Integrand size = 25, antiderivative size = 304 \[ \int (A+B x) (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=-\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}{3 e^6}-\frac {\left (c d^2-b d e+a e^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right ) (d+e x)^4}{4 e^6}-\frac {\left (B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )\right ) (d+e x)^5}{5 e^6}-\frac {\left (2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )\right ) (d+e x)^6}{6 e^6}-\frac {c (5 B c d-2 b B e-A c e) (d+e x)^7}{7 e^6}+\frac {B c^2 (d+e x)^8}{8 e^6} \]
-1/3*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^3/e^6-1/4*(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)^4/e^6-1/5*(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)^5/e^6-1/6*(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)^6/e^6-1/7*c*(-A*c*e-2*B*b*e+5*B*c*d )*(e*x+d)^7/e^6+1/8*B*c^2*(e*x+d)^8/e^6
Time = 0.10 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.99 \[ \int (A+B x) (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=a^2 A d^2 x+\frac {1}{2} a d (a B d+2 A (b d+a e)) x^2+\frac {1}{3} \left (2 a B d (b d+a e)+A \left (b^2 d^2+4 a b d e+a \left (2 c d^2+a e^2\right )\right )\right ) x^3+\frac {1}{4} \left (b^2 d (B d+2 A e)+2 b \left (A c d^2+2 a B d e+a A e^2\right )+a \left (2 B c d^2+4 A c d e+a B e^2\right )\right ) x^4+\frac {1}{5} \left (b^2 e (2 B d+A e)+c \left (A c d^2+4 a B d e+2 a A e^2\right )+2 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) x^5+\frac {1}{6} \left (2 A c e (c d+b e)+B \left (c^2 d^2+b^2 e^2+2 c e (2 b d+a e)\right )\right ) x^6+\frac {1}{7} c e (A c e+2 B (c d+b e)) x^7+\frac {1}{8} B c^2 e^2 x^8 \]
a^2*A*d^2*x + (a*d*(a*B*d + 2*A*(b*d + a*e))*x^2)/2 + ((2*a*B*d*(b*d + a*e ) + A*(b^2*d^2 + 4*a*b*d*e + a*(2*c*d^2 + a*e^2)))*x^3)/3 + ((b^2*d*(B*d + 2*A*e) + 2*b*(A*c*d^2 + 2*a*B*d*e + a*A*e^2) + a*(2*B*c*d^2 + 4*A*c*d*e + a*B*e^2))*x^4)/4 + ((b^2*e*(2*B*d + A*e) + c*(A*c*d^2 + 4*a*B*d*e + 2*a*A *e^2) + 2*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*x^5)/5 + ((2*A*c*e*(c*d + b*e ) + B*(c^2*d^2 + b^2*e^2 + 2*c*e*(2*b*d + a*e)))*x^6)/6 + (c*e*(A*c*e + 2* B*(c*d + b*e))*x^7)/7 + (B*c^2*e^2*x^8)/8
Time = 0.65 (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.
