Integrand size = 27, antiderivative size = 289 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \left (A+B x+C x^2\right ) \, dx=a^3 A d^2 x+\frac {1}{3} a^2 \left (a d (C d+2 B e)+A \left (3 c d^2+a e^2\right )\right ) x^3+\frac {1}{4} a^3 e (2 C d+B e) x^4+\frac {1}{5} a \left (3 A c \left (c d^2+a e^2\right )+a \left (a C e^2+3 c d (C d+2 B e)\right )\right ) x^5+\frac {1}{2} a^2 c e (2 C d+B e) x^6+\frac {1}{7} c \left (A c \left (c d^2+3 a e^2\right )+3 a \left (a C e^2+c d (C d+2 B e)\right )\right ) x^7+\frac {3}{8} a c^2 e (2 C d+B e) x^8+\frac {1}{9} c^2 \left (3 a C e^2+c \left (C d^2+e (2 B d+A e)\right )\right ) x^9+\frac {1}{10} c^3 e (2 C d+B e) x^{10}+\frac {1}{11} c^3 C e^2 x^{11}+\frac {d (B d+2 A e) \left (a+c x^2\right )^4}{8 c} \]
a^3*A*d^2*x+1/3*a^2*(a*d*(2*B*e+C*d)+A*(a*e^2+3*c*d^2))*x^3+1/4*a^3*e*(B*e +2*C*d)*x^4+1/5*a*(3*A*c*(a*e^2+c*d^2)+a*(a*C*e^2+3*c*d*(2*B*e+C*d)))*x^5+ 1/2*a^2*c*e*(B*e+2*C*d)*x^6+1/7*c*(A*c*(3*a*e^2+c*d^2)+3*a*(a*C*e^2+c*d*(2 *B*e+C*d)))*x^7+3/8*a*c^2*e*(B*e+2*C*d)*x^8+1/9*c^2*(3*a*C*e^2+c*(C*d^2+e* (A*e+2*B*d)))*x^9+1/10*c^3*e*(B*e+2*C*d)*x^10+1/11*c^3*C*e^2*x^11+1/8*d*(2 *A*e+B*d)*(c*x^2+a)^4/c
Time = 0.08 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.14 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \left (A+B x+C x^2\right ) \, dx=a^3 A d^2 x+\frac {1}{2} a^3 d (B d+2 A e) x^2+\frac {1}{3} a^2 \left (a d (C d+2 B e)+A \left (3 c d^2+a e^2\right )\right ) x^3+\frac {1}{4} a^2 \left (3 B c d^2+6 A c d e+2 a C d e+a B e^2\right ) x^4+\frac {1}{5} a \left (3 A c \left (c d^2+a e^2\right )+a \left (a C e^2+3 c d (C d+2 B e)\right )\right ) x^5+\frac {1}{2} a c \left (2 (A c+a C) d e+B \left (c d^2+a e^2\right )\right ) x^6+\frac {1}{7} c \left (A c \left (c d^2+3 a e^2\right )+3 a \left (a C e^2+c d (C d+2 B e)\right )\right ) x^7+\frac {1}{8} c^2 \left (B c d^2+2 A c d e+6 a C d e+3 a B e^2\right ) x^8+\frac {1}{9} c^2 \left (c C d^2+3 a C e^2+c e (2 B d+A e)\right ) x^9+\frac {1}{10} c^3 e (2 C d+B e) x^{10}+\frac {1}{11} c^3 C e^2 x^{11} \]
a^3*A*d^2*x + (a^3*d*(B*d + 2*A*e)*x^2)/2 + (a^2*(a*d*(C*d + 2*B*e) + A*(3 *c*d^2 + a*e^2))*x^3)/3 + (a^2*(3*B*c*d^2 + 6*A*c*d*e + 2*a*C*d*e + a*B*e^ 2)*x^4)/4 + (a*(3*A*c*(c*d^2 + a*e^2) + a*(a*C*e^2 + 3*c*d*(C*d + 2*B*e))) *x^5)/5 + (a*c*(2*(A*c + a*C)*d*e + B*(c*d^2 + a*e^2))*x^6)/2 + (c*(A*c*(c *d^2 + 3*a*e^2) + 3*a*(a*C*e^2 + c*d*(C*d + 2*B*e)))*x^7)/7 + (c^2*(B*c*d^ 2 + 2*A*c*d*e + 6*a*C*d*e + 3*a*B*e^2)*x^8)/8 + (c^2*(c*C*d^2 + 3*a*C*e^2 + c*e*(2*B*d + A*e))*x^9)/9 + (c^3*e*(2*C*d + B*e)*x^10)/10 + (c^3*C*e^2*x ^11)/11
Time = 0.71 (sec) , antiderivative size = 288, 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.
