Integrand size = 27, antiderivative size = 217 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \left (A+B x+C x^2\right ) \, dx=a^2 A d^2 x+\frac {1}{3} a \left (a d (C d+2 B e)+A \left (2 c d^2+a e^2\right )\right ) x^3+\frac {1}{4} a^2 e (2 C d+B e) x^4+\frac {1}{5} \left (A c \left (c d^2+2 a e^2\right )+a \left (a C e^2+2 c d (C d+2 B e)\right )\right ) x^5+\frac {1}{3} a c e (2 C d+B e) x^6+\frac {1}{7} c \left (2 a C e^2+c \left (C d^2+e (2 B d+A e)\right )\right ) x^7+\frac {1}{8} c^2 e (2 C d+B e) x^8+\frac {1}{9} c^2 C e^2 x^9+\frac {d (B d+2 A e) \left (a+c x^2\right )^3}{6 c} \]
a^2*A*d^2*x+1/3*a*(a*d*(2*B*e+C*d)+A*(a*e^2+2*c*d^2))*x^3+1/4*a^2*e*(B*e+2 *C*d)*x^4+1/5*(A*c*(2*a*e^2+c*d^2)+a*(a*C*e^2+2*c*d*(2*B*e+C*d)))*x^5+1/3* a*c*e*(B*e+2*C*d)*x^6+1/7*c*(2*a*C*e^2+c*(C*d^2+e*(A*e+2*B*d)))*x^7+1/8*c^ 2*e*(B*e+2*C*d)*x^8+1/9*c^2*C*e^2*x^9+1/6*d*(2*A*e+B*d)*(c*x^2+a)^3/c
Time = 0.06 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.11 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \left (A+B x+C x^2\right ) \, dx=a^2 A d^2 x+\frac {1}{2} a^2 d (B d+2 A e) x^2+\frac {1}{3} a \left (a d (C d+2 B e)+A \left (2 c d^2+a e^2\right )\right ) x^3+\frac {1}{4} a \left (2 B c d^2+4 A c d e+2 a C d e+a B e^2\right ) x^4+\frac {1}{5} \left (A c \left (c d^2+2 a e^2\right )+a \left (a C e^2+2 c d (C d+2 B e)\right )\right ) x^5+\frac {1}{6} c \left (B c d^2+2 A c d e+4 a C d e+2 a B e^2\right ) x^6+\frac {1}{7} c \left (c C d^2+2 a C e^2+c e (2 B d+A e)\right ) x^7+\frac {1}{8} c^2 e (2 C d+B e) x^8+\frac {1}{9} c^2 C e^2 x^9 \]
a^2*A*d^2*x + (a^2*d*(B*d + 2*A*e)*x^2)/2 + (a*(a*d*(C*d + 2*B*e) + A*(2*c *d^2 + a*e^2))*x^3)/3 + (a*(2*B*c*d^2 + 4*A*c*d*e + 2*a*C*d*e + a*B*e^2)*x ^4)/4 + ((A*c*(c*d^2 + 2*a*e^2) + a*(a*C*e^2 + 2*c*d*(C*d + 2*B*e)))*x^5)/ 5 + (c*(B*c*d^2 + 2*A*c*d*e + 4*a*C*d*e + 2*a*B*e^2)*x^6)/6 + (c*(c*C*d^2 + 2*a*C*e^2 + c*e*(2*B*d + A*e))*x^7)/7 + (c^2*e*(2*C*d + B*e)*x^8)/8 + (c ^2*C*e^2*x^9)/9
Time = 0.58 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2017, 2341, 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.
