Integrand size = 28, antiderivative size = 244 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=a^2 c^2 e^2 x+\frac {2}{3} a c e (b c e+a d e+a c f) x^3+\frac {1}{5} \left (b^2 c^2 e^2+4 a b c e (d e+c f)+a^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^5+\frac {2}{7} \left (b^2 c e (d e+c f)+a^2 d f (d e+c f)+a b \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^7+\frac {1}{9} \left (a^2 d^2 f^2+4 a b d f (d e+c f)+b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^9+\frac {2}{11} b d f (b d e+b c f+a d f) x^{11}+\frac {1}{13} b^2 d^2 f^2 x^{13} \] Output:
a^2*c^2*e^2*x+2/3*a*c*e*(a*c*f+a*d*e+b*c*e)*x^3+1/5*(b^2*c^2*e^2+4*a*b*c*e *(c*f+d*e)+a^2*(c^2*f^2+4*c*d*e*f+d^2*e^2))*x^5+2/7*(b^2*c*e*(c*f+d*e)+a^2 *d*f*(c*f+d*e)+a*b*(c^2*f^2+4*c*d*e*f+d^2*e^2))*x^7+1/9*(a^2*d^2*f^2+4*a*b *d*f*(c*f+d*e)+b^2*(c^2*f^2+4*c*d*e*f+d^2*e^2))*x^9+2/11*b*d*f*(a*d*f+b*c* f+b*d*e)*x^11+1/13*b^2*d^2*f^2*x^13
Time = 0.09 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=a^2 c^2 e^2 x+\frac {2}{3} a c e (b c e+a d e+a c f) x^3+\frac {1}{5} \left (b^2 c^2 e^2+4 a b c e (d e+c f)+a^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^5+\frac {2}{7} \left (b^2 c e (d e+c f)+a^2 d f (d e+c f)+a b \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^7+\frac {1}{9} \left (a^2 d^2 f^2+4 a b d f (d e+c f)+b^2 \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^9+\frac {2}{11} b d f (b d e+b c f+a d f) x^{11}+\frac {1}{13} b^2 d^2 f^2 x^{13} \] Input:
Integrate[(a + b*x^2)^2*(c + d*x^2)^2*(e + f*x^2)^2,x]
Output:
a^2*c^2*e^2*x + (2*a*c*e*(b*c*e + a*d*e + a*c*f)*x^3)/3 + ((b^2*c^2*e^2 + 4*a*b*c*e*(d*e + c*f) + a^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^5)/5 + (2*( b^2*c*e*(d*e + c*f) + a^2*d*f*(d*e + c*f) + a*b*(d^2*e^2 + 4*c*d*e*f + c^2 *f^2))*x^7)/7 + ((a^2*d^2*f^2 + 4*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + 4*c *d*e*f + c^2*f^2))*x^9)/9 + (2*b*d*f*(b*d*e + b*c*f + a*d*f)*x^11)/11 + (b ^2*d^2*f^2*x^13)/13
Time = 0.48 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {396, 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.
