Integrand size = 22, antiderivative size = 206 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^2 \, dx=-\frac {(B d-A e) \left (c d^2+a e^2\right )^2 (d+e x)^4}{4 e^6}+\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right ) (d+e x)^5}{5 e^6}-\frac {c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^6}{3 e^6}+\frac {2 c \left (5 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^7}{7 e^6}-\frac {c^2 (5 B d-A e) (d+e x)^8}{8 e^6}+\frac {B c^2 (d+e x)^9}{9 e^6} \]
-1/4*(-A*e+B*d)*(a*e^2+c*d^2)^2*(e*x+d)^4/e^6+1/5*(a*e^2+c*d^2)*(-4*A*c*d* e+B*a*e^2+5*B*c*d^2)*(e*x+d)^5/e^6-1/3*c*(-A*a*e^3-3*A*c*d^2*e+3*B*a*d*e^2 +5*B*c*d^3)*(e*x+d)^6/e^6+2/7*c*(-2*A*c*d*e+B*a*e^2+5*B*c*d^2)*(e*x+d)^7/e ^6-1/8*c^2*(-A*e+5*B*d)*(e*x+d)^8/e^6+1/9*B*c^2*(e*x+d)^9/e^6
Time = 0.04 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.18 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^2 \, dx=a^2 A d^3 x+\frac {1}{2} a^2 d^2 (B d+3 A e) x^2+\frac {1}{3} a d \left (2 A c d^2+3 a B d e+3 a A e^2\right ) x^3+\frac {1}{4} a \left (2 B c d^3+6 A c d^2 e+3 a B d e^2+a A e^3\right ) x^4+\frac {1}{5} \left (A c^2 d^3+6 a B c d^2 e+6 a A c d e^2+a^2 B e^3\right ) x^5+\frac {1}{6} c \left (B c d^3+3 A c d^2 e+6 a B d e^2+2 a A e^3\right ) x^6+\frac {1}{7} c e \left (3 B c d^2+3 A c d e+2 a B e^2\right ) x^7+\frac {1}{8} c^2 e^2 (3 B d+A e) x^8+\frac {1}{9} B c^2 e^3 x^9 \]
a^2*A*d^3*x + (a^2*d^2*(B*d + 3*A*e)*x^2)/2 + (a*d*(2*A*c*d^2 + 3*a*B*d*e + 3*a*A*e^2)*x^3)/3 + (a*(2*B*c*d^3 + 6*A*c*d^2*e + 3*a*B*d*e^2 + a*A*e^3) *x^4)/4 + ((A*c^2*d^3 + 6*a*B*c*d^2*e + 6*a*A*c*d*e^2 + a^2*B*e^3)*x^5)/5 + (c*(B*c*d^3 + 3*A*c*d^2*e + 6*a*B*d*e^2 + 2*a*A*e^3)*x^6)/6 + (c*e*(3*B* c*d^2 + 3*A*c*d*e + 2*a*B*e^2)*x^7)/7 + (c^2*e^2*(3*B*d + A*e)*x^8)/8 + (B *c^2*e^3*x^9)/9
Time = 0.44 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {652, 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.