| \(\displaystyle \int \left (a+c x^2\right )^2 (d+e x)^2 \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2017 |
| \(\displaystyle \int \left (c x^2+a\right )^2 \left ((d+e x)^2 \left (C x^2+B x+A\right )-\left (B d^2+2 A e d\right ) x\right )dx+\frac {d \left (a+c x^2\right )^3 (2 A e+B d)}{6 c}\) |
| \(\Big \downarrow \) 2341 |
| \(\displaystyle \int \left (c^2 C e^2 x^8+c^2 e (2 C d+B e) x^7+c \left (c C d^2+2 a C e^2+c e (2 B d+A e)\right ) x^6+2 a c e (2 C d+B e) x^5+\left (A c \left (c d^2+2 a e^2\right )+a \left (a C e^2+2 c d (C d+2 B e)\right )\right ) x^4+a^2 e (2 C d+B e) x^3+a \left (a d (C d+2 B e)+A \left (2 c d^2+a e^2\right )\right ) x^2+a^2 A d^2\right )dx+\frac {d \left (a+c x^2\right )^3 (2 A e+B d)}{6 c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 A d^2 x+\frac {1}{4} a^2 e x^4 (B e+2 C d)+\frac {1}{7} c x^7 \left (2 a C e^2+c e (A e+2 B d)+c C d^2\right )+\frac {1}{5} x^5 \left (A c \left (2 a e^2+c d^2\right )+a \left (a C e^2+2 c d (2 B e+C d)\right )\right )+\frac {1}{3} a x^3 \left (A \left (a e^2+2 c d^2\right )+a d (2 B e+C d)\right )+\frac {d \left (a+c x^2\right )^3 (2 A e+B d)}{6 c}+\frac {1}{3} a c e x^6 (B e+2 C d)+\frac {1}{8} c^2 e x^8 (B e+2 C d)+\frac {1}{9} c^2 C e^2 x^9\) |
a^2*A*d^2*x + (a*(a*d*(C*d + 2*B*e) + A*(2*c*d^2 + a*e^2))*x^3)/3 + (a^2*e *(2*C*d + B*e)*x^4)/4 + ((A*c*(c*d^2 + 2*a*e^2) + a*(a*C*e^2 + 2*c*d*(C*d + 2*B*e)))*x^5)/5 + (a*c*e*(2*C*d + B*e)*x^6)/3 + (c*(c*C*d^2 + 2*a*C*e^2 + c*e*(2*B*d + A*e))*x^7)/7 + (c^2*e*(2*C*d + B*e)*x^8)/8 + (c^2*C*e^2*x^9 )/9 + (d*(B*d + 2*A*e)*(a + c*x^2)^3)/(6*c)
3.1.26.3.1 Defintions of rubi rules used
Int[(Px_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Coeff[Px, x, n - 1]*((a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] + Int[(Px - Coeff[Px, x, n - 1] *x^(n - 1))*(a + b*x^n)^p, x] /; FreeQ[{a, b}, x] && PolyQ[Px, x] && IGtQ[p , 1] && IGtQ[n, 1] && NeQ[Coeff[Px, x, n - 1], 0] && NeQ[Px, Coeff[Px, x, n - 1]*x^(n - 1)] && !MatchQ[Px, (Qx_.)*((c_) + (d_.)*x^(m_))^(q_) /; FreeQ [{c, d}, x] && PolyQ[Qx, x] && IGtQ[q, 1] && IGtQ[m, 1] && NeQ[Coeff[Qx*(a + b*x^n)^p, x, m - 1], 0] && GtQ[m*q, n*p]]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq* (a + b*x^2)^p, x], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.48 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.21
| method | result | size |