| \(\displaystyle \int (A+B x) (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(d+e x)^5 \left (B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )-2 A c e (2 c d-b e)\right )}{e^5}+\frac {(d+e x)^4 \left (A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )-B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )\right )}{e^5}+\frac {(d+e x)^3 \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{e^5}+\frac {(d+e x)^2 (A e-B d) \left (a e^2-b d e+c d^2\right )^2}{e^5}+\frac {c (d+e x)^6 (A c e+2 b B e-5 B c d)}{e^5}+\frac {B c^2 (d+e x)^7}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {(d+e x)^6 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{6 e^6}-\frac {(d+e x)^5 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{5 e^6}+\frac {(d+e x)^4 \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{4 e^6}-\frac {(d+e x)^3 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6}-\frac {c (d+e x)^7 (-A c e-2 b B e+5 B c d)}{7 e^6}+\frac {B c^2 (d+e x)^8}{8 e^6}\) |
-1/3*((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^3)/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)^4)/(4*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)^5)/(5 *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)^6)/(6*e^6) - (c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^7) /(7*e^6) + (B*c^2*(d + e*x)^8)/(8*e^6)
3.24.21.3.1 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 = 0.37 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(\frac {B \,c^{2} e^{2} x^{8}}{8}+\frac {\left (\left (A \,e^{2}+2 B d e \right ) c^{2}+2 B \,e^{2} b c \right ) x^{7}}{7}+\frac {\left (\left (2 A d e +B \,d^{2}\right ) c^{2}+2 \left (A \,e^{2}+2 B d e \right ) b c +B \,e^{2} \left (2 a c +b^{2}\right )\right ) x^{6}}{6}+\frac {\left (A \,c^{2} d^{2}+2 \left (2 A d e +B \,d^{2}\right ) b c +\left (A \,e^{2}+2 B d e \right ) \left (2 a c +b^{2}\right )+2 B a b \,e^{2}\right ) x^{5}}{5}+\frac {\left (2 A \,d^{2} b c +\left (2 A d e +B \,d^{2}\right ) \left (2 a c +b^{2}\right )+2 \left (A \,e^{2}+2 B d e \right ) b a +a^{2} B \,e^{2}\right ) x^{4}}{4}+\frac {\left (A \,d^{2} \left (2 a c +b^{2}\right )+2 \left (2 A d e +B \,d^{2}\right ) b a +\left (A \,e^{2}+2 B d e \right ) a^{2}\right ) x^{3}}{3}+\frac {\left (2 A a b \,d^{2}+\left (2 A d e +B \,d^{2}\right ) a^{2}\right ) x^{2}}{2}+A \,d^{2} a^{2} x\) | \(293\) |
