*c^2*d^2*e^2)/7 + (8*A*b*c*d*e^3)/7 + ( 8*B*a*c*d*e^3)/7 + (12*B*b*c*d^2*e^2)/7) + x^5*((A*a^2*e^4)/5 + (A*c^2*d^4 )/5 + (2*B*b*c*d^4)/5 + (4*B*a^2*d*e^3)/5 + (4*B*b^2*d^3*e)/5 + (6*A*b^2*d ^2*e^2)/5 + (8*A*a*b*d*e^3)/5 + (8*A*b*c*d^3*e)/5 + (8*B*a*c*d^3*e)/5 + (1 2*A*a*c*d^2*e^2)/5 + (12*B*a*b*d^2*e^2)/5) + x^6*((B*a^2*e^4)/6 + (B*c^2*d ^4)/6 + (A*a*b*e^4)/3 + (2*A*b^2*d*e^3)/3 + (2*A*c^2*d^3*e)/3 + B*b^2*d^2* e^2 + (4*A*a*c*d*e^3)/3 + (4*B*a*b*d*e^3)/3 + (4*B*b*c*d^3*e)/3 + 2*A*b*c* d^2*e^2 + 2*B*a*c*d^2*e^2) + A*a^2*d^4*x + (a*d^3*x^2*(4*A*a*e + 2*A*b*d + B*a*d))/2 + (c*e^3*x^9*(A*c*e + 2*B*b*e + 4*B*c*d))/9 + (B*c^2*e^4*x^10)/ 10