| norman | \(\frac {c^{2} C \,e^{2} x^{9}}{9}+\left (\frac {1}{8} B \,c^{2} e^{2}+\frac {1}{4} c^{2} d e C \right ) x^{8}+\left (\frac {1}{7} A \,c^{2} e^{2}+\frac {2}{7} B \,c^{2} d e +\frac {2}{7} C a c \,e^{2}+\frac {1}{7} C \,c^{2} d^{2}\right ) x^{7}+\left (\frac {1}{3} A \,c^{2} d e +\frac {1}{3} B \,e^{2} a c +\frac {1}{6} B \,c^{2} d^{2}+\frac {2}{3} a c d e C \right ) x^{6}+\left (\frac {2}{5} A a c \,e^{2}+\frac {1}{5} A \,c^{2} d^{2}+\frac {4}{5} B a c d e +\frac {1}{5} a^{2} C \,e^{2}+\frac {2}{5} C a c \,d^{2}\right ) x^{5}+\left (A a c d e +\frac {1}{4} a^{2} B \,e^{2}+\frac {1}{2} B a c \,d^{2}+\frac {1}{2} d e \,a^{2} C \right ) x^{4}+\left (\frac {1}{3} A \,a^{2} e^{2}+\frac {2}{3} A \,d^{2} a c +\frac {2}{3} B \,a^{2} d e +\frac {1}{3} a^{2} d^{2} C \right ) x^{3}+\left (d e \,a^{2} A +\frac {1}{2} a^{2} d^{2} B \right ) x^{2}+A \,d^{2} a^{2} x\) | \(263\) |
| default | \(\frac {c^{2} C \,e^{2} x^{9}}{9}+\frac {\left (B \,c^{2} e^{2}+2 c^{2} d e C \right ) x^{8}}{8}+\frac {\left (\left (2 a c \,e^{2}+c^{2} d^{2}\right ) C +2 B \,c^{2} d e +A \,c^{2} e^{2}\right ) x^{7}}{7}+\frac {\left (4 a c d e C +\left (2 a c \,e^{2}+c^{2} d^{2}\right ) B +2 A \,c^{2} d e \right ) x^{6}}{6}+\frac {\left (\left (e^{2} a^{2}+2 c \,d^{2} a \right ) C +4 B a c d e +\left (2 a c \,e^{2}+c^{2} d^{2}\right ) A \right ) x^{5}}{5}+\frac {\left (2 d e \,a^{2} C +\left (e^{2} a^{2}+2 c \,d^{2} a \right ) B +4 A a c d e \right ) x^{4}}{4}+\frac {\left (a^{2} d^{2} C +2 B \,a^{2} d e +\left (e^{2} a^{2}+2 c \,d^{2} a \right ) A \right ) x^{3}}{3}+\frac {\left (2 d e \,a^{2} A +a^{2} d^{2} B \right ) x^{2}}{2}+A \,d^{2} a^{2} x\) | \(268\) |
| gosper | \(\frac {2}{3} x^{6} a c d e C +\frac {4}{5} x^{5} B a c d e +x^{4} A a c d e +\frac {1}{2} x^{4} B a c \,d^{2}+\frac {1}{8} B \,c^{2} e^{2} x^{8}+\frac {2}{3} x^{3} A \,d^{2} a c +\frac {2}{3} x^{3} B \,a^{2} d e +x^{2} d e \,a^{2} A +\frac {1}{2} x^{4} d e \,a^{2} C +\frac {1}{3} x^{3} a^{2} d^{2} C +\frac {1}{2} x^{2} a^{2} d^{2} B +\frac {1}{4} x^{4} a^{2} B \,e^{2}+\frac {1}{3} x^{3} A \,a^{2} e^{2}+\frac {1}{5} x^{5} A \,c^{2} d^{2}+\frac {1}{5} x^{5} a^{2} C \,e^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{2}+\frac {1}{7} x^{7} A \,c^{2} e^{2}+\frac {1}{7} x^{7} C \,c^{2} d^{2}+A \,d^{2} a^{2} x +\frac {2}{7} x^{7} C a c \,e^{2}+\frac {1}{3} x^{6} A \,c^{2} d e +\frac {1}{3} x^{6} B \,e^{2} a c +\frac {2}{5} x^{5} A a c \,e^{2}+\frac {2}{5} x^{5} C a c \,d^{2}+\frac {1}{4} x^{8} c^{2} d e C +\frac {2}{7} x^{7} B \,c^{2} d e +\frac {1}{9} c^{2} C \,e^{2} x^{9}\) | \(303\) |