| norman | \(\frac {B \,c^{2} e^{2} x^{8}}{8}+\left (\frac {1}{7} A \,c^{2} e^{2}+\frac {2}{7} B \,e^{2} b c +\frac {2}{7} B \,c^{2} d e \right ) x^{7}+\left (\frac {1}{3} A b c \,e^{2}+\frac {1}{3} A \,c^{2} d e +\frac {1}{3} B \,e^{2} a c +\frac {1}{6} B \,b^{2} e^{2}+\frac {2}{3} B b c d e +\frac {1}{6} B \,c^{2} d^{2}\right ) x^{6}+\left (\frac {2}{5} A a c \,e^{2}+\frac {1}{5} A \,b^{2} e^{2}+\frac {4}{5} A b c d e +\frac {1}{5} A \,c^{2} d^{2}+\frac {2}{5} B a b \,e^{2}+\frac {4}{5} B a c d e +\frac {2}{5} B \,b^{2} d e +\frac {2}{5} c \,d^{2} B b \right ) x^{5}+\left (\frac {1}{2} A a b \,e^{2}+A a c d e +\frac {1}{2} A \,b^{2} d e +\frac {1}{2} A \,d^{2} b c +\frac {1}{4} a^{2} B \,e^{2}+B a b d e +\frac {1}{2} B a c \,d^{2}+\frac {1}{4} B \,b^{2} d^{2}\right ) x^{4}+\left (\frac {1}{3} A \,a^{2} e^{2}+\frac {4}{3} A a b d e +\frac {2}{3} A \,d^{2} a c +\frac {1}{3} A \,b^{2} d^{2}+\frac {2}{3} B \,a^{2} d e +\frac {2}{3} B a b \,d^{2}\right ) x^{3}+\left (A \,a^{2} d e +A a b \,d^{2}+\frac {1}{2} B \,a^{2} d^{2}\right ) x^{2}+A \,d^{2} a^{2} x\) | \(325\) |
| gosper | \(\frac {1}{8} B \,c^{2} e^{2} x^{8}+\frac {1}{3} A \,b^{2} d^{2} x^{3}+\frac {4}{3} x^{3} A a b d e +x^{4} B a b d e +x^{4} A a c d e +\frac {1}{2} x^{2} B \,a^{2} d^{2}+\frac {1}{4} x^{4} B \,b^{2} d^{2}+\frac {1}{3} x^{3} A \,a^{2} e^{2}+\frac {1}{5} x^{5} A \,c^{2} d^{2}+\frac {1}{4} x^{4} a^{2} B \,e^{2}+\frac {1}{7} x^{7} A \,c^{2} e^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{2}+\frac {1}{5} x^{5} A \,b^{2} e^{2}+\frac {4}{5} x^{5} B a c d e +\frac {2}{3} x^{6} B b c d e +\frac {4}{5} x^{5} A b c d e +A \,d^{2} a^{2} x +\frac {1}{6} B \,b^{2} e^{2} x^{6}+\frac {1}{2} x^{4} A \,d^{2} b c +\frac {1}{2} x^{4} B a c \,d^{2}+\frac {2}{3} x^{3} A \,d^{2} a c +\frac {2}{3} x^{3} B \,a^{2} d e +\frac {2}{3} x^{3} B a b \,d^{2}+x^{2} A \,a^{2} d e +x^{2} A a b \,d^{2}+\frac {1}{2} x^{4} A \,b^{2} d e +\frac {2}{5} x^{5} c \,d^{2} B b +\frac {1}{2} x^{4} A a b \,e^{2}+\frac {1}{3} x^{6} A \,c^{2} d e +\frac {1}{3} x^{6} B \,e^{2} a c +\frac {2}{5} x^{5} A a c \,e^{2}+\frac {2}{5} x^{5} B a b \,e^{2}+\frac {2}{5} x^{5} B \,b^{2} d e +\frac {2}{7} x^{7} B \,c^{2} d e +\frac {1}{3} x^{6} A b c \,e^{2}+\frac {2}{7} x^{7} B \,e^{2} b c\) | \(397\) |
| risch | \(\frac {1}{8} B \,c^{2} e^{2} x^{8}+\frac {1}{3} A \,b^{2} d^{2} x^{3}+\frac {4}{3} x^{3} A a b d e +x^{4} B a b d e +x^{4} A a c d e +\frac {1}{2} x^{2} B \,a^{2} d^{2}+\frac {1}{4} x^{4} B \,b^{2} d^{2}+\frac {1}{3} x^{3} A \,a^{2} e^{2}+\frac {1}{5} x^{5} A \,c^{2} d^{2}+\frac {1}{4} x^{4} a^{2} B \,e^{2}+\frac {1}{7} x^{7} A \,c^{2} e^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{2}+\frac {1}{5} x^{5} A \,b^{2} e^{2}+\frac {4}{5} x^{5} B a c d e +\frac {2}{3} x^{6} B b c d e +\frac {4}{5} x^{5} A b c d e +A \,d^{2} a^{2} x +\frac {1}{6} B \,b^{2} e^{2} x^{6}+\frac {1}{2} x^{4} A \,d^{2} b c +\frac {1}{2} x^{4} B a c \,d^{2}+\frac {2}{3} x^{3} A \,d^{2} a c +\frac {2}{3} x^{3} B \,a^{2} d e +\frac {2}{3} x^{3} B a b \,d^{2}+x^{2} A \,a^{2} d e +x^{2} A a b \,d^{2}+\frac {1}{2} x^{4} A \,b^{2} d e +\frac {2}{5} x^{5} c \,d^{2} B b +\frac {1}{2} x^{4} A a b \,e^{2}+\frac {1}{3} x^{6} A \,c^{2} d e +\frac {1}{3} x^{6} B \,e^{2} a c +\frac {2}{5} x^{5} A a c \,e^{2}+\frac {2}{5} x^{5} B a b \,e^{2}+\frac {2}{5} x^{5} B \,b^{2} d e +\frac {2}{7} x^{7} B \,c^{2} d e +\frac {1}{3} x^{6} A b c \,e^{2}+\frac {2}{7} x^{7} B \,e^{2} b c\) | \(397\) |
| parallelrisch | \(\frac {1}{8} B \,c^{2} e^{2} x^{8}+\frac {1}{3} A \,b^{2} d^{2} x^{3}+\frac {4}{3} x^{3} A a b d e +x^{4} B a b d e +x^{4} A a c d e +\frac {1}{2} x^{2} B \,a^{2} d^{2}+\frac {1}{4} x^{4} B \,b^{2} d^{2}+\frac {1}{3} x^{3} A \,a^{2} e^{2}+\frac {1}{5} x^{5} A \,c^{2} d^{2}+\frac {1}{4} x^{4} a^{2} B \,e^{2}+\frac {1}{7} x^{7} A \,c^{2} e^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{2}+\frac {1}{5} x^{5} A \,b^{2} e^{2}+\frac {4}{5} x^{5} B a c d e +\frac {2}{3} x^{6} B b c d e +\frac {4}{5} x^{5} A b c d e +A \,d^{2} a^{2} x +\frac {1}{6} B \,b^{2} e^{2} x^{6}+\frac {1}{2} x^{4} A \,d^{2} b c +\frac {1}{2} x^{4} B a c \,d^{2}+\frac {2}{3} x^{3} A \,d^{2} a c +\frac {2}{3} x^{3} B \,a^{2} d e +\frac {2}{3} x^{3} B a b \,d^{2}+x^{2} A \,a^{2} d e +x^{2} A a b \,d^{2}+\frac {1}{2} x^{4} A \,b^{2} d e +\frac {2}{5} x^{5} c \,d^{2} B b +\frac {1}{2} x^{4} A a b \,e^{2}+\frac {1}{3} x^{6} A \,c^{2} d e +\frac {1}{3} x^{6} B \,e^{2} a c +\frac {2}{5} x^{5} A a c \,e^{2}+\frac {2}{5} x^{5} B a b \,e^{2}+\frac {2}{5} x^{5} B \,b^{2} d e +\frac {2}{7} x^{7} B \,c^{2} d e +\frac {1}{3} x^{6} A b c \,e^{2}+\frac {2}{7} x^{7} B \,e^{2} b c\) | \(397\) |
1/8*B*c^2*e^2*x^8+1/7*((A*e^2+2*B*d*e)*c^2+2*B*e^2*b*c)*x^7+1/6*((2*A*d*e+ B*d^2)*c^2+2*(A*e^2+2*B*d*e)*b*c+B*e^2*(2*a*c+b^2))*x^6+1/5*(A*c^2*d^2+2*( 2*A*d*e+B*d^2)*b*c+(A*e^2+2*B*d*e)*(2*a*c+b^2)+2*B*a*b*e^2)*x^5+1/4*(2*A*d ^2*b*c+(2*A*d*e+B*d^2)*(2*a*c+b^2)+2*(A*e^2+2*B*d*e)*b*a+a^2*B*e^2)*x^4+1/ 3*(A*d^2*(2*a*c+b^2)+2*(2*A*d*e+B*d^2)*b*a+(A*e^2+2*B*d*e)*a^2)*x^3+1/2*(2 *A*a*b*d^2+(2*A*d*e+B*d^2)*a^2)*x^2+A*d^2*a^2*x
Time = 0.31 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.99 \[ \int (A+B x) (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{8} \, B c^{2} e^{2} x^{8} + \frac {1}{7} \, {\left (2 \, B c^{2} d e + {\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{2} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{2}\right )} x^{6} + A a^{2} d^{2} x + \frac {1}{5} \, {\left ({\left (2 \, B b c + A c^{2}\right )} d^{2} + 2 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left ({\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} + 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{2} e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{2} d e + {\left (B a^{2} + 2 \, A a b\right )} d^{2}\right )} x^{2} \]