| \(\displaystyle \int \left (a+c x^2\right )^3 (d+e x)^2 \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2017 |
| \(\displaystyle \int \left (c x^2+a\right )^3 \left ((d+e x)^2 \left (C x^2+B x+A\right )-\left (B d^2+2 A e d\right ) x\right )dx+\frac {d \left (a+c x^2\right )^4 (2 A e+B d)}{8 c}\) |
| \(\Big \downarrow \) 2341 |
| \(\displaystyle \int \left (c^3 C e^2 x^{10}+c^3 e (2 C d+B e) x^9+c^2 \left (c C d^2+3 a C e^2+c e (2 B d+A e)\right ) x^8+3 a c^2 e (2 C d+B e) x^7+c \left (A c \left (c d^2+3 a e^2\right )+3 a \left (a C e^2+c d (C d+2 B e)\right )\right ) x^6+3 a^2 c e (2 C d+B e) x^5+a \left (3 A c \left (c d^2+a e^2\right )+a \left (a C e^2+3 c d (C d+2 B e)\right )\right ) x^4+a^3 e (2 C d+B e) x^3+a^2 \left (a d (C d+2 B e)+A \left (3 c d^2+a e^2\right )\right ) x^2+a^3 A d^2\right )dx+\frac {d \left (a+c x^2\right )^4 (2 A e+B d)}{8 c}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^3 A d^2 x+\frac {1}{4} a^3 e x^4 (B e+2 C d)+\frac {1}{3} a^2 x^3 \left (A \left (a e^2+3 c d^2\right )+a d (2 B e+C d)\right )+\frac {1}{2} a^2 c e x^6 (B e+2 C d)+\frac {1}{9} c^2 x^9 \left (3 a C e^2+c e (A e+2 B d)+c C d^2\right )+\frac {1}{7} c x^7 \left (A c \left (3 a e^2+c d^2\right )+3 a \left (a C e^2+c d (2 B e+C d)\right )\right )+\frac {1}{5} a x^5 \left (3 A c \left (a e^2+c d^2\right )+a \left (a C e^2+3 c d (2 B e+C d)\right )\right )+\frac {d \left (a+c x^2\right )^4 (2 A e+B d)}{8 c}+\frac {3}{8} a c^2 e x^8 (B e+2 C d)+\frac {1}{10} c^3 e x^{10} (B e+2 C d)+\frac {1}{11} c^3 C e^2 x^{11}\) |
a^3*A*d^2*x + (a^2*(a*d*(C*d + 2*B*e) + A*(3*c*d^2 + a*e^2))*x^3)/3 + (a^3 *e*(2*C*d + B*e)*x^4)/4 + (a*(3*A*c*(c*d^2 + a*e^2) + a*(a*C*e^2 + 3*c*d*( C*d + 2*B*e)))*x^5)/5 + (a^2*c*e*(2*C*d + B*e)*x^6)/2 + (c*(A*c*(c*d^2 + 3 *a*e^2) + 3*a*(a*C*e^2 + c*d*(C*d + 2*B*e)))*x^7)/7 + (3*a*c^2*e*(2*C*d + B*e)*x^8)/8 + (c^2*(c*C*d^2 + 3*a*C*e^2 + c*e*(2*B*d + A*e))*x^9)/9 + (c^3 *e*(2*C*d + B*e)*x^10)/10 + (c^3*C*e^2*x^11)/11 + (d*(B*d + 2*A*e)*(a + c* x^2)^4)/(8*c)
3.1.33.3.1 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 = 0.50 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.30
| method | result | size |
| norman | \(\frac {c^{3} C \,e^{2} x^{11}}{11}+\left (\frac {1}{10} c^{3} e^{2} B +\frac {1}{5} c^{3} d e C \right ) x^{10}+\left (\frac {1}{9} c^{3} e^{2} A +\frac {2}{9} c^{3} d e B +\frac {1}{3} C a \,c^{2} e^{2}+\frac {1}{9} C \,c^{3} d^{2}\right ) x^{9}+\left (\frac {1}{4} c^{3} d e A +\frac {3}{8} B \,e^{2} a \,c^{2}+\frac {1}{8} B \,c^{3} d^{2}+\frac {3}{4} a \,c^{2} d e C \right ) x^{8}+\left (\frac {3}{7} A a \,c^{2} e^{2}+\frac {1}{7} A \,c^{3} d^{2}+\frac {6}{7} B