| \(\displaystyle \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 396 |
| \(\displaystyle \int \left (x^8 \left (a^2 d^2 f^2+4 a b d f (c f+d e)+b^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+2 x^6 \left (a^2 d f (c f+d e)+a b \left (c^2 f^2+4 c d e f+d^2 e^2\right )+b^2 c e (c f+d e)\right )+x^4 \left (a^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )+4 a b c e (c f+d e)+b^2 c^2 e^2\right )+a^2 c^2 e^2+2 b d f x^{10} (a d f+b c f+b d e)+2 a c e x^2 (a c f+a d e+b c e)+b^2 d^2 f^2 x^{12}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{9} x^9 \left (a^2 d^2 f^2+4 a b d f (c f+d e)+b^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+\frac {2}{7} x^7 \left (a^2 d f (c f+d e)+a b \left (c^2 f^2+4 c d e f+d^2 e^2\right )+b^2 c e (c f+d e)\right )+\frac {1}{5} x^5 \left (a^2 \left (c^2 f^2+4 c d e f+d^2 e^2\right )+4 a b c e (c f+d e)+b^2 c^2 e^2\right )+a^2 c^2 e^2 x+\frac {2}{11} b d f x^{11} (a d f+b c f+b d e)+\frac {2}{3} a c e x^3 (a c f+a d e+b c e)+\frac {1}{13} b^2 d^2 f^2 x^{13}\) |
Input:
Int[(a + b*x^2)^2*(c + d*x^2)^2*(e + f*x^2)^2,x]
Output:
a^2*c^2*e^2*x + (2*a*c*e*(b*c*e + a*d*e + a*c*f)*x^3)/3 + ((b^2*c^2*e^2 + 4*a*b*c*e*(d*e + c*f) + a^2*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^5)/5 + (2*( b^2*c*e*(d*e + c*f) + a^2*d*f*(d*e + c*f) + a*b*(d^2*e^2 + 4*c*d*e*f + c^2 *f^2))*x^7)/7 + ((a^2*d^2*f^2 + 4*a*b*d*f*(d*e + c*f) + b^2*(d^2*e^2 + 4*c *d*e*f + c^2*f^2))*x^9)/9 + (2*b*d*f*(b*d*e + b*c*f + a*d*f)*x^11)/11 + (b ^2*d^2*f^2*x^13)/13
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2)^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2)^q* (e + f*x^2)^r, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IGtQ [q, 0] && IGtQ[r, 0]
Time = 0.58 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.17
| method | result | size |
| default | \(\frac {b^{2} d^{2} f^{2} x^{13}}{13}+\frac {\left (\left (2 a b \,d^{2}+2 b^{2} c d \right ) f^{2}+2 b^{2} d^{2} e f \right ) x^{11}}{11}+\frac {\left (\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) f^{2}+2 \left (2 a b \,d^{2}+2 b^{2} c d \right ) e f +b^{2} d^{2} e^{2}\right ) x^{9}}{9}+\frac {\left (\left (2 a^{2} c d +2 b \,c^{2} a \right ) f^{2}+2 \left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) e f +\left (2 a b \,d^{2}+2 b^{2} c d \right ) e^{2}\right ) x^{7}}{7}+\frac {\left (a^{2} c^{2} f^{2}+2 \left (2 a^{2} c d +2 b \,c^{2} a \right ) e f +\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) e^{2}\right ) x^{5}}{5}+\frac {\left (2 a^{2} c^{2} e f +\left (2 a^{2} c d +2 b \,c^{2} a \right ) e^{2}\right ) x^{3}}{3}+a^{2} c^{2} e^{2} x\) | \(286\) |