1/8*B*c^2*e^2*x^8 + 1/7*(2*B*c^2*d*e + (2*B*b*c + A*c^2)*e^2)*x^7 + 1/6*(B *c^2*d^2 + 2*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*(B*a + A*b)*c)*e^2)*x^6 + A*a^2*d^2*x + 1/5*((2*B*b*c + A*c^2)*d^2 + 2*(B*b^2 + 2*(B*a + A*b)*c)*d*e + (2*B*a*b + A*b^2 + 2*A*a*c)*e^2)*x^5 + 1/4*((B*b^2 + 2*(B*a + A*b)*c)*d ^2 + 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e + (B*a^2 + 2*A*a*b)*e^2)*x^4 + 1/3* (A*a^2*e^2 + (2*B*a*b + A*b^2 + 2*A*a*c)*d^2 + 2*(B*a^2 + 2*A*a*b)*d*e)*x^ 3 + 1/2*(2*A*a^2*d*e + (B*a^2 + 2*A*a*b)*d^2)*x^2
Time = 0.05 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.33 \[ \int (A+B x) (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=A a^{2} d^{2} x + \frac {B c^{2} e^{2} x^{8}}{8} + x^{7} \left (\frac {A c^{2} e^{2}}{7} + \frac {2 B b c e^{2}}{7} + \frac {2 B c^{2} d e}{7}\right ) + x^{6} \left (\frac {A b c e^{2}}{3} + \frac {A c^{2} d e}{3} + \frac {B a c e^{2}}{3} + \frac {B b^{2} e^{2}}{6} + \frac {2 B b c d e}{3} + \frac {B c^{2} d^{2}}{6}\right ) + x^{5} \cdot \left (\frac {2 A a c e^{2}}{5} + \frac {A b^{2} e^{2}}{5} + \frac {4 A b c d e}{5} + \frac {A c^{2} d^{2}}{5} + \frac {2 B a b e^{2}}{5} + \frac {4 B a c d e}{5} + \frac {2 B b^{2} d e}{5} + \frac {2 B b c d^{2}}{5}\right ) + x^{4} \left (\frac {A a b e^{2}}{2} + A a c d e + \frac {A b^{2} d e}{2} + \frac {A b c d^{2}}{2} + \frac {B a^{2} e^{2}}{4} + B a b d e + \frac {B a c d^{2}}{2} + \frac {B b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac {A a^{2} e^{2}}{3} + \frac {4 A a b d e}{3} + \frac {2 A a c d^{2}}{3} + \frac {A b^{2} d^{2}}{3} + \frac {2 B a^{2} d e}{3} + \frac {2 B a b d^{2}}{3}\right ) + x^{2} \left (A a^{2} d e + A a b d^{2} + \frac {B a^{2} d^{2}}{2}\right ) \]
A*a**2*d**2*x + B*c**2*e**2*x**8/8 + x**7*(A*c**2*e**2/7 + 2*B*b*c*e**2/7 + 2*B*c**2*d*e/7) + x**6*(A*b*c*e**2/3 + A*c**2*d*e/3 + B*a*c*e**2/3 + B*b **2*e**2/6 + 2*B*b*c*d*e/3 + B*c**2*d**2/6) + x**5*(2*A*a*c*e**2/5 + A*b** 2*e**2/5 + 4*A*b*c*d*e/5 + A*c**2*d**2/5 + 2*B*a*b*e**2/5 + 4*B*a*c*d*e/5 + 2*B*b**2*d*e/5 + 2*B*b*c*d**2/5) + x**4*(A*a*b*e**2/2 + A*a*c*d*e + A*b* *2*d*e/2 + A*b*c*d**2/2 + B*a**2*e**2/4 + B*a*b*d*e + B*a*c*d**2/2 + B*b** 2*d**2/4) + x**3*(A*a**2*e**2/3 + 4*A*a*b*d*e/3 + 2*A*a*c*d**2/3 + A*b**2* d**2/3 + 2*B*a**2*d*e/3 + 2*B*a*b*d**2/3) + x**2*(A*a**2*d*e + A*a*b*d**2 + B*a**2*d**2/2)