| risch | \(\frac {2}{3} x^{6} a c d e C +\frac {4}{5} x^{5} B a c d e +x^{4} A a c d e +\frac {1}{2} x^{4} B a c \,d^{2}+\frac {1}{8} B \,c^{2} e^{2} x^{8}+\frac {2}{3} x^{3} A \,d^{2} a c +\frac {2}{3} x^{3} B \,a^{2} d e +x^{2} d e \,a^{2} A +\frac {1}{2} x^{4} d e \,a^{2} C +\frac {1}{3} x^{3} a^{2} d^{2} C +\frac {1}{2} x^{2} a^{2} d^{2} B +\frac {1}{4} x^{4} a^{2} B \,e^{2}+\frac {1}{3} x^{3} A \,a^{2} e^{2}+\frac {1}{5} x^{5} A \,c^{2} d^{2}+\frac {1}{5} x^{5} a^{2} C \,e^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{2}+\frac {1}{7} x^{7} A \,c^{2} e^{2}+\frac {1}{7} x^{7} C \,c^{2} d^{2}+A \,d^{2} a^{2} x +\frac {2}{7} x^{7} C a c \,e^{2}+\frac {1}{3} x^{6} A \,c^{2} d e +\frac {1}{3} x^{6} B \,e^{2} a c +\frac {2}{5} x^{5} A a c \,e^{2}+\frac {2}{5} x^{5} C a c \,d^{2}+\frac {1}{4} x^{8} c^{2} d e C +\frac {2}{7} x^{7} B \,c^{2} d e +\frac {1}{9} c^{2} C \,e^{2} x^{9}\) | \(303\) |
| parallelrisch | \(\frac {2}{3} x^{6} a c d e C +\frac {4}{5} x^{5} B a c d e +x^{4} A a c d e +\frac {1}{2} x^{4} B a c \,d^{2}+\frac {1}{8} B \,c^{2} e^{2} x^{8}+\frac {2}{3} x^{3} A \,d^{2} a c +\frac {2}{3} x^{3} B \,a^{2} d e +x^{2} d e \,a^{2} A +\frac {1}{2} x^{4} d e \,a^{2} C +\frac {1}{3} x^{3} a^{2} d^{2} C +\frac {1}{2} x^{2} a^{2} d^{2} B +\frac {1}{4} x^{4} a^{2} B \,e^{2}+\frac {1}{3} x^{3} A \,a^{2} e^{2}+\frac {1}{5} x^{5} A \,c^{2} d^{2}+\frac {1}{5} x^{5} a^{2} C \,e^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{2}+\frac {1}{7} x^{7} A \,c^{2} e^{2}+\frac {1}{7} x^{7} C \,c^{2} d^{2}+A \,d^{2} a^{2} x +\frac {2}{7} x^{7} C a c \,e^{2}+\frac {1}{3} x^{6} A \,c^{2} d e +\frac {1}{3} x^{6} B \,e^{2} a c +\frac {2}{5} x^{5} A a c \,e^{2}+\frac {2}{5} x^{5} C a c \,d^{2}+\frac {1}{4} x^{8} c^{2} d e C +\frac {2}{7} x^{7} B \,c^{2} d e +\frac {1}{9} c^{2} C \,e^{2} x^{9}\) | \(303\) |
1/9*c^2*C*e^2*x^9+(1/8*B*c^2*e^2+1/4*c^2*d*e*C)*x^8+(1/7*A*c^2*e^2+2/7*B*c ^2*d*e+2/7*C*a*c*e^2+1/7*C*c^2*d^2)*x^7+(1/3*A*c^2*d*e+1/3*B*e^2*a*c+1/6*B *c^2*d^2+2/3*a*c*d*e*C)*x^6+(2/5*A*a*c*e^2+1/5*A*c^2*d^2+4/5*B*a*c*d*e+1/5 *a^2*C*e^2+2/5*C*a*c*d^2)*x^5+(A*a*c*d*e+1/4*a^2*B*e^2+1/2*B*a*c*d^2+1/2*d *e*a^2*C)*x^4+(1/3*A*a^2*e^2+2/3*A*d^2*a*c+2/3*B*a^2*d*e+1/3*a^2*d^2*C)*x^ 3+(d*e*a^2*A+1/2*a^2*d^2*B)*x^2+A*d^2*a^2*x