| \(\displaystyle \int \left (a+c x^2\right )^2 (A+B x) (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (-\frac {2 c (d+e x)^6 \left (-a B e^2+2 A c d e-5 B c d^2\right )}{e^5}+\frac {(d+e x)^4 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^5}+\frac {(d+e x)^3 \left (a e^2+c d^2\right )^2 (A e-B d)}{e^5}+\frac {2 c (d+e x)^5 \left (a A e^3-3 a B d e^2+3 A c d^2 e-5 B c d^3\right )}{e^5}+\frac {c^2 (d+e x)^7 (A e-5 B d)}{e^5}+\frac {B c^2 (d+e x)^8}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 c (d+e x)^7 \left (a B e^2-2 A c d e+5 B c d^2\right )}{7 e^6}+\frac {(d+e x)^5 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6}-\frac {(d+e x)^4 \left (a e^2+c d^2\right )^2 (B d-A e)}{4 e^6}-\frac {c (d+e x)^6 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{3 e^6}-\frac {c^2 (d+e x)^8 (5 B d-A e)}{8 e^6}+\frac {B c^2 (d+e x)^9}{9 e^6}\) |
-1/4*((B*d - A*e)*(c*d^2 + a*e^2)^2*(d + e*x)^4)/e^6 + ((c*d^2 + a*e^2)*(5 *B*c*d^2 - 4*A*c*d*e + a*B*e^2)*(d + e*x)^5)/(5*e^6) - (c*(5*B*c*d^3 - 3*A *c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^6)/(3*e^6) + (2*c*(5*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^7)/(7*e^6) - (c^2*(5*B*d - A*e)*(d + e*x)^ 8)/(8*e^6) + (B*c^2*(d + e*x)^9)/(9*e^6)
3.13.99.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Time = 0.27 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.22
| method | result | size |
| default | \(\frac {B \,c^{2} e^{3} x^{9}}{9}+\frac {\left (A \,e^{3}+3 B d \,e^{2}\right ) c^{2} x^{8}}{8}+\frac {\left (\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) c^{2}+2 B \,e^{3} a c \right ) x^{7}}{7}+\frac {\left (\left (3 A \,d^{2} e +B \,d^{3}\right ) c^{2}+2 \left (A \,e^{3}+3 B d \,e^{2}\right ) a c \right ) x^{6}}{6}+\frac {\left (A \,d^{3} c^{2}+2 \left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a c +a^{2} B \,e^{3}\right ) x^{5}}{5}+\frac {\left (2 \left (3 A \,d^{2} e +B \,d^{3}\right ) a c +\left (A \,e^{3}+3 B d \,e^{2}\right ) a^{2}\right ) x^{4}}{4}+\frac {\left (2 A \,d^{3} a c +\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a^{2}\right ) x^{3}}{3}+\frac {\left (3 A \,d^{2} e +B \,d^{3}\right ) a^{2} x^{2}}{2}+A \,d^{3} a^{2} x\) | \(252\) |