a \,c^{2} d e +\frac {3}{7} C \,a^{2} c \,e^{2}+\frac {3}{7} C a \,c^{2} d^{2}\right ) x^{7}+\left (A a \,c^{2} d e +\frac {1}{2} B \,e^{2} c \,a^{2}+\frac {1}{2} B a \,c^{2} d^{2}+a^{2} c d e C \right ) x^{6}+\left (\frac {3}{5} A \,a^{2} c \,e^{2}+\frac {3}{5} A \,d^{2} a \,c^{2}+\frac {6}{5} a^{2} c d e B +\frac {1}{5} C \,a^{3} e^{2}+\frac {3}{5} C \,a^{2} c \,d^{2}\right ) x^{5}+\left (\frac {3}{2} A \,a^{2} c d e +\frac {1}{4} B \,e^{2} a^{3}+\frac {3}{4} B \,a^{2} c \,d^{2}+\frac {1}{2} d e \,a^{3} C \right ) x^{4}+\left (\frac {1}{3} A \,a^{3} e^{2}+A \,d^{2} c \,a^{2}+\frac {2}{3} d e \,a^{3} B +\frac {1}{3} d^{2} a^{3} C \right ) x^{3}+\left (A \,a^{3} d e +\frac {1}{2} B \,a^{3} d^{2}\right ) x^{2}+A \,d^{2} a^{3} x\) | \(376\) |
| default | \(\frac {c^{3} C \,e^{2} x^{11}}{11}+\frac {\left (c^{3} e^{2} B +2 c^{3} d e C \right ) x^{10}}{10}+\frac {\left (\left (3 a \,c^{2} e^{2}+c^{3} d^{2}\right ) C +2 c^{3} d e B +c^{3} e^{2} A \right ) x^{9}}{9}+\frac {\left (6 a \,c^{2} d e C +\left (3 a \,c^{2} e^{2}+c^{3} d^{2}\right ) B +2 c^{3} d e A \right ) x^{8}}{8}+\frac {\left (\left (3 a^{2} c \,e^{2}+3 a \,c^{2} d^{2}\right ) C +6 B a \,c^{2} d e +\left (3 a \,c^{2} e^{2}+c^{3} d^{2}\right ) A \right ) x^{7}}{7}+\frac {\left (6 a^{2} c d e C +\left (3 a^{2} c \,e^{2}+3 a \,c^{2} d^{2}\right ) B +6 A a \,c^{2} d e \right ) x^{6}}{6}+\frac {\left (\left (e^{2} a^{3}+3 d^{2} c \,a^{2}\right ) C +6 a^{2} c d e B +\left (3 a^{2} c \,e^{2}+3 a \,c^{2} d^{2}\right ) A \right ) x^{5}}{5}+\frac {\left (2 d e \,a^{3} C +\left (e^{2} a^{3}+3 d^{2} c \,a^{2}\right ) B +6 A \,a^{2} c d e \right ) x^{4}}{4}+\frac {\left (d^{2} a^{3} C +2 d e \,a^{3} B +\left (e^{2} a^{3}+3 d^{2} c \,a^{2}\right ) A \right ) x^{3}}{3}+\frac {\left (2 A \,a^{3} d e +B \,a^{3} d^{2}\right ) x^{2}}{2}+A \,d^{2} a^{3} x\) | \(388\) |
| gosper | \(\frac {1}{10} B \,c^{3} e^{2} x^{10}+\frac {1}{3} x^{3} d^{2} a^{3} C +\frac {1}{2} x^{2} B \,a^{3} d^{2}+\frac {1}{3} x^{3} A \,a^{3} e^{2}+\frac {1}{4} x^{4} B \,e^{2} a^{3}+\frac {1}{8} x^{8} B \,c^{3} d^{2}+\frac {1}{7} x^{7} A \,c^{3} d^{2}+\frac {1}{5} x^{5} C \,a^{3} e^{2}+\frac {1}{9} x^{9} c^{3} e^{2} A +\frac {1}{9} x^{9} C \,c^{3} d^{2}+\frac {3}{2} x^{4} A \,a^{2} c d e +\frac {3}{4} x^{8} a \,c^{2} d e C +\frac {6}{7} x^{7} B a \,c^{2} d e +x^{6} A a \,c^{2} d e +x^{6} a^{2} c d e C +\frac {6}{5} x^{5} a^{2} c d e B +A \,d^{2} a^{3} x +\frac {3}{4} x^{4} B \,a^{2} c \,d^{2}+\frac {2}{9} x^{9} c^{3} d e B +\frac {1}{3} x^{9} C a \,c^{2} e^{2}+\frac {1}{4} x^{8} c^{3} d e A +\frac {3}{8} x^{8} B \,e^{2} a \,c^{2}+\frac {3}{7} x^{7} A a \,c^{2} e^{2}+\frac {3}{5} x^{5} A \,a^{2} c \,e^{2}+\frac {3}{5} x^{5} A \,d^{2} a \,c^{2}+\frac {2}{3} x^{3} d e \,a^{3} B +\frac {3}{7} x^{7} C \,a^{2} c \,e^{2}+\frac {3}{7} x^{7} C a \,c^{2} d^{2}+x^{3} A \,d^{2} c \,a^{2}+x^{2} A \,a^{3} d e +\frac {1}{2} x^{4} d e \,a^{3} C +\frac {3}{5} x^{5} C \,a^{2} c \,d^{2}+\frac {1}{5} x^{10} c^{3} d e C +\frac {1}{2} x^{6} B \,e^{2} c \,a^{2}+\frac {1}{2} x^{6} B a \,c^{2} d^{2}+\frac {1}{11} c^{3} C \,e^{2} x^{11}\) | \(433\) |
| risch | \(\frac {1}{10} B \,c^{3} e^{2} x^{10}+\frac {1}{3} x^{3} d^{2} a^{3} C +\frac {1}{2} x^{2} B \,a^{3} d^{2}+\frac {1}{3} x^{3} A \,a^{3} e^{2}+\frac {1}{4} x^{4} B \,e^{2} a^{3}+\frac {1}{8} x^{8} B \,c^{3} d^{2}+\frac {1}{7} x^{7} A \,c^{3} d^{2}+\frac {1}{5} x^{5} C \,a^{3} e^{2}+\frac {1}{9} x^{9} c^{3} e^{2} A +\frac {1}{9} x^{9} C \,c^{3} d^{2}+\frac {3}{2} x^{4} A \,a^{2} c d e +\frac {3}{4} x^{8} a \,c^{2} d e C +\frac {6}{7} x^{7} B a \,c^{2} d e +x^{6} A a \,c^{2} d e +x^{6} a^{2} c d e C +\frac {6}{5} x^{5} a^{2} c d e B +A \,d^{2} a^{3} x +\frac {3}{4} x^{4} B \,a^{2} c \,d^{2}+\frac {2}{9} x^{9} c^{3} d e B +\frac {1}{3} x^{9} C a \,c^{2} e^{2}+\frac {1}{4} x^{8} c^{3} d e A +\frac {3}{8} x^{8} B \,e^{2} a \,c^{2}+\frac {3}{7} x^{7} A a \,c^{2} e^{2}+\frac {3}{5} x^{5} A \,a^{2} c \,e^{2}+\frac {3}{5} x^{5} A \,d^{2} a \,c^{2}+\frac {2}{3} x^{3} d e \,a^{3} B +\frac {3}{7} x^{7} C \,a^{2} c \,e^{2}+\frac {3}{7} x^{7} C a \,c^{2} d^{2}+x^{3} A \,d^{2} c \,a^{2}+x^{2} A \,a^{3} d e +\frac {1}{2} x^{4} d e \,a^{3} C +\frac {3}{5} x^{5} C \,a^{2} c \,d^{2}+\frac {1}{5} x^{10} c^{3} d e C +\frac {1}{2} x^{6} B \,e^{2} c \,a^{2}+\frac {1}{2} x^{6} B a \,c^{2} d^{2}+\frac {1}{11} c^{3} C \,e^{2} x^{11}\) | \(433\) |
| parallelrisch | \(\frac {1}{10} B \,c^{3} e^{2} x^{10}+\frac {1}{3} x^{3} d^{2} a^{3} C +\frac {1}{2} x^{2} B \,a^{3} d^{2}+\frac {1}{3} x^{3} A \,a^{3} e^{2}+\frac {1}{4} x^{4} B \,e^{2} a^{3}+\frac {1}{8} x^{8} B \,c^{3} d^{2}+\frac {1}{7} x^{7} A \,c^{3} d^{2}+\frac {1}{5} x^{5} C \,a^{3} e^{2}+\frac {1}{9} x^{9} c^{3} e^{2} A +\frac {1}{9} x^{9} C \,c^{3} d^{2}+\frac {3}{2} x^{4} A \,a^{2} c d e +\frac {3}{4} x^{8} a \,c^{2} d e C +\frac {6}{7} x^{7} B a \,c^{2} d e +x^{6} A a \,c^{2} d e +x^{6} a^{2} c d e C +\frac {6}{5} x^{5} a^{2} c d e B +A \,d^{2} a^{3} x +\frac {3}{4} x^{4} B \,a^{2} c \,d^{2}+\frac {2}{9} x^{9} c^{3} d e B +\frac {1}{3} x^{9} C a \,c^{2} e^{2}+\frac {1}{4} x^{8} c^{3} d e A +\frac {3}{8} x^{8} B \,e^{2} a \,c^{2}+\frac {3}{7} x^{7} A a \,c^{2} e^{2}+\frac {3}{5} x^{5} A \,a^{2} c \,e^{2}+\frac {3}{5} x^{5} A \,d^{2} a \,c^{2}+\frac {2}{3} x^{3} d e \,a^{3} B +\frac {3}{7} x^{7} C \,a^{2} c \,e^{2}+\frac {3}{7} x^{7} C a \,c^{2} d^{2}+x^{3} A \,d^{2} c \,a^{2}+x^{2} A \,a^{3} d e +\frac {1}{2} x^{4} d e \,a^{3} C +\frac {3}{5} x^{5} C \,a^{2} c \,d^{2}+\frac {1}{5} x^{10} c^{3} d e C +\frac {1}{2} x^{6} B \,e^{2} c \,a^{2}+\frac {1}{2} x^{6} B a \,c^{2} d^{2}+\frac {1}{11} c^{3} C \,e^{2} x^{11}\) | \(433\) |