Time = 0.19 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.99 \[ \int (A+B x) (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{8} \, B c^{2} e^{2} x^{8} + \frac {1}{7} \, {\left (2 \, B c^{2} d e + {\left (2 \, B b c + A c^{2}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{2} + 2 \, {\left (2 \, B b c + A c^{2}\right )} d e + {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} e^{2}\right )} x^{6} + A a^{2} d^{2} x + \frac {1}{5} \, {\left ({\left (2 \, B b c + A c^{2}\right )} d^{2} + 2 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d e + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left ({\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} d^{2} + 2 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{2} e^{2} + {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{2} d e + {\left (B a^{2} + 2 \, A a b\right )} d^{2}\right )} x^{2} \]
1/8*B*c^2*e^2*x^8 + 1/7*(2*B*c^2*d*e + (2*B*b*c + A*c^2)*e^2)*x^7 + 1/6*(B *c^2*d^2 + 2*(2*B*b*c + A*c^2)*d*e + (B*b^2 + 2*(B*a + A*b)*c)*e^2)*x^6 + A*a^2*d^2*x + 1/5*((2*B*b*c + A*c^2)*d^2 + 2*(B*b^2 + 2*(B*a + A*b)*c)*d*e + (2*B*a*b + A*b^2 + 2*A*a*c)*e^2)*x^5 + 1/4*((B*b^2 + 2*(B*a + A*b)*c)*d ^2 + 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e + (B*a^2 + 2*A*a*b)*e^2)*x^4 + 1/3* (A*a^2*e^2 + (2*B*a*b + A*b^2 + 2*A*a*c)*d^2 + 2*(B*a^2 + 2*A*a*b)*d*e)*x^ 3 + 1/2*(2*A*a^2*d*e + (B*a^2 + 2*A*a*b)*d^2)*x^2
Time = 0.27 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.30 \[ \int (A+B x) (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=\frac {1}{8} \, B c^{2} e^{2} x^{8} + \frac {2}{7} \, B c^{2} d e x^{7} + \frac {2}{7} \, B b c e^{2} x^{7} + \frac {1}{7} \, A c^{2} e^{2} x^{7} + \frac {1}{6} \, B c^{2} d^{2} x^{6} + \frac {2}{3} \, B b c d e x^{6} + \frac {1}{3} \, A c^{2} d e x^{6} + \frac {1}{6} \, B b^{2} e^{2} x^{6} + \frac {1}{3} \, B a c e^{2} x^{6} + \frac {1}{3} \, A b c e^{2} x^{6} + \frac {2}{5} \, B b c d^{2} x^{5} + \frac {1}{5} \, A c^{2} d^{2} x^{5} + \frac {2}{5} \, B b^{2} d e x^{5} + \frac {4}{5} \, B a c d e x^{5} + \frac {4}{5} \, A b c d e x^{5} + \frac {2}{5} \, B a b e^{2} x^{5} + \frac {1}{5} \, A b^{2} e^{2} x^{5} + \frac {2}{5} \, A a c e^{2} x^{5} + \frac {1}{4} \, B b^{2} d^{2} x^{4} + \frac {1}{2} \, B a c d^{2} x^{4} + \frac {1}{2} \, A b c d^{2} x^{4} + B a b d e x^{4} + \frac {1}{2} \, A b^{2} d e x^{4} + A a c d e x^{4} + \frac {1}{4} \, B a^{2} e^{2} x^{4} + \frac {1}{2} \, A a b e^{2} x^{4} + \frac {2}{3} \, B a b d^{2} x^{3} + \frac {1}{3} \, A b^{2} d^{2} x^{3} + \frac {2}{3} \, A a c d^{2} x^{3} + \frac {2}{3} \, B a^{2} d e x^{3} + \frac {4}{3} \, A a b d e x^{3} + \frac {1}{3} \, A a^{2} e^{2} x^{3} + \frac {1}{2} \, B a^{2} d^{2} x^{2} + A a b d^{2} x^{2} + A a^{2} d e x^{2} + A a^{2} d^{2} x \]