| norman | \(\frac {B \,c^{2} e^{3} x^{9}}{9}+\left (\frac {1}{8} A \,c^{2} e^{3}+\frac {3}{8} B \,c^{2} d \,e^{2}\right ) x^{8}+\left (\frac {3}{7} A \,c^{2} d \,e^{2}+\frac {2}{7} B \,e^{3} a c +\frac {3}{7} B \,c^{2} d^{2} e \right ) x^{7}+\left (\frac {1}{3} A a c \,e^{3}+\frac {1}{2} A \,c^{2} d^{2} e +B a c d \,e^{2}+\frac {1}{6} B \,c^{2} d^{3}\right ) x^{6}+\left (\frac {6}{5} A a c d \,e^{2}+\frac {1}{5} A \,d^{3} c^{2}+\frac {1}{5} a^{2} B \,e^{3}+\frac {6}{5} B a c \,d^{2} e \right ) x^{5}+\left (\frac {1}{4} A \,a^{2} e^{3}+\frac {3}{2} A a c \,d^{2} e +\frac {3}{4} B \,a^{2} d \,e^{2}+\frac {1}{2} B a c \,d^{3}\right ) x^{4}+\left (A \,a^{2} d \,e^{2}+\frac {2}{3} A \,d^{3} a c +B \,a^{2} d^{2} e \right ) x^{3}+\left (\frac {3}{2} A \,a^{2} d^{2} e +\frac {1}{2} B \,a^{2} d^{3}\right ) x^{2}+A \,d^{3} a^{2} x\) | \(257\) |
| gosper | \(\frac {1}{9} B \,c^{2} e^{3} x^{9}+\frac {1}{8} x^{8} A \,c^{2} e^{3}+\frac {3}{8} x^{8} B \,c^{2} d \,e^{2}+\frac {3}{7} x^{7} A \,c^{2} d \,e^{2}+\frac {2}{7} x^{7} B \,e^{3} a c +\frac {3}{7} x^{7} B \,c^{2} d^{2} e +\frac {1}{3} x^{6} A a c \,e^{3}+\frac {1}{2} x^{6} A \,c^{2} d^{2} e +x^{6} B a c d \,e^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{3}+\frac {6}{5} x^{5} A a c d \,e^{2}+\frac {1}{5} x^{5} A \,d^{3} c^{2}+\frac {1}{5} x^{5} a^{2} B \,e^{3}+\frac {6}{5} x^{5} B a c \,d^{2} e +\frac {1}{4} x^{4} A \,a^{2} e^{3}+\frac {3}{2} x^{4} A a c \,d^{2} e +\frac {3}{4} x^{4} B \,a^{2} d \,e^{2}+\frac {1}{2} x^{4} B a c \,d^{3}+x^{3} A \,a^{2} d \,e^{2}+\frac {2}{3} x^{3} A \,d^{3} a c +x^{3} B \,a^{2} d^{2} e +\frac {3}{2} x^{2} A \,a^{2} d^{2} e +\frac {1}{2} x^{2} B \,a^{2} d^{3}+A \,d^{3} a^{2} x\) | \(288\) |
| risch | \(\frac {1}{9} B \,c^{2} e^{3} x^{9}+\frac {1}{8} x^{8} A \,c^{2} e^{3}+\frac {3}{8} x^{8} B \,c^{2} d \,e^{2}+\frac {3}{7} x^{7} A \,c^{2} d \,e^{2}+\frac {2}{7} x^{7} B \,e^{3} a c +\frac {3}{7} x^{7} B \,c^{2} d^{2} e +\frac {1}{3} x^{6} A a c \,e^{3}+\frac {1}{2} x^{6} A \,c^{2} d^{2} e +x^{6} B a c d \,e^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{3}+\frac {6}{5} x^{5} A a c d \,e^{2}+\frac {1}{5} x^{5} A \,d^{3} c^{2}+\frac {1}{5} x^{5} a^{2} B \,e^{3}+\frac {6}{5} x^{5} B a c \,d^{2} e +\frac {1}{4} x^{4} A \,a^{2} e^{3}+\frac {3}{2} x^{4} A a c \,d^{2} e +\frac {3}{4} x^{4} B \,a^{2} d \,e^{2}+\frac {1}{2} x^{4} B a c \,d^{3}+x^{3} A \,a^{2} d \,e^{2}+\frac {2}{3} x^{3} A \,d^{3} a c +x^{3} B \,a^{2} d^{2} e +\frac {3}{2} x^{2} A \,a^{2} d^{2} e +\frac {1}{2} x^{2} B \,a^{2} d^{3}+A \,d^{3} a^{2} x\) | \(288\) |