1/11*c^3*C*e^2*x^11+(1/10*c^3*e^2*B+1/5*c^3*d*e*C)*x^10+(1/9*c^3*e^2*A+2/9 *c^3*d*e*B+1/3*C*a*c^2*e^2+1/9*C*c^3*d^2)*x^9+(1/4*c^3*d*e*A+3/8*B*e^2*a*c ^2+1/8*B*c^3*d^2+3/4*a*c^2*d*e*C)*x^8+(3/7*A*a*c^2*e^2+1/7*A*c^3*d^2+6/7*B *a*c^2*d*e+3/7*C*a^2*c*e^2+3/7*C*a*c^2*d^2)*x^7+(A*a*c^2*d*e+1/2*B*e^2*c*a ^2+1/2*B*a*c^2*d^2+a^2*c*d*e*C)*x^6+(3/5*A*a^2*c*e^2+3/5*A*d^2*a*c^2+6/5*a ^2*c*d*e*B+1/5*C*a^3*e^2+3/5*C*a^2*c*d^2)*x^5+(3/2*A*a^2*c*d*e+1/4*B*e^2*a ^3+3/4*B*a^2*c*d^2+1/2*d*e*a^3*C)*x^4+(1/3*A*a^3*e^2+A*d^2*c*a^2+2/3*d*e*a ^3*B+1/3*d^2*a^3*C)*x^3+(A*a^3*d*e+1/2*B*a^3*d^2)*x^2+A*d^2*a^3*x
Time = 0.27 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.27 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \left (A+B x+C x^2\right ) \, dx=\frac {1}{11} \, C c^{3} e^{2} x^{11} + \frac {1}{10} \, {\left (2 \, C c^{3} d e + B c^{3} e^{2}\right )} x^{10} + \frac {1}{9} \, {\left (C c^{3} d^{2} + 2 \, B c^{3} d e + {\left (3 \, C a c^{2} + A c^{3}\right )} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{2} + 3 \, B a c^{2} e^{2} + 2 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d e\right )} x^{8} + \frac {1}{7} \, {\left (6 \, B a c^{2} d e + {\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} e^{2}\right )} x^{7} + A a^{3} d^{2} x + \frac {1}{2} \, {\left (B a c^{2} d^{2} + B a^{2} c e^{2} + 2 \, {\left (C a^{2} c + A a c^{2}\right )} d e\right )} x^{6} + \frac {1}{5} \, {\left (6 \, B a^{2} c d e + 3 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} + {\left (C a^{3} + 3 \, A a^{2} c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{2} + B a^{3} e^{2} + 2 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (2 \, B a^{3} d e + A a^{3} e^{2} + {\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d^{2} + 2 \, A a^{3} d e\right )} x^{2} \]
1/11*C*c^3*e^2*x^11 + 1/10*(2*C*c^3*d*e + B*c^3*e^2)*x^10 + 1/9*(C*c^3*d^2 + 2*B*c^3*d*e + (3*C*a*c^2 + A*c^3)*e^2)*x^9 + 1/8*(B*c^3*d^2 + 3*B*a*c^2 *e^2 + 2*(3*C*a*c^2 + A*c^3)*d*e)*x^8 + 1/7*(6*B*a*c^2*d*e + (3*C*a*c^2 + A*c^3)*d^2 + 3*(C*a^2*c + A*a*c^2)*e^2)*x^7 + A*a^3*d^2*x + 1/2*(B*a*c^2*d ^2 + B*a^2*c*e^2 + 2*(C*a^2*c + A*a*c^2)*d*e)*x^6 + 1/5*(6*B*a^2*c*d*e + 3 *(C*a^2*c + A*a*c^2)*d^2 + (C*a^3 + 3*A*a^2*c)*e^2)*x^5 + 1/4*(3*B*a^2*c*d ^2 + B*a^3*e^2 + 2*(C*a^3 + 3*A*a^2*c)*d*e)*x^4 + 1/3*(2*B*a^3*d*e + A*a^3 *e^2 + (C*a^3 + 3*A*a^2*c)*d^2)*x^3 + 1/2*(B*a^3*d^2 + 2*A*a^3*d*e)*x^2