| norman | \(\frac {b^{2} d^{2} f^{2} x^{13}}{13}+\left (\frac {2}{11} a \,d^{2} f^{2} b +\frac {2}{11} b^{2} c d \,f^{2}+\frac {2}{11} b^{2} d^{2} e f \right ) x^{11}+\left (\frac {1}{9} a^{2} d^{2} f^{2}+\frac {4}{9} a b c d \,f^{2}+\frac {4}{9} a b \,d^{2} e f +\frac {1}{9} b^{2} c^{2} f^{2}+\frac {4}{9} b^{2} c d e f +\frac {1}{9} b^{2} d^{2} e^{2}\right ) x^{9}+\left (\frac {2}{7} a^{2} c d \,f^{2}+\frac {2}{7} a^{2} d^{2} e f +\frac {2}{7} a b \,c^{2} f^{2}+\frac {8}{7} a b c d e f +\frac {2}{7} a b \,d^{2} e^{2}+\frac {2}{7} b^{2} c^{2} e f +\frac {2}{7} b^{2} d \,e^{2} c \right ) x^{7}+\left (\frac {1}{5} a^{2} c^{2} f^{2}+\frac {4}{5} a^{2} c d e f +\frac {1}{5} a^{2} d^{2} e^{2}+\frac {4}{5} a b \,c^{2} e f +\frac {4}{5} a b c d \,e^{2}+\frac {1}{5} b^{2} c^{2} e^{2}\right ) x^{5}+\left (\frac {2}{3} a^{2} c^{2} e f +\frac {2}{3} a^{2} c d \,e^{2}+\frac {2}{3} a b \,c^{2} e^{2}\right ) x^{3}+a^{2} c^{2} e^{2} x\) | \(300\) |
| gosper | \(\frac {1}{5} x^{5} a^{2} c^{2} f^{2}+\frac {2}{3} x^{3} a^{2} c^{2} e f +\frac {2}{7} x^{7} a b \,c^{2} f^{2}+\frac {2}{7} x^{7} a b \,d^{2} e^{2}+\frac {2}{7} x^{7} b^{2} c^{2} e f +\frac {1}{5} x^{5} a^{2} d^{2} e^{2}+\frac {2}{11} x^{11} a \,d^{2} f^{2} b +\frac {2}{11} x^{11} b^{2} c d \,f^{2}+\frac {2}{3} x^{3} a b \,c^{2} e^{2}+\frac {1}{5} x^{5} b^{2} c^{2} e^{2}+\frac {1}{9} x^{9} b^{2} d^{2} e^{2}+\frac {2}{7} x^{7} a^{2} d^{2} e f +\frac {4}{9} x^{9} a b c d \,f^{2}+\frac {4}{9} x^{9} a b \,d^{2} e f +\frac {8}{7} x^{7} a b c d e f +\frac {4}{9} x^{9} b^{2} c d e f +\frac {4}{5} x^{5} a^{2} c d e f +\frac {4}{5} x^{5} a b \,c^{2} e f +\frac {4}{5} x^{5} a b c d \,e^{2}+\frac {2}{7} x^{7} b^{2} d \,e^{2} c +a^{2} c^{2} e^{2} x +\frac {2}{3} x^{3} a^{2} c d \,e^{2}+\frac {1}{9} x^{9} a^{2} d^{2} f^{2}+\frac {2}{11} x^{11} b^{2} d^{2} e f +\frac {2}{7} x^{7} a^{2} c d \,f^{2}+\frac {1}{9} x^{9} b^{2} c^{2} f^{2}+\frac {1}{13} b^{2} d^{2} f^{2} x^{13}\) | \(350\) |
| risch | \(\frac {1}{5} x^{5} a^{2} c^{2} f^{2}+\frac {2}{3} x^{3} a^{2} c^{2} e f +\frac {2}{7} x^{7} a b \,c^{2} f^{2}+\frac {2}{7} x^{7} a b \,d^{2} e^{2}+\frac {2}{7} x^{7} b^{2} c^{2} e f +\frac {1}{5} x^{5} a^{2} d^{2} e^{2}+\frac {2}{11} x^{11} a \,d^{2} f^{2} b +\frac {2}{11} x^{11} b^{2} c d \,f^{2}+\frac {2}{3} x^{3} a b \,c^{2} e^{2}+\frac {1}{5} x^{5} b^{2} c^{2} e^{2}+\frac {1}{9} x^{9} b^{2} d^{2} e^{2}+\frac {2}{7} x^{7} a^{2} d^{2} e f +\frac {4}{9} x^{9} a