| parallelrisch | \(\frac {1}{9} B \,c^{2} e^{3} x^{9}+\frac {1}{8} x^{8} A \,c^{2} e^{3}+\frac {3}{8} x^{8} B \,c^{2} d \,e^{2}+\frac {3}{7} x^{7} A \,c^{2} d \,e^{2}+\frac {2}{7} x^{7} B \,e^{3} a c +\frac {3}{7} x^{7} B \,c^{2} d^{2} e +\frac {1}{3} x^{6} A a c \,e^{3}+\frac {1}{2} x^{6} A \,c^{2} d^{2} e +x^{6} B a c d \,e^{2}+\frac {1}{6} x^{6} B \,c^{2} d^{3}+\frac {6}{5} x^{5} A a c d \,e^{2}+\frac {1}{5} x^{5} A \,d^{3} c^{2}+\frac {1}{5} x^{5} a^{2} B \,e^{3}+\frac {6}{5} x^{5} B a c \,d^{2} e +\frac {1}{4} x^{4} A \,a^{2} e^{3}+\frac {3}{2} x^{4} A a c \,d^{2} e +\frac {3}{4} x^{4} B \,a^{2} d \,e^{2}+\frac {1}{2} x^{4} B a c \,d^{3}+x^{3} A \,a^{2} d \,e^{2}+\frac {2}{3} x^{3} A \,d^{3} a c +x^{3} B \,a^{2} d^{2} e +\frac {3}{2} x^{2} A \,a^{2} d^{2} e +\frac {1}{2} x^{2} B \,a^{2} d^{3}+A \,d^{3} a^{2} x\) | \(288\) |
1/9*B*c^2*e^3*x^9+1/8*(A*e^3+3*B*d*e^2)*c^2*x^8+1/7*((3*A*d*e^2+3*B*d^2*e) *c^2+2*B*e^3*a*c)*x^7+1/6*((3*A*d^2*e+B*d^3)*c^2+2*(A*e^3+3*B*d*e^2)*a*c)* x^6+1/5*(A*d^3*c^2+2*(3*A*d*e^2+3*B*d^2*e)*a*c+a^2*B*e^3)*x^5+1/4*(2*(3*A* d^2*e+B*d^3)*a*c+(A*e^3+3*B*d*e^2)*a^2)*x^4+1/3*(2*A*d^3*a*c+(3*A*d*e^2+3* B*d^2*e)*a^2)*x^3+1/2*(3*A*d^2*e+B*d^3)*a^2*x^2+A*d^3*a^2*x
Time = 0.25 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.26 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^2 \, dx=\frac {1}{9} \, B c^{2} e^{3} x^{9} + \frac {1}{8} \, {\left (3 \, B c^{2} d e^{2} + A c^{2} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, B c^{2} d^{2} e + 3 \, A c^{2} d e^{2} + 2 \, B a c e^{3}\right )} x^{7} + A a^{2} d^{3} x + \frac {1}{6} \, {\left (B c^{2} d^{3} + 3 \, A c^{2} d^{2} e + 6 \, B a c d e^{2} + 2 \, A a c e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (A c^{2} d^{3} + 6 \, B a c d^{2} e + 6 \, A a c d e^{2} + B a^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, B a c d^{3} + 6 \, A a c d^{2} e + 3 \, B a^{2} d e^{2} + A a^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (2 \, A a c d^{3} + 3 \, B a^{2} d^{2} e + 3 \, A a^{2} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} d^{3} + 3 \, A a^{2} d^{2} e\right )} x^{2} \]
1/9*B*c^2*e^3*x^9 + 1/8*(3*B*c^2*d*e^2 + A*c^2*e^3)*x^8 + 1/7*(3*B*c^2*d^2 *e + 3*A*c^2*d*e^2 + 2*B*a*c*e^3)*x^7 + A*a^2*d^3*x + 1/6*(B*c^2*d^3 + 3*A *c^2*d^2*e + 6*B*a*c*d*e^2 + 2*A*a*c*e^3)*x^6 + 1/5*(A*c^2*d^3 + 6*B*a*c*d ^2*e + 6*A*a*c*d*e^2 + B*a^2*e^3)*x^5 + 1/4*(2*B*a*c*d^3 + 6*A*a*c*d^2*e + 3*B*a^2*d*e^2 + A*a^2*e^3)*x^4 + 1/3*(2*A*a*c*d^3 + 3*B*a^2*d^2*e + 3*A*a ^2*d*e^2)*x^3 + 1/2*(B*a^2*d^3 + 3*A*a^2*d^2*e)*x^2