Time = 0.05 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.55 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \left (A+B x+C x^2\right ) \, dx=A a^{3} d^{2} x + \frac {C c^{3} e^{2} x^{11}}{11} + x^{10} \left (\frac {B c^{3} e^{2}}{10} + \frac {C c^{3} d e}{5}\right ) + x^{9} \left (\frac {A c^{3} e^{2}}{9} + \frac {2 B c^{3} d e}{9} + \frac {C a c^{2} e^{2}}{3} + \frac {C c^{3} d^{2}}{9}\right ) + x^{8} \left (\frac {A c^{3} d e}{4} + \frac {3 B a c^{2} e^{2}}{8} + \frac {B c^{3} d^{2}}{8} + \frac {3 C a c^{2} d e}{4}\right ) + x^{7} \cdot \left (\frac {3 A a c^{2} e^{2}}{7} + \frac {A c^{3} d^{2}}{7} + \frac {6 B a c^{2} d e}{7} + \frac {3 C a^{2} c e^{2}}{7} + \frac {3 C a c^{2} d^{2}}{7}\right ) + x^{6} \left (A a c^{2} d e + \frac {B a^{2} c e^{2}}{2} + \frac {B a c^{2} d^{2}}{2} + C a^{2} c d e\right ) + x^{5} \cdot \left (\frac {3 A a^{2} c e^{2}}{5} + \frac {3 A a c^{2} d^{2}}{5} + \frac {6 B a^{2} c d e}{5} + \frac {C a^{3} e^{2}}{5} + \frac {3 C a^{2} c d^{2}}{5}\right ) + x^{4} \cdot \left (\frac {3 A a^{2} c d e}{2} + \frac {B a^{3} e^{2}}{4} + \frac {3 B a^{2} c d^{2}}{4} + \frac {C a^{3} d e}{2}\right ) + x^{3} \left (\frac {A a^{3} e^{2}}{3} + A a^{2} c d^{2} + \frac {2 B a^{3} d e}{3} + \frac {C a^{3} d^{2}}{3}\right ) + x^{2} \left (A a^{3} d e + \frac {B a^{3} d^{2}}{2}\right ) \]
A*a**3*d**2*x + C*c**3*e**2*x**11/11 + x**10*(B*c**3*e**2/10 + C*c**3*d*e/ 5) + x**9*(A*c**3*e**2/9 + 2*B*c**3*d*e/9 + C*a*c**2*e**2/3 + C*c**3*d**2/ 9) + x**8*(A*c**3*d*e/4 + 3*B*a*c**2*e**2/8 + B*c**3*d**2/8 + 3*C*a*c**2*d *e/4) + x**7*(3*A*a*c**2*e**2/7 + A*c**3*d**2/7 + 6*B*a*c**2*d*e/7 + 3*C*a **2*c*e**2/7 + 3*C*a*c**2*d**2/7) + x**6*(A*a*c**2*d*e + B*a**2*c*e**2/2 + B*a*c**2*d**2/2 + C*a**2*c*d*e) + x**5*(3*A*a**2*c*e**2/5 + 3*A*a*c**2*d* *2/5 + 6*B*a**2*c*d*e/5 + C*a**3*e**2/5 + 3*C*a**2*c*d**2/5) + x**4*(3*A*a **2*c*d*e/2 + B*a**3*e**2/4 + 3*B*a**2*c*d**2/4 + C*a**3*d*e/2) + x**3*(A* a**3*e**2/3 + A*a**2*c*d**2 + 2*B*a**3*d*e/3 + C*a**3*d**2/3) + x**2*(A*a* *3*d*e + B*a**3*d**2/2)
Time = 0.18 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.27 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \left (A+B x+C x^2\right ) \, dx=\frac {1}{11} \, C c^{3} e^{2} x^{11} + \frac {1}{10} \, {\left (2 \, C c^{3} d e + B c^{3} e^{2}\right )} x^{10} + \frac {1}{9} \, {\left (C c^{3} d^{2} + 2 \, B c^{3} d e + {\left (3 \, C a c^{2} + A c^{3}\right )} e^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{2} + 3 \, B a c^{2} e^{2} + 2 \, {\left (3 \, C a c^{2} + A c^{3}\right )} d e\right )} x^{8} + \frac {1}{7} \, {\left (6 \, B a c^{2} d e + {\left (3 \, C a c^{2} + A c^{3}\right )} d^{2} + 3 \, {\left (C a^{2} c + A a c^{2}\right )} e^{2}\right )} x^{7} + A a^{3} d^{2} x + \frac {1}{2} \, {\left (B a c^{2} d^{2} + B a^{2} c e^{2} + 2 \, {\left (C a^{2} c + A a c^{2}\right )} d e\right )} x^{6} + \frac {1}{5} \, {\left (6 \, B a^{2} c d e + 3 \, {\left (C a^{2} c + A a c^{2}\right )} d^{2} + {\left (C a^{3} + 3 \, A a^{2} c\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{2} + B a^{3} e^{2} + 2 \, {\left (C a^{3} + 3 \, A a^{2} c\right )} d e\right )} x^{4} + \frac {1}{3} \, {\left (2 \, B a^{3} d e + A a^{3} e^{2} + {\left (C a^{3} + 3 \, A a^{2} c\right )} d^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d^{2} + 2 \, A a^{3} d e\right )} x^{2} \]
1/11*C*c^3*e^2*x^11 + 1/10*(2*C*c^3*d*e + B*c^3*e^2)*x^10 + 1/9*(C*c^3*d^2 + 2*B*c^3*d*e + (3*C*a*c^2 + A*c^3)*e^2)*x^9 + 1/8*(B*c^3*d^2 + 3*B*a*c^2 *e^2 + 2*(3*C*a*c^2 + A*c^3)*d*e)*x^8 + 1/7*(6*B*a*c^2*d*e + (3*C*a*c^2 + A*c^3)*d^2 + 3*(C*a^2*c + A*a*c^2)*e^2)*x^7 + A*a^3*d^2*x + 1/2*(B*a*c^2*d ^2 + B*a^2*c*e^2 + 2*(C*a^2*c + A*a*c^2)*d*e)*x^6 + 1/5*(6*B*a^2*c*d*e + 3 *(C*a^2*c + A*a*c^2)*d^2 + (C*a^3 + 3*A*a^2*c)*e^2)*x^5 + 1/4*(3*B*a^2*c*d ^2 + B*a^3*e^2 + 2*(C*a^3 + 3*A*a^2*c)*d*e)*x^4 + 1/3*(2*B*a^3*d*e + A*a^3 *e^2 + (C*a^3 + 3*A*a^2*c)*d^2)*x^3 + 1/2*(B*a^3*d^2 + 2*A*a^3*d*e)*x^2
Time = 0.26 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.49 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \left (A+B x+C x^2\right ) \, dx=\frac {1}{11} \, C c^{3} e^{2} x^{11} + \frac {1}{5} \, C c^{3} d e x^{10} + \frac {1}{10} \, B c^{3} e^{2} x^{10} + \frac {1}{9} \, C c^{3} d^{2} x^{9} + \frac {2}{9} \, B c^{3} d e x^{9} + \frac {1}{3} \, C a c^{2} e^{2} x^{9} + \frac {1}{9} \, A c^{3} e^{2} x^{9} + \frac {1}{8} \, B c^{3} d^{2} x^{8} + \frac {3}{4} \, C a c^{2} d e x^{8} + \frac {1}{4} \, A c^{3} d e x^{8} + \frac {3}{8} \, B a c^{2} e^{2} x^{8} + \frac {3}{7} \, C a c^{2} d^{2} x^{7} + \frac {1}{7} \, A c^{3} d^{2} x^{7} + \frac {6}{7} \, B a c^{2} d e x^{7} + \frac {3}{7} \, C a^{2} c e^{2} x^{7} + \frac {3}{7} \, A a c^{2} e^{2} x^{7} + \frac {1}{2} \, B a c^{2} d^{2} x^{6} + C a^{2} c d e x^{6} + A a c^{2} d e x^{6} + \frac {1}{2} \, B a^{2} c e^{2} x^{6} + \frac {3}{5} \, C a^{2} c d^{2} x^{5} + \frac {3}{5} \, A a c^{2} d^{2} x^{5} + \frac {6}{5} \, B a^{2} c d e x^{5} + \frac {1}{5} \, C a^{3} e^{2} x^{5} + \frac {3}{5} \, A a^{2} c e^{2} x^{5} + \frac {3}{4} \, B a^{2} c d^{2} x^{4} + \frac {1}{2} \, C a^{3} d e x^{4} + \frac {3}{2} \, A a^{2} c d e x^{4} + \frac {1}{4} \, B a^{3} e^{2} x^{4} + \frac {1}{3} \, C a^{3} d^{2} x^{3} + A a^{2} c d^{2} x^{3} + \frac {2}{3} \, B a^{3} d e x^{3} + \frac {1}{3} \, A a^{3} e^{2} x^{3} + \frac {1}{2} \, B a^{3} d^{2} x^{2} + A a^{3} d e x^{2} + A a^{3} d^{2} x \]