b c d \,f^{2}+\frac {4}{9} x^{9} a b \,d^{2} e f +\frac {8}{7} x^{7} a b c d e f +\frac {4}{9} x^{9} b^{2} c d e f +\frac {4}{5} x^{5} a^{2} c d e f +\frac {4}{5} x^{5} a b \,c^{2} e f +\frac {4}{5} x^{5} a b c d \,e^{2}+\frac {2}{7} x^{7} b^{2} d \,e^{2} c +a^{2} c^{2} e^{2} x +\frac {2}{3} x^{3} a^{2} c d \,e^{2}+\frac {1}{9} x^{9} a^{2} d^{2} f^{2}+\frac {2}{11} x^{11} b^{2} d^{2} e f +\frac {2}{7} x^{7} a^{2} c d \,f^{2}+\frac {1}{9} x^{9} b^{2} c^{2} f^{2}+\frac {1}{13} b^{2} d^{2} f^{2} x^{13}\) | \(350\) |
| parallelrisch | \(\frac {1}{5} x^{5} a^{2} c^{2} f^{2}+\frac {2}{3} x^{3} a^{2} c^{2} e f +\frac {2}{7} x^{7} a b \,c^{2} f^{2}+\frac {2}{7} x^{7} a b \,d^{2} e^{2}+\frac {2}{7} x^{7} b^{2} c^{2} e f +\frac {1}{5} x^{5} a^{2} d^{2} e^{2}+\frac {2}{11} x^{11} a \,d^{2} f^{2} b +\frac {2}{11} x^{11} b^{2} c d \,f^{2}+\frac {2}{3} x^{3} a b \,c^{2} e^{2}+\frac {1}{5} x^{5} b^{2} c^{2} e^{2}+\frac {1}{9} x^{9} b^{2} d^{2} e^{2}+\frac {2}{7} x^{7} a^{2} d^{2} e f +\frac {4}{9} x^{9} a b c d \,f^{2}+\frac {4}{9} x^{9} a b \,d^{2} e f +\frac {8}{7} x^{7} a b c d e f +\frac {4}{9} x^{9} b^{2} c d e f +\frac {4}{5} x^{5} a^{2} c d e f +\frac {4}{5} x^{5} a b \,c^{2} e f +\frac {4}{5} x^{5} a b c d \,e^{2}+\frac {2}{7} x^{7} b^{2} d \,e^{2} c +a^{2} c^{2} e^{2} x +\frac {2}{3} x^{3} a^{2} c d \,e^{2}+\frac {1}{9} x^{9} a^{2} d^{2} f^{2}+\frac {2}{11} x^{11} b^{2} d^{2} e f +\frac {2}{7} x^{7} a^{2} c d \,f^{2}+\frac {1}{9} x^{9} b^{2} c^{2} f^{2}+\frac {1}{13} b^{2} d^{2} f^{2} x^{13}\) | \(350\) |
| orering | \(\frac {x \left (3465 b^{2} d^{2} f^{2} x^{12}+8190 a b \,d^{2} f^{2} x^{10}+8190 b^{2} c d \,f^{2} x^{10}+8190 b^{2} d^{2} e f \,x^{10}+5005 a^{2} d^{2} f^{2} x^{8}+20020 a b c d \,f^{2} x^{8}+20020 a b \,d^{2} e f \,x^{8}+5005 b^{2} c^{2} f^{2} x^{8}+20020 b^{2} c d e f \,x^{8}+5005 b^{2} d^{2} e^{2} x^{8}+12870 a^{2} c d \,f^{2} x^{6}+12870 a^{2} d^{2} e f \,x^{6}+12870 a b \,c^{2} f^{2} x^{6}+51480 a b c d e f \,x^{6}+12870 a b \,d^{2} e^{2} x^{6}+12870 b^{2} c^{2} e f \,x^{6}+12870 b^{2} c d \,e^{2} x^{6}+9009 a^{2} c^{2} f^{2} x^{4}+36036 a^{2} c d e f \,x^{4}+9009 a^{2} d^{2} e^{2} x^{4}+36036 a b \,c^{2} e f \,x^{4}+36036 a b c d \,e^{2} x^{4}+9009 b^{2} c^{2} e^{2} x^{4}+30030 a^{2} c^{2} e f \,x^{2}+30030 a^{2} c d \,e^{2} x^{2}+30030 a b \,c^{2} e^{2} x^{2}+45045 a^{2} c^{2} e^{2}\right )}{45045}\) | \(353\) |
Input:
int((b*x^2+a)^2*(d*x^2+c)^2*(f*x^2+e)^2,x,method=_RETURNVERBOSE)
Output:
1/13*b^2*d^2*f^2*x^13+1/11*((2*a*b*d^2+2*b^2*c*d)*f^2+2*b^2*d^2*e*f)*x^11+ 1/9*((a^2*d^2+4*a*b*c*d+b^2*c^2)*f^2+2*(2*a*b*d^2+2*b^2*c*d)*e*f+b^2*d^2*e ^2)*x^9+1/7*((2*a^2*c*d+2*a*b*c^2)*f^2+2*(a^2*d^2+4*a*b*c*d+b^2*c^2)*e*f+( 2*a*b*d^2+2*b^2*c*d)*e^2)*x^7+1/5*(a^2*c^2*f^2+2*(2*a^2*c*d+2*a*b*c^2)*e*f +(a^2*d^2+4*a*b*c*d+b^2*c^2)*e^2)*x^5+1/3*(2*a^2*c^2*e*f+(2*a^2*c*d+2*a*b* c^2)*e^2)*x^3+a^2*c^2*e^2*x
Time = 0.07 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.11 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {1}{13} \, b^{2} d^{2} f^{2} x^{13} + \frac {2}{11} \, {\left (b^{2} d^{2} e f + {\left (b^{2} c d + a b d^{2}\right )} f^{2}\right )} x^{11} + \frac {1}{9} \, {\left (b^{2} d^{2} e^{2} + 4 \, {\left (b^{2} c d + a b d^{2}\right )} e f + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2}\right )} x^{9} + \frac {2}{7} \, {\left ({\left (b^{2} c d + a b d^{2}\right )} e^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e f + {\left (a b c^{2} + a^{2} c d\right )} f^{2}\right )} x^{7} + a^{2} c^{2} e^{2} x + \frac {1}{5} \, {\left (a^{2} c^{2} f^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e^{2} + 4 \, {\left (a b c^{2} + a^{2} c d\right )} e f\right )} x^{5} + \frac {2}{3} \, {\left (a^{2} c^{2} e f + {\left (a b c^{2} + a^{2} c d\right )} e^{2}\right )} x^{3} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^2*(f*x^2+e)^2,x, algorithm="fricas")
Output:
1/13*b^2*d^2*f^2*x^13 + 2/11*(b^2*d^2*e*f + (b^2*c*d + a*b*d^2)*f^2)*x^11 + 1/9*(b^2*d^2*e^2 + 4*(b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 + 4*a*b*c*d + a^ 2*d^2)*f^2)*x^9 + 2/7*((b^2*c*d + a*b*d^2)*e^2 + (b^2*c^2 + 4*a*b*c*d + a^ 2*d^2)*e*f + (a*b*c^2 + a^2*c*d)*f^2)*x^7 + a^2*c^2*e^2*x + 1/5*(a^2*c^2*f ^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2 + 4*(a*b*c^2 + a^2*c*d)*e*f)*x^5 + 2/3*(a^2*c^2*e*f + (a*b*c^2 + a^2*c*d)*e^2)*x^3
Time = 0.04 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.52 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=a^{2} c^{2} e^{2} x + \frac {b^{2} d^{2} f^{2} x^{13}}{13} + x^{11} \cdot \left (\frac {2 a b d^{2} f^{2}}{11} + \frac {2 b^{2} c d f^{2}}{11} + \frac {2 b^{2} d^{2} e f}{11}\right ) + x^{9} \left (\frac {a^{2} d^{2} f^{2}}{9} + \frac {4 a b c d f^{2}}{9} + \frac {4 a b d^{2} e f}{9} + \frac {b^{2} c^{2} f^{2}}{9} + \frac {4 b^{2} c d e f}{9} + \frac {b^{2} d^{2} e^{2}}{9}\right ) + x^{7} \cdot \left (\frac {2 a^{2} c d f^{2}}{7} + \frac {2 a^{2} d^{2} e f}{7} + \frac {2 a b c^{2} f^{2}}{7} + \frac {8 a b c d e f}{7} + \frac {2 a b d^{2} e^{2}}{7} + \frac {2 b^{2} c^{2} e f}{7} + \frac {2 b^{2} c d e^{2}}{7}\right ) + x^{5} \left (\frac {a^{2} c^{2} f^{2}}{5} + \frac {4 a^{2} c d e f}{5} + \frac {a^{2} d^{2} e^{2}}{5} + \frac {4 a b c^{2} e f}{5} + \frac {4 a b c d e^{2}}{5} + \frac {b^{2} c^{2} e^{2}}{5}\right ) + x^{3} \cdot \left (\frac {2 a^{2} c^{2} e f}{3} + \frac {2 a^{2} c d e^{2}}{3} + \frac {2 a b c^{2} e^{2}}{3}\right ) \] Input:
integrate((b*x**2+a)**2*(d*x**2+c)**2*(f*x**2+e)**2,x)
Output:
a**2*c**2*e**2*x + b**2*d**2*f**2*x**13/13 + x**11*(2*a*b*d**2*f**2/11 + 2 *b**2*c*d*f**2/11 + 2*b**2*d**2*e*f/11) + x**9*(a**2*d**2*f**2/9 + 4*a*b*c *d*f**2/9 + 4*a*b*d**2*e*f/9 + b**2*c**2*f**2/9 + 4*b**2*c*d*e*f/9 + b**2* d**2*e**2/9) + x**7*(2*a**2*c*d*f**2/7 + 2*a**2*d**2*e*f/7 + 2*a*b*c**2*f* *2/7 + 8*a*b*c*d*e*f/7 + 2*a*b*d**2*e**2/7 + 2*b**2*c**2*e*f/7 + 2*b**2*c* d*e**2/7) + x**5*(a**2*c**2*f**2/5 + 4*a**2*c*d*e*f/5 + a**2*d**2*e**2/5 + 4*a*b*c**2*e*f/5 + 4*a*b*c*d*e**2/5 + b**2*c**2*e**2/5) + x**3*(2*a**2*c* *2*e*f/3 + 2*a**2*c*d*e**2/3 + 2*a*b*c**2*e**2/3)
Time = 0.03 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.11 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {1}{13} \, b^{2} d^{2} f^{2} x^{13} + \frac {2}{11} \, {\left (b^{2} d^{2} e f + {\left (b^{2} c d + a b d^{2}\right )} f^{2}\right )} x^{11} + \frac {1}{9} \, {\left (b^{2} d^{2} e^{2} + 4 \, {\left (b^{2} c d + a b d^{2}\right )} e f + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2}\right )} x^{9} + \frac {2}{7} \, {\left ({\left (b^{2} c d + a b d^{2}\right )} e^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e f + {\left (a b c^{2} + a^{2} c d\right )} f^{2}\right )} x^{7} + a^{2} c^{2} e^{2} x + \frac {1}{5} \, {\left (a^{2} c^{2} f^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} e^{2} + 4 \, {\left (a b c^{2} + a^{2} c d\right )} e f\right )} x^{5} + \frac {2}{3} \, {\left (a^{2} c^{2} e f + {\left (a b c^{2} + a^{2} c d\right )} e^{2}\right )} x^{3} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^2*(f*x^2+e)^2,x, algorithm="maxima")
Output:
1/13*b^2*d^2*f^2*x^13 + 2/11*(b^2*d^2*e*f + (b^2*c*d + a*b*d^2)*f^2)*x^11 + 1/9*(b^2*d^2*e^2 + 4*(b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 + 4*a*b*c*d + a^ 2*d^2)*f^2)*x^9 + 2/7*((b^2*c*d + a*b*d^2)*e^2 + (b^2*c^2 + 4*a*b*c*d + a^ 2*d^2)*e*f + (a*b*c^2 + a^2*c*d)*f^2)*x^7 + a^2*c^2*e^2*x + 1/5*(a^2*c^2*f ^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*e^2 + 4*(a*b*c^2 + a^2*c*d)*e*f)*x^5 + 2/3*(a^2*c^2*e*f + (a*b*c^2 + a^2*c*d)*e^2)*x^3