Time = 0.04 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.47 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^2 \, dx=A a^{2} d^{3} x + \frac {B c^{2} e^{3} x^{9}}{9} + x^{8} \left (\frac {A c^{2} e^{3}}{8} + \frac {3 B c^{2} d e^{2}}{8}\right ) + x^{7} \cdot \left (\frac {3 A c^{2} d e^{2}}{7} + \frac {2 B a c e^{3}}{7} + \frac {3 B c^{2} d^{2} e}{7}\right ) + x^{6} \left (\frac {A a c e^{3}}{3} + \frac {A c^{2} d^{2} e}{2} + B a c d e^{2} + \frac {B c^{2} d^{3}}{6}\right ) + x^{5} \cdot \left (\frac {6 A a c d e^{2}}{5} + \frac {A c^{2} d^{3}}{5} + \frac {B a^{2} e^{3}}{5} + \frac {6 B a c d^{2} e}{5}\right ) + x^{4} \left (\frac {A a^{2} e^{3}}{4} + \frac {3 A a c d^{2} e}{2} + \frac {3 B a^{2} d e^{2}}{4} + \frac {B a c d^{3}}{2}\right ) + x^{3} \left (A a^{2} d e^{2} + \frac {2 A a c d^{3}}{3} + B a^{2} d^{2} e\right ) + x^{2} \cdot \left (\frac {3 A a^{2} d^{2} e}{2} + \frac {B a^{2} d^{3}}{2}\right ) \]
A*a**2*d**3*x + B*c**2*e**3*x**9/9 + x**8*(A*c**2*e**3/8 + 3*B*c**2*d*e**2 /8) + x**7*(3*A*c**2*d*e**2/7 + 2*B*a*c*e**3/7 + 3*B*c**2*d**2*e/7) + x**6 *(A*a*c*e**3/3 + A*c**2*d**2*e/2 + B*a*c*d*e**2 + B*c**2*d**3/6) + x**5*(6 *A*a*c*d*e**2/5 + A*c**2*d**3/5 + B*a**2*e**3/5 + 6*B*a*c*d**2*e/5) + x**4 *(A*a**2*e**3/4 + 3*A*a*c*d**2*e/2 + 3*B*a**2*d*e**2/4 + B*a*c*d**3/2) + x **3*(A*a**2*d*e**2 + 2*A*a*c*d**3/3 + B*a**2*d**2*e) + x**2*(3*A*a**2*d**2 *e/2 + B*a**2*d**3/2)
Time = 0.20 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.26 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^2 \, dx=\frac {1}{9} \, B c^{2} e^{3} x^{9} + \frac {1}{8} \, {\left (3 \, B c^{2} d e^{2} + A c^{2} e^{3}\right )} x^{8} + \frac {1}{7} \, {\left (3 \, B c^{2} d^{2} e + 3 \, A c^{2} d e^{2} + 2 \, B a c e^{3}\right )} x^{7} + A a^{2} d^{3} x + \frac {1}{6} \, {\left (B c^{2} d^{3} + 3 \, A c^{2} d^{2} e + 6 \, B a c d e^{2} + 2 \, A a c e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (A c^{2} d^{3} + 6 \, B a c d^{2} e + 6 \, A a c d e^{2} + B a^{2} e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (2 \, B a c d^{3} + 6 \, A a c d^{2} e + 3 \, B a^{2} d e^{2} + A a^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (2 \, A a c d^{3} + 3 \, B a^{2} d^{2} e + 3 \, A a^{2} d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{2} d^{3} + 3 \, A a^{2} d^{2} e\right )} x^{2} \]
1/9*B*c^2*e^3*x^9 + 1/8*(3*B*c^2*d*e^2 + A*c^2*e^3)*x^8 + 1/7*(3*B*c^2*d^2 *e + 3*A*c^2*d*e^2 + 2*B*a*c*e^3)*x^7 + A*a^2*d^3*x + 1/6*(B*c^2*d^3 + 3*A *c^2*d^2*e + 6*B*a*c*d*e^2 + 2*A*a*c*e^3)*x^6 + 1/5*(A*c^2*d^3 + 6*B*a*c*d ^2*e + 6*A*a*c*d*e^2 + B*a^2*e^3)*x^5 + 1/4*(2*B*a*c*d^3 + 6*A*a*c*d^2*e + 3*B*a^2*d*e^2 + A*a^2*e^3)*x^4 + 1/3*(2*A*a*c*d^3 + 3*B*a^2*d^2*e + 3*A*a ^2*d*e^2)*x^3 + 1/2*(B*a^2*d^3 + 3*A*a^2*d^2*e)*x^2