1/8*B*c^2*e^2*x^8 + 2/7*B*c^2*d*e*x^7 + 2/7*B*b*c*e^2*x^7 + 1/7*A*c^2*e^2* x^7 + 1/6*B*c^2*d^2*x^6 + 2/3*B*b*c*d*e*x^6 + 1/3*A*c^2*d*e*x^6 + 1/6*B*b^ 2*e^2*x^6 + 1/3*B*a*c*e^2*x^6 + 1/3*A*b*c*e^2*x^6 + 2/5*B*b*c*d^2*x^5 + 1/ 5*A*c^2*d^2*x^5 + 2/5*B*b^2*d*e*x^5 + 4/5*B*a*c*d*e*x^5 + 4/5*A*b*c*d*e*x^ 5 + 2/5*B*a*b*e^2*x^5 + 1/5*A*b^2*e^2*x^5 + 2/5*A*a*c*e^2*x^5 + 1/4*B*b^2* d^2*x^4 + 1/2*B*a*c*d^2*x^4 + 1/2*A*b*c*d^2*x^4 + B*a*b*d*e*x^4 + 1/2*A*b^ 2*d*e*x^4 + A*a*c*d*e*x^4 + 1/4*B*a^2*e^2*x^4 + 1/2*A*a*b*e^2*x^4 + 2/3*B* a*b*d^2*x^3 + 1/3*A*b^2*d^2*x^3 + 2/3*A*a*c*d^2*x^3 + 2/3*B*a^2*d*e*x^3 + 4/3*A*a*b*d*e*x^3 + 1/3*A*a^2*e^2*x^3 + 1/2*B*a^2*d^2*x^2 + A*a*b*d^2*x^2 + A*a^2*d*e*x^2 + A*a^2*d^2*x
Time = 11.04 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.02 \[ \int (A+B x) (d+e x)^2 \left (a+b x+c x^2\right )^2 \, dx=x^3\,\left (\frac {2\,B\,a^2\,d\,e}{3}+\frac {A\,a^2\,e^2}{3}+\frac {2\,B\,a\,b\,d^2}{3}+\frac {4\,A\,a\,b\,d\,e}{3}+\frac {2\,A\,c\,a\,d^2}{3}+\frac {A\,b^2\,d^2}{3}\right )+x^6\,\left (\frac {B\,b^2\,e^2}{6}+\frac {2\,B\,b\,c\,d\,e}{3}+\frac {A\,b\,c\,e^2}{3}+\frac {B\,c^2\,d^2}{6}+\frac {A\,c^2\,d\,e}{3}+\frac {B\,a\,c\,e^2}{3}\right )+x^4\,\left (\frac {B\,a^2\,e^2}{4}+B\,a\,b\,d\,e+\frac {A\,a\,b\,e^2}{2}+\frac {B\,c\,a\,d^2}{2}+A\,c\,a\,d\,e+\frac {B\,b^2\,d^2}{4}+\frac {A\,b^2\,d\,e}{2}+\frac {A\,c\,b\,d^2}{2}\right )+x^5\,\left (\frac {2\,B\,b^2\,d\,e}{5}+\frac {A\,b^2\,e^2}{5}+\frac {2\,B\,b\,c\,d^2}{5}+\frac {4\,A\,b\,c\,d\,e}{5}+\frac {2\,B\,a\,b\,e^2}{5}+\frac {A\,c^2\,d^2}{5}+\frac {4\,B\,a\,c\,d\,e}{5}+\frac {2\,A\,a\,c\,e^2}{5}\right )+\frac {a\,d\,x^2\,\left (2\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {c\,e\,x^7\,\left (A\,c\,e+2\,B\,b\,e+2\,B\,c\,d\right )}{7}+A\,a^2\,d^2\,x+\frac {B\,c^2\,e^2\,x^8}{8} \]
x^3*((A*a^2*e^2)/3 + (A*b^2*d^2)/3 + (2*A*a*c*d^2)/3 + (2*B*a*b*d^2)/3 + ( 2*B*a^2*d*e)/3 + (4*A*a*b*d*e)/3) + x^6*((B*b^2*e^2)/6 + (B*c^2*d^2)/6 + ( A*b*c*e^2)/3 + (B*a*c*e^2)/3 + (A*c^2*d*e)/3 + (2*B*b*c*d*e)/3) + x^4*((B* a^2*e^2)/4 + (B*b^2*d^2)/4 + (A*a*b*e^2)/2 + (A*b*c*d^2)/2 + (B*a*c*d^2)/2 + (A*b^2*d*e)/2 + A*a*c*d*e + B*a*b*d*e) + x^5*((A*b^2*e^2)/5 + (A*c^2*d^ 2)/5 + (2*A*a*c*e^2)/5 + (2*B*a*b*e^2)/5 + (2*B*b*c*d^2)/5 + (2*B*b^2*d*e) /5 + (4*A*b*c*d*e)/5 + (4*B*a*c*d*e)/5) + (a*d*x^2*(2*A*a*e + 2*A*b*d + B* a*d))/2 + (c*e*x^7*(A*c*e + 2*B*b*e + 2*B*c*d))/7 + A*a^2*d^2*x + (B*c^2*e ^2*x^8)/8