1/11*C*c^3*e^2*x^11 + 1/5*C*c^3*d*e*x^10 + 1/10*B*c^3*e^2*x^10 + 1/9*C*c^3 *d^2*x^9 + 2/9*B*c^3*d*e*x^9 + 1/3*C*a*c^2*e^2*x^9 + 1/9*A*c^3*e^2*x^9 + 1 /8*B*c^3*d^2*x^8 + 3/4*C*a*c^2*d*e*x^8 + 1/4*A*c^3*d*e*x^8 + 3/8*B*a*c^2*e ^2*x^8 + 3/7*C*a*c^2*d^2*x^7 + 1/7*A*c^3*d^2*x^7 + 6/7*B*a*c^2*d*e*x^7 + 3 /7*C*a^2*c*e^2*x^7 + 3/7*A*a*c^2*e^2*x^7 + 1/2*B*a*c^2*d^2*x^6 + C*a^2*c*d *e*x^6 + A*a*c^2*d*e*x^6 + 1/2*B*a^2*c*e^2*x^6 + 3/5*C*a^2*c*d^2*x^5 + 3/5 *A*a*c^2*d^2*x^5 + 6/5*B*a^2*c*d*e*x^5 + 1/5*C*a^3*e^2*x^5 + 3/5*A*a^2*c*e ^2*x^5 + 3/4*B*a^2*c*d^2*x^4 + 1/2*C*a^3*d*e*x^4 + 3/2*A*a^2*c*d*e*x^4 + 1 /4*B*a^3*e^2*x^4 + 1/3*C*a^3*d^2*x^3 + A*a^2*c*d^2*x^3 + 2/3*B*a^3*d*e*x^3 + 1/3*A*a^3*e^2*x^3 + 1/2*B*a^3*d^2*x^2 + A*a^3*d*e*x^2 + A*a^3*d^2*x
Time = 12.38 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.19 \[ \int (d+e x)^2 \left (a+c x^2\right )^3 \left (A+B x+C x^2\right ) \, dx=x^3\,\left (\frac {C\,a^3\,d^2}{3}+\frac {2\,B\,a^3\,d\,e}{3}+\frac {A\,a^3\,e^2}{3}+A\,c\,a^2\,d^2\right )+x^9\,\left (\frac {C\,c^3\,d^2}{9}+\frac {2\,B\,c^3\,d\,e}{9}+\frac {A\,c^3\,e^2}{9}+\frac {C\,a\,c^2\,e^2}{3}\right )+x^5\,\left (\frac {C\,a^3\,e^2}{5}+\frac {3\,C\,a^2\,c\,d^2}{5}+\frac {6\,B\,a^2\,c\,d\,e}{5}+\frac {3\,A\,a^2\,c\,e^2}{5}+\frac {3\,A\,a\,c^2\,d^2}{5}\right )+x^7\,\left (\frac {3\,C\,a^2\,c\,e^2}{7}+\frac {3\,C\,a\,c^2\,d^2}{7}+\frac {6\,B\,a\,c^2\,d\,e}{7}+\frac {3\,A\,a\,c^2\,e^2}{7}+\frac {A\,c^3\,d^2}{7}\right )+\frac {a^2\,x^4\,\left (B\,a\,e^2+3\,B\,c\,d^2+6\,A\,c\,d\,e+2\,C\,a\,d\,e\right )}{4}+\frac {c^2\,x^8\,\left (3\,B\,a\,e^2+B\,c\,d^2+2\,A\,c\,d\,e+6\,C\,a\,d\,e\right )}{8}+\frac {C\,c^3\,e^2\,x^{11}}{11}+\frac {a\,c\,x^6\,\left (B\,a\,e^2+B\,c\,d^2+2\,A\,c\,d\,e+2\,C\,a\,d\,e\right )}{2}+A\,a^3\,d^2\,x+\frac {a^3\,d\,x^2\,\left (2\,A\,e+B\,d\right )}{2}+\frac {c^3\,e\,x^{10}\,\left (B\,e+2\,C\,d\right )}{10} \]
x^3*((A*a^3*e^2)/3 + (C*a^3*d^2)/3 + (2*B*a^3*d*e)/3 + A*a^2*c*d^2) + x^9* ((A*c^3*e^2)/9 + (C*c^3*d^2)/9 + (2*B*c^3*d*e)/9 + (C*a*c^2*e^2)/3) + x^5* ((C*a^3*e^2)/5 + (3*A*a*c^2*d^2)/5 + (3*A*a^2*c*e^2)/5 + (3*C*a^2*c*d^2)/5 + (6*B*a^2*c*d*e)/5) + x^7*((A*c^3*d^2)/7 + (3*A*a*c^2*e^2)/7 + (3*C*a*c^ 2*d^2)/7 + (3*C*a^2*c*e^2)/7 + (6*B*a*c^2*d*e)/7) + (a^2*x^4*(B*a*e^2 + 3* B*c*d^2 + 6*A*c*d*e + 2*C*a*d*e))/4 + (c^2*x^8*(3*B*a*e^2 + B*c*d^2 + 2*A* c*d*e + 6*C*a*d*e))/8 + (C*c^3*e^2*x^11)/11 + (a*c*x^6*(B*a*e^2 + B*c*d^2 + 2*A*c*d*e + 2*C*a*d*e))/2 + A*a^3*d^2*x + (a^3*d*x^2*(2*A*e + B*d))/2 + (c^3*e*x^10*(B*e + 2*C*d))/10