Integrand size = 28, antiderivative size = 493 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=a^2 c^3 e^3 x+\frac {1}{3} a c^2 e^2 (2 b c e+3 a (d e+c f)) x^3+\frac {1}{5} c e \left (b^2 c^2 e^2+6 a b c e (d e+c f)+3 a^2 \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (3 b^2 c^2 e^2 (d e+c f)+6 a b c e \left (d^2 e^2+3 c d e f+c^2 f^2\right )+a^2 \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^7+\frac {1}{9} \left (3 b^2 c e \left (d^2 e^2+3 c d e f+c^2 f^2\right )+3 a^2 d f \left (d^2 e^2+3 c d e f+c^2 f^2\right )+2 a b \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^9+\frac {1}{11} \left (3 a^2 d^2 f^2 (d e+c f)+6 a b d f \left (d^2 e^2+3 c d e f+c^2 f^2\right )+b^2 \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^{11}+\frac {1}{13} d f \left (a^2 d^2 f^2+6 a b d f (d e+c f)+3 b^2 \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^{13}+\frac {1}{15} b d^2 f^2 (2 a d f+3 b (d e+c f)) x^{15}+\frac {1}{17} b^2 d^3 f^3 x^{17} \] Output:
a^2*c^3*e^3*x+1/3*a*c^2*e^2*(2*b*c*e+3*a*(c*f+d*e))*x^3+1/5*c*e*(b^2*c^2*e ^2+6*a*b*c*e*(c*f+d*e)+3*a^2*(c^2*f^2+3*c*d*e*f+d^2*e^2))*x^5+1/7*(3*b^2*c ^2*e^2*(c*f+d*e)+6*a*b*c*e*(c^2*f^2+3*c*d*e*f+d^2*e^2)+a^2*(c^3*f^3+9*c^2* d*e*f^2+9*c*d^2*e^2*f+d^3*e^3))*x^7+1/9*(3*b^2*c*e*(c^2*f^2+3*c*d*e*f+d^2* e^2)+3*a^2*d*f*(c^2*f^2+3*c*d*e*f+d^2*e^2)+2*a*b*(c^3*f^3+9*c^2*d*e*f^2+9* c*d^2*e^2*f+d^3*e^3))*x^9+1/11*(3*a^2*d^2*f^2*(c*f+d*e)+6*a*b*d*f*(c^2*f^2 +3*c*d*e*f+d^2*e^2)+b^2*(c^3*f^3+9*c^2*d*e*f^2+9*c*d^2*e^2*f+d^3*e^3))*x^1 1+1/13*d*f*(a^2*d^2*f^2+6*a*b*d*f*(c*f+d*e)+3*b^2*(c^2*f^2+3*c*d*e*f+d^2*e ^2))*x^13+1/15*b*d^2*f^2*(2*a*d*f+3*b*(c*f+d*e))*x^15+1/17*b^2*d^3*f^3*x^1 7
Time = 0.17 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=a^2 c^3 e^3 x+\frac {1}{3} a c^2 e^2 (2 b c e+3 a (d e+c f)) x^3+\frac {1}{5} c e \left (b^2 c^2 e^2+6 a b c e (d e+c f)+3 a^2 \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (3 b^2 c^2 e^2 (d e+c f)+6 a b c e \left (d^2 e^2+3 c d e f+c^2 f^2\right )+a^2 \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^7+\frac {1}{9} \left (3 b^2 c e \left (d^2 e^2+3 c d e f+c^2 f^2\right )+3 a^2 d f \left (d^2 e^2+3 c d e f+c^2 f^2\right )+2 a b \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^9+\frac {1}{11} \left (3 a^2 d^2 f^2 (d e+c f)+6 a b d f \left (d^2 e^2+3 c d e f+c^2 f^2\right )+b^2 \left (d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+c^3 f^3\right )\right ) x^{11}+\frac {1}{13} d f \left (a^2 d^2 f^2+6 a b d f (d e+c f)+3 b^2 \left (d^2 e^2+3 c d e f+c^2 f^2\right )\right ) x^{13}+\frac {1}{15} b d^2 f^2 (2 a d f+3 b (d e+c f)) x^{15}+\frac {1}{17} b^2 d^3 f^3 x^{17} \] Input:
Integrate[(a + b*x^2)^2*(c + d*x^2)^3*(e + f*x^2)^3,x]
Output:
a^2*c^3*e^3*x + (a*c^2*e^2*(2*b*c*e + 3*a*(d*e + c*f))*x^3)/3 + (c*e*(b^2* c^2*e^2 + 6*a*b*c*e*(d*e + c*f) + 3*a^2*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x ^5)/5 + ((3*b^2*c^2*e^2*(d*e + c*f) + 6*a*b*c*e*(d^2*e^2 + 3*c*d*e*f + c^2 *f^2) + a^2*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3*f^3))*x^7)/7 + ((3*b^2*c*e*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + 3*a^2*d*f*(d^2*e^2 + 3*c*d*e *f + c^2*f^2) + 2*a*b*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3*f^3)) *x^9)/9 + ((3*a^2*d^2*f^2*(d*e + c*f) + 6*a*b*d*f*(d^2*e^2 + 3*c*d*e*f + c ^2*f^2) + b^2*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3*f^3))*x^11)/1 1 + (d*f*(a^2*d^2*f^2 + 6*a*b*d*f*(d*e + c*f) + 3*b^2*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x^13)/13 + (b*d^2*f^2*(2*a*d*f + 3*b*(d*e + c*f))*x^15)/15 + (b^2*d^3*f^3*x^17)/17
Time = 0.82 (sec) , antiderivative size = 493, 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.