Time = 0.27 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.18 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \left (A+B x+C x^2\right ) \, dx=\frac {1}{9} \, C c^{2} e^{2} x^{9} + \frac {1}{8} \, {\left (2 \, C c^{2} d e + B c^{2} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (C c^{2} d^{2} + 2 \, B c^{2} d e + {\left (2 \, C a c + A c^{2}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{2} + 2 \, B a c e^{2} + 2 \, {\left (2 \, C a c + A c^{2}\right )} d e\right )} x^{6} + A a^{2} d^{2} x + \frac {1}{5} \, {\left (4 \, B a c d e + {\left (2 \, C a c + A c^{2}\right )} d^{2} + {\left (C a^{2} + 2 \, A a c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, B a c d^{2} + B a^{2} e^{2} + 2 \, {\left (C a^{2} + 2 \, A a c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (2 \, B a^{2} d e + A a^{2} e^{2} + {\left (C a^{2} + 2 \, A a c\right )} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} d^{2} + 2 \, A a^{2} d e\right )} x^{2} \]
1/9*C*c^2*e^2*x^9 + 1/8*(2*C*c^2*d*e + B*c^2*e^2)*x^8 + 1/7*(C*c^2*d^2 + 2 *B*c^2*d*e + (2*C*a*c + A*c^2)*e^2)*x^7 + 1/6*(B*c^2*d^2 + 2*B*a*c*e^2 + 2 *(2*C*a*c + A*c^2)*d*e)*x^6 + A*a^2*d^2*x + 1/5*(4*B*a*c*d*e + (2*C*a*c + A*c^2)*d^2 + (C*a^2 + 2*A*a*c)*e^2)*x^5 + 1/4*(2*B*a*c*d^2 + B*a^2*e^2 + 2 *(C*a^2 + 2*A*a*c)*d*e)*x^4 + 1/3*(2*B*a^2*d*e + A*a^2*e^2 + (C*a^2 + 2*A* a*c)*d^2)*x^3 + 1/2*(B*a^2*d^2 + 2*A*a^2*d*e)*x^2
Time = 0.03 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.43 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \left (A+B x+C x^2\right ) \, dx=A a^{2} d^{2} x + \frac {C c^{2} e^{2} x^{9}}{9} + x^{8} \left (\frac {B c^{2} e^{2}}{8} + \frac {C c^{2} d e}{4}\right ) + x^{7} \left (\frac {A c^{2} e^{2}}{7} + \frac {2 B c^{2} d e}{7} + \frac {2 C a c e^{2}}{7} + \frac {C c^{2} d^{2}}{7}\right ) + x^{6} \left (\frac {A c^{2} d e}{3} + \frac {B a c e^{2}}{3} + \frac {B c^{2} d^{2}}{6} + \frac {2 C a c d e}{3}\right ) + x^{5} \cdot \left (\frac {2 A a c e^{2}}{5} + \frac {A c^{2} d^{2}}{5} + \frac {4 B a c d e}{5} + \frac {C a^{2} e^{2}}{5} + \frac {2 C a c d^{2}}{5}\right ) + x^{4} \left (A a c d e + \frac {B a^{2} e^{2}}{4} + \frac {B a c d^{2}}{2} + \frac {C a^{2} d e}{2}\right ) + x^{3} \left (\frac {A a^{2} e^{2}}{3} + \frac {2 A a c d^{2}}{3} + \frac {2 B a^{2} d e}{3} + \frac {C a^{2} d^{2}}{3}\right ) + x^{2} \left (A a^{2} d e + \frac {B a^{2} d^{2}}{2}\right ) \]