Time = 0.27 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.39 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^2 \, dx=\frac {1}{9} \, B c^{2} e^{3} x^{9} + \frac {3}{8} \, B c^{2} d e^{2} x^{8} + \frac {1}{8} \, A c^{2} e^{3} x^{8} + \frac {3}{7} \, B c^{2} d^{2} e x^{7} + \frac {3}{7} \, A c^{2} d e^{2} x^{7} + \frac {2}{7} \, B a c e^{3} x^{7} + \frac {1}{6} \, B c^{2} d^{3} x^{6} + \frac {1}{2} \, A c^{2} d^{2} e x^{6} + B a c d e^{2} x^{6} + \frac {1}{3} \, A a c e^{3} x^{6} + \frac {1}{5} \, A c^{2} d^{3} x^{5} + \frac {6}{5} \, B a c d^{2} e x^{5} + \frac {6}{5} \, A a c d e^{2} x^{5} + \frac {1}{5} \, B a^{2} e^{3} x^{5} + \frac {1}{2} \, B a c d^{3} x^{4} + \frac {3}{2} \, A a c d^{2} e x^{4} + \frac {3}{4} \, B a^{2} d e^{2} x^{4} + \frac {1}{4} \, A a^{2} e^{3} x^{4} + \frac {2}{3} \, A a c d^{3} x^{3} + B a^{2} d^{2} e x^{3} + A a^{2} d e^{2} x^{3} + \frac {1}{2} \, B a^{2} d^{3} x^{2} + \frac {3}{2} \, A a^{2} d^{2} e x^{2} + A a^{2} d^{3} x \]
1/9*B*c^2*e^3*x^9 + 3/8*B*c^2*d*e^2*x^8 + 1/8*A*c^2*e^3*x^8 + 3/7*B*c^2*d^ 2*e*x^7 + 3/7*A*c^2*d*e^2*x^7 + 2/7*B*a*c*e^3*x^7 + 1/6*B*c^2*d^3*x^6 + 1/ 2*A*c^2*d^2*e*x^6 + B*a*c*d*e^2*x^6 + 1/3*A*a*c*e^3*x^6 + 1/5*A*c^2*d^3*x^ 5 + 6/5*B*a*c*d^2*e*x^5 + 6/5*A*a*c*d*e^2*x^5 + 1/5*B*a^2*e^3*x^5 + 1/2*B* a*c*d^3*x^4 + 3/2*A*a*c*d^2*e*x^4 + 3/4*B*a^2*d*e^2*x^4 + 1/4*A*a^2*e^3*x^ 4 + 2/3*A*a*c*d^3*x^3 + B*a^2*d^2*e*x^3 + A*a^2*d*e^2*x^3 + 1/2*B*a^2*d^3* x^2 + 3/2*A*a^2*d^2*e*x^2 + A*a^2*d^3*x
Time = 10.56 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.11 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right )^2 \, dx=x^5\,\left (\frac {B\,a^2\,e^3}{5}+\frac {6\,B\,a\,c\,d^2\,e}{5}+\frac {6\,A\,a\,c\,d\,e^2}{5}+\frac {A\,c^2\,d^3}{5}\right )+\frac {a\,x^4\,\left (2\,B\,c\,d^3+6\,A\,c\,d^2\,e+3\,B\,a\,d\,e^2+A\,a\,e^3\right )}{4}+\frac {c\,x^6\,\left (B\,c\,d^3+3\,A\,c\,d^2\,e+6\,B\,a\,d\,e^2+2\,A\,a\,e^3\right )}{6}+\frac {a^2\,d^2\,x^2\,\left (3\,A\,e+B\,d\right )}{2}+\frac {c^2\,e^2\,x^8\,\left (A\,e+3\,B\,d\right )}{8}+A\,a^2\,d^3\,x+\frac {a\,d\,x^3\,\left (2\,A\,c\,d^2+3\,B\,a\,d\,e+3\,A\,a\,e^2\right )}{3}+\frac {c\,e\,x^7\,\left (3\,B\,c\,d^2+3\,A\,c\,d\,e+2\,B\,a\,e^2\right )}{7}+\frac {B\,c^2\,e^3\,x^9}{9} \]
x^5*((A*c^2*d^3)/5 + (B*a^2*e^3)/5 + (6*A*a*c*d*e^2)/5 + (6*B*a*c*d^2*e)/5 ) + (a*x^4*(A*a*e^3 + 2*B*c*d^3 + 3*B*a*d*e^2 + 6*A*c*d^2*e))/4 + (c*x^6*( 2*A*a*e^3 + B*c*d^3 + 6*B*a*d*e^2 + 3*A*c*d^2*e))/6 + (a^2*d^2*x^2*(3*A*e + B*d))/2 + (c^2*e^2*x^8*(A*e + 3*B*d))/8 + A*a^2*d^3*x + (a*d*x^3*(3*A*a* e^2 + 2*A*c*d^2 + 3*B*a*d*e))/3 + (c*e*x^7*(2*B*a*e^2 + 3*B*c*d^2 + 3*A*c* d*e))/7 + (B*c^2*e^3*x^9)/9