| \(\displaystyle \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 396 |
| \(\displaystyle \int \left (d f x^{12} \left (a^2 d^2 f^2+6 a b d f (c f+d e)+3 b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+c e x^4 \left (3 a^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )+6 a b c e (c f+d e)+b^2 c^2 e^2\right )+x^{10} \left (3 a^2 d^2 f^2 (c f+d e)+6 a b d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b^2 \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )\right )+x^8 \left (3 a^2 d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+2 a b \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )+3 b^2 c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+x^6 \left (a^2 \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )+6 a b c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )+3 b^2 c^2 e^2 (c f+d e)\right )+a^2 c^3 e^3+a c^2 e^2 x^2 (3 a (c f+d e)+2 b c e)+b d^2 f^2 x^{14} (2 a d f+3 b (c f+d e))+b^2 d^3 f^3 x^{16}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{13} d f x^{13} \left (a^2 d^2 f^2+6 a b d f (c f+d e)+3 b^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac {1}{5} c e x^5 \left (3 a^2 \left (c^2 f^2+3 c d e f+d^2 e^2\right )+6 a b c e (c f+d e)+b^2 c^2 e^2\right )+\frac {1}{11} x^{11} \left (3 a^2 d^2 f^2 (c f+d e)+6 a b d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+b^2 \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )\right )+\frac {1}{9} x^9 \left (3 a^2 d f \left (c^2 f^2+3 c d e f+d^2 e^2\right )+2 a b \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )+3 b^2 c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )\right )+\frac {1}{7} x^7 \left (a^2 \left (c^3 f^3+9 c^2 d e f^2+9 c d^2 e^2 f+d^3 e^3\right )+6 a b c e \left (c^2 f^2+3 c d e f+d^2 e^2\right )+3 b^2 c^2 e^2 (c f+d e)\right )+a^2 c^3 e^3 x+\frac {1}{3} a c^2 e^2 x^3 (3 a (c f+d e)+2 b c e)+\frac {1}{15} b d^2 f^2 x^{15} (2 a d f+3 b (c f+d e))+\frac {1}{17} b^2 d^3 f^3 x^{17}\) |
Input:
Int[(a + b*x^2)^2*(c + d*x^2)^3*(e + f*x^2)^3,x]
Output:
a^2*c^3*e^3*x + (a*c^2*e^2*(2*b*c*e + 3*a*(d*e + c*f))*x^3)/3 + (c*e*(b^2* c^2*e^2 + 6*a*b*c*e*(d*e + c*f) + 3*a^2*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x ^5)/5 + ((3*b^2*c^2*e^2*(d*e + c*f) + 6*a*b*c*e*(d^2*e^2 + 3*c*d*e*f + c^2 *f^2) + a^2*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3*f^3))*x^7)/7 + ((3*b^2*c*e*(d^2*e^2 + 3*c*d*e*f + c^2*f^2) + 3*a^2*d*f*(d^2*e^2 + 3*c*d*e *f + c^2*f^2) + 2*a*b*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3*f^3)) *x^9)/9 + ((3*a^2*d