A*a**2*d**2*x + C*c**2*e**2*x**9/9 + x**8*(B*c**2*e**2/8 + C*c**2*d*e/4) + x**7*(A*c**2*e**2/7 + 2*B*c**2*d*e/7 + 2*C*a*c*e**2/7 + C*c**2*d**2/7) + x**6*(A*c**2*d*e/3 + B*a*c*e**2/3 + B*c**2*d**2/6 + 2*C*a*c*d*e/3) + x**5* (2*A*a*c*e**2/5 + A*c**2*d**2/5 + 4*B*a*c*d*e/5 + C*a**2*e**2/5 + 2*C*a*c* d**2/5) + x**4*(A*a*c*d*e + B*a**2*e**2/4 + B*a*c*d**2/2 + C*a**2*d*e/2) + x**3*(A*a**2*e**2/3 + 2*A*a*c*d**2/3 + 2*B*a**2*d*e/3 + C*a**2*d**2/3) + x**2*(A*a**2*d*e + B*a**2*d**2/2)
Time = 0.19 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.18 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \left (A+B x+C x^2\right ) \, dx=\frac {1}{9} \, C c^{2} e^{2} x^{9} + \frac {1}{8} \, {\left (2 \, C c^{2} d e + B c^{2} e^{2}\right )} x^{8} + \frac {1}{7} \, {\left (C c^{2} d^{2} + 2 \, B c^{2} d e + {\left (2 \, C a c + A c^{2}\right )} e^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B c^{2} d^{2} + 2 \, B a c e^{2} + 2 \, {\left (2 \, C a c + A c^{2}\right )} d e\right )} x^{6} + A a^{2} d^{2} x + \frac {1}{5} \, {\left (4 \, B a c d e + {\left (2 \, C a c + A c^{2}\right )} d^{2} + {\left (C a^{2} + 2 \, A a c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, B a c d^{2} + B a^{2} e^{2} + 2 \, {\left (C a^{2} + 2 \, A a c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (2 \, B a^{2} d e + A a^{2} e^{2} + {\left (C a^{2} + 2 \, A a c\right )} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} d^{2} + 2 \, A a^{2} d e\right )} x^{2} \]
1/9*C*c^2*e^2*x^9 + 1/8*(2*C*c^2*d*e + B*c^2*e^2)*x^8 + 1/7*(C*c^2*d^2 + 2 *B*c^2*d*e + (2*C*a*c + A*c^2)*e^2)*x^7 + 1/6*(B*c^2*d^2 + 2*B*a*c*e^2 + 2 *(2*C*a*c + A*c^2)*d*e)*x^6 + A*a^2*d^2*x + 1/5*(4*B*a*c*d*e + (2*C*a*c + A*c^2)*d^2 + (C*a^2 + 2*A*a*c)*e^2)*x^5 + 1/4*(2*B*a*c*d^2 + B*a^2*e^2 + 2 *(C*a^2 + 2*A*a*c)*d*e)*x^4 + 1/3*(2*B*a^2*d*e + A*a^2*e^2 + (C*a^2 + 2*A* a*c)*d^2)*x^3 + 1/2*(B*a^2*d^2 + 2*A*a^2*d*e)*x^2
Time = 0.27 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.39 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \left (A+B x+C x^2\right ) \, dx=\frac {1}{9} \, C c^{2} e^{2} x^{9} + \frac {1}{4} \, C c^{2} d e x^{8} + \frac {1}{8} \, B c^{2} e^{2} x^{8} + \frac {1}{7} \, C c^{2} d^{2} x^{7} + \frac {2}{7} \, B c^{2} d e x^{7} + \frac {2}{7} \, C a c e^{2} x^{7} + \frac {1}{7} \, A c^{2} e^{2} x^{7} + \frac {1}{6} \, B c^{2} d^{2} x^{6} + \frac {2}{3} \, C a c d e x^{6} + \frac {1}{3} \, A c^{2} d e x^{6} + \frac {1}{3} \, B a c e^{2} x^{6} + \frac {2}{5} \, C a c d^{2} x^{5} + \frac {1}{5} \, A c^{2} d^{2} x^{5} + \frac {4}{5} \, B a c d e x^{5} + \frac {1}{5} \, C a^{2} e^{2} x^{5} + \frac {2}{5} \, A a c e^{2} x^{5} + \frac {1}{2} \, B a c d^{2} x^{4} + \frac {1}{2} \, C a^{2} d e x^{4} + A a c d e x^{4} + \frac {1}{4} \, B a^{2} e^{2} x^{4} + \frac {1}{3} \, C a^{2} d^{2} x^{3} + \frac {2}{3} \, A a c d^{2} x^{3} + \frac {2}{3} \, B a^{2} d e x^{3} + \frac {1}{3} \, A a^{2} e^{2} x^{3} + \frac {1}{2} \, B a^{2} d^{2} x^{2} + A a^{2} d e x^{2} + A a^{2} d^{2} x \]
1/9*C*c^2*e^2*x^9 + 1/4*C*c^2*d*e*x^8 + 1/8*B*c^2*e^2*x^8 + 1/7*C*c^2*d^2* x^7 + 2/7*B*c^2*d*e*x^7 + 2/7*C*a*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*C*a*c*d*e*x^6 + 1/3*A*c^2*d*e*x^6 + 1/3*B*a*c*e^2*x^6 + 2/ 5*C*a*c*d^2*x^5 + 1/5*A*c^2*d^2*x^5 + 4/5*B*a*c*d*e*x^5 + 1/5*C*a^2*e^2*x^ 5 + 2/5*A*a*c*e^2*x^5 + 1/2*B*a*c*d^2*x^4 + 1/2*C*a^2*d*e*x^4 + A*a*c*d*e* x^4 + 1/4*B*a^2*e^2*x^4 + 1/3*C*a^2*d^2*x^3 + 2/3*A*a*c*d^2*x^3 + 2/3*B*a^ 2*d*e*x^3 + 1/3*A*a^2*e^2*x^3 + 1/2*B*a^2*d^2*x^2 + A*a^2*d*e*x^2 + A*a^2* d^2*x
Time = 0.10 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.12 \[ \int (d+e x)^2 \left (a+c x^2\right )^2 \left (A+B x+C x^2\right ) \, dx=x^3\,\left (\frac {C\,a^2\,d^2}{3}+\frac {2\,B\,a^2\,d\,e}{3}+\frac {A\,a^2\,e^2}{3}+\frac {2\,A\,c\,a\,d^2}{3}\right )+x^7\,\left (\frac {C\,c^2\,d^2}{7}+\frac {2\,B\,c^2\,d\,e}{7}+\frac {A\,c^2\,e^2}{7}+\frac {2\,C\,a\,c\,e^2}{7}\right )+x^5\,\left (\frac {C\,a^2\,e^2}{5}+\frac {2\,C\,a\,c\,d^2}{5}+\frac {4\,B\,a\,c\,d\,e}{5}+\frac {2\,A\,a\,c\,e^2}{5}+\frac {A\,c^2\,d^2}{5}\right )+\frac {a\,x^4\,\left (B\,a\,e^2+2\,B\,c\,d^2+4\,A\,c\,d\,e+2\,C\,a\,d\,e\right )}{4}+\frac {c\,x^6\,\left (2\,B\,a\,e^2+B\,c\,d^2+2\,A\,c\,d\,e+4\,C\,a\,d\,e\right )}{6}+\frac {C\,c^2\,e^2\,x^9}{9}+A\,a^2\,d^2\,x+\frac {a^2\,d\,x^2\,\left (2\,A\,e+B\,d\right )}{2}+\frac {c^2\,e\,x^8\,\left (B\,e+2\,C\,d\right )}{8} \]
x^3*((A*a^2*e^2)/3 + (C*a^2*d^2)/3 + (2*A*a*c*d^2)/3 + (2*B*a^2*d*e)/3) + x^7*((A*c^2*e^2)/7 + (C*c^2*d^2)/7 + (2*C*a*c*e^2)/7 + (2*B*c^2*d*e)/7) + x^5*((A*c^2*d^2)/5 + (C*a^2*e^2)/5 + (2*A*a*c*e^2)/5 + (2*C*a*c*d^2)/5 + ( 4*B*a*c*d*e)/5) + (a*x^4*(B*a*e^2 + 2*B*c*d^2 + 4*A*c*d*e + 2*C*a*d*e))/4 + (c*x^6*(2*B*a*e^2 + B*c*d^2 + 2*A*c*d*e + 4*C*a*d*e))/6 + (C*c^2*e^2*x^9 )/9 + A*a^2*d^2*x + (a^2*d*x^2*(2*A*e + B*d))/2 + (c^2*e*x^8*(B*e + 2*C*d) )/8