Time = 0.13 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.43 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {1}{13} \, b^{2} d^{2} f^{2} x^{13} + \frac {2}{11} \, b^{2} d^{2} e f x^{11} + \frac {2}{11} \, b^{2} c d f^{2} x^{11} + \frac {2}{11} \, a b d^{2} f^{2} x^{11} + \frac {1}{9} \, b^{2} d^{2} e^{2} x^{9} + \frac {4}{9} \, b^{2} c d e f x^{9} + \frac {4}{9} \, a b d^{2} e f x^{9} + \frac {1}{9} \, b^{2} c^{2} f^{2} x^{9} + \frac {4}{9} \, a b c d f^{2} x^{9} + \frac {1}{9} \, a^{2} d^{2} f^{2} x^{9} + \frac {2}{7} \, b^{2} c d e^{2} x^{7} + \frac {2}{7} \, a b d^{2} e^{2} x^{7} + \frac {2}{7} \, b^{2} c^{2} e f x^{7} + \frac {8}{7} \, a b c d e f x^{7} + \frac {2}{7} \, a^{2} d^{2} e f x^{7} + \frac {2}{7} \, a b c^{2} f^{2} x^{7} + \frac {2}{7} \, a^{2} c d f^{2} x^{7} + \frac {1}{5} \, b^{2} c^{2} e^{2} x^{5} + \frac {4}{5} \, a b c d e^{2} x^{5} + \frac {1}{5} \, a^{2} d^{2} e^{2} x^{5} + \frac {4}{5} \, a b c^{2} e f x^{5} + \frac {4}{5} \, a^{2} c d e f x^{5} + \frac {1}{5} \, a^{2} c^{2} f^{2} x^{5} + \frac {2}{3} \, a b c^{2} e^{2} x^{3} + \frac {2}{3} \, a^{2} c d e^{2} x^{3} + \frac {2}{3} \, a^{2} c^{2} e f x^{3} + a^{2} c^{2} e^{2} x \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^2*(f*x^2+e)^2,x, algorithm="giac")
Output:
1/13*b^2*d^2*f^2*x^13 + 2/11*b^2*d^2*e*f*x^11 + 2/11*b^2*c*d*f^2*x^11 + 2/ 11*a*b*d^2*f^2*x^11 + 1/9*b^2*d^2*e^2*x^9 + 4/9*b^2*c*d*e*f*x^9 + 4/9*a*b* d^2*e*f*x^9 + 1/9*b^2*c^2*f^2*x^9 + 4/9*a*b*c*d*f^2*x^9 + 1/9*a^2*d^2*f^2* x^9 + 2/7*b^2*c*d*e^2*x^7 + 2/7*a*b*d^2*e^2*x^7 + 2/7*b^2*c^2*e*f*x^7 + 8/ 7*a*b*c*d*e*f*x^7 + 2/7*a^2*d^2*e*f*x^7 + 2/7*a*b*c^2*f^2*x^7 + 2/7*a^2*c* d*f^2*x^7 + 1/5*b^2*c^2*e^2*x^5 + 4/5*a*b*c*d*e^2*x^5 + 1/5*a^2*d^2*e^2*x^ 5 + 4/5*a*b*c^2*e*f*x^5 + 4/5*a^2*c*d*e*f*x^5 + 1/5*a^2*c^2*f^2*x^5 + 2/3* a*b*c^2*e^2*x^3 + 2/3*a^2*c*d*e^2*x^3 + 2/3*a^2*c^2*e*f*x^3 + a^2*c^2*e^2* x
Time = 2.08 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.11 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=x^7\,\left (\frac {2\,a^2\,c\,d\,f^2}{7}+\frac {2\,a^2\,d^2\,e\,f}{7}+\frac {2\,a\,b\,c^2\,f^2}{7}+\frac {8\,a\,b\,c\,d\,e\,f}{7}+\frac {2\,a\,b\,d^2\,e^2}{7}+\frac {2\,b^2\,c^2\,e\,f}{7}+\frac {2\,b^2\,c\,d\,e^2}{7}\right )+x^5\,\left (\frac {a^2\,c^2\,f^2}{5}+\frac {4\,a^2\,c\,d\,e\,f}{5}+\frac {a^2\,d^2\,e^2}{5}+\frac {4\,a\,b\,c^2\,e\,f}{5}+\frac {4\,a\,b\,c\,d\,e^2}{5}+\frac {b^2\,c^2\,e^2}{5}\right )+x^9\,\left (\frac {a^2\,d^2\,f^2}{9}+\frac {4\,a\,b\,c\,d\,f^2}{9}+\frac {4\,a\,b\,d^2\,e\,f}{9}+\frac {b^2\,c^2\,f^2}{9}+\frac {4\,b^2\,c\,d\,e\,f}{9}+\frac {b^2\,d^2\,e^2}{9}\right )+a^2\,c^2\,e^2\,x+\frac {b^2\,d^2\,f^2\,x^{13}}{13}+\frac {2\,a\,c\,e\,x^3\,\left (a\,c\,f+a\,d\,e+b\,c\,e\right )}{3}+\frac {2\,b\,d\,f\,x^{11}\,\left (a\,d\,f+b\,c\,f+b\,d\,e\right )}{11} \] Input:
int((a + b*x^2)^2*(c + d*x^2)^2*(e + f*x^2)^2,x)
Output:
x^7*((2*a*b*c^2*f^2)/7 + (2*a*b*d^2*e^2)/7 + (2*a^2*c*d*f^2)/7 + (2*b^2*c* d*e^2)/7 + (2*a^2*d^2*e*f)/7 + (2*b^2*c^2*e*f)/7 + (8*a*b*c*d*e*f)/7) + x^ 5*((a^2*c^2*f^2)/5 + (a^2*d^2*e^2)/5 + (b^2*c^2*e^2)/5 + (4*a*b*c*d*e^2)/5 + (4*a*b*c^2*e*f)/5 + (4*a^2*c*d*e*f)/5) + x^9*((a^2*d^2*f^2)/9 + (b^2*c^ 2*f^2)/9 + (b^2*d^2*e^2)/9 + (4*a*b*c*d*f^2)/9 + (4*a*b*d^2*e*f)/9 + (4*b^ 2*c*d*e*f)/9) + a^2*c^2*e^2*x + (b^2*d^2*f^2*x^13)/13 + (2*a*c*e*x^3*(a*c* f + a*d*e + b*c*e))/3 + (2*b*d*f*x^11*(a*d*f + b*c*f + b*d*e))/11
Time = 0.15 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.44 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {x \left (3465 b^{2} d^{2} f^{2} x^{12}+8190 a b \,d^{2} f^{2} x^{10}+8190 b^{2} c d \,f^{2} x^{10}+8190 b^{2} d^{2} e f \,x^{10}+5005 a^{2} d^{2} f^{2} x^{8}+20020 a b c d \,f^{2} x^{8}+20020 a b \,d^{2} e f \,x^{8}+5005 b^{2} c^{2} f^{2} x^{8}+20020 b^{2} c d e f \,x^{8}+5005 b^{2} d^{2} e^{2} x^{8}+12870 a^{2} c d \,f^{2} x^{6}+12870 a^{2} d^{2} e f \,x^{6}+12870 a b \,c^{2} f^{2} x^{6}+51480 a b c d e f \,x^{6}+12870 a b \,d^{2} e^{2} x^{6}+12870 b^{2} c^{2} e f \,x^{6}+12870 b^{2} c d \,e^{2} x^{6}+9009 a^{2} c^{2} f^{2} x^{4}+36036 a^{2} c d e f \,x^{4}+9009 a^{2} d^{2} e^{2} x^{4}+36036 a b \,c^{2} e f \,x^{4}+36036 a b c d \,e^{2} x^{4}+9009 b^{2} c^{2} e^{2} x^{4}+30030 a^{2} c^{2} e f \,x^{2}+30030 a^{2} c d \,e^{2} x^{2}+30030 a b \,c^{2} e^{2} x^{2}+45045 a^{2} c^{2} e^{2}\right )}{45045} \] Input:
int((b*x^2+a)^2*(d*x^2+c)^2*(f*x^2+e)^2,x)
Output:
(x*(45045*a**2*c**2*e**2 + 30030*a**2*c**2*e*f*x**2 + 9009*a**2*c**2*f**2* x**4 + 30030*a**2*c*d*e**2*x**2 + 36036*a**2*c*d*e*f*x**4 + 12870*a**2*c*d *f**2*x**6 + 9009*a**2*d**2*e**2*x**4 + 12870*a**2*d**2*e*f*x**6 + 5005*a* *2*d**2*f**2*x**8 + 30030*a*b*c**2*e**2*x**2 + 36036*a*b*c**2*e*f*x**4 + 1 2870*a*b*c**2*f**2*x**6 + 36036*a*b*c*d*e**2*x**4 + 51480*a*b*c*d*e*f*x**6 + 20020*a*b*c*d*f**2*x**8 + 12870*a*b*d**2*e**2*x**6 + 20020*a*b*d**2*e*f *x**8 + 8190*a*b*d**2*f**2*x**10 + 9009*b**2*c**2*e**2*x**4 + 12870*b**2*c **2*e*f*x**6 + 5005*b**2*c**2*f**2*x**8 + 12870*b**2*c*d*e**2*x**6 + 20020 *b**2*c*d*e*f*x**8 + 8190*b**2*c*d*f**2*x**10 + 5005*b**2*d**2*e**2*x**8 + 8190*b**2*d**2*e*f*x**10 + 3465*b**2*d**2*f**2*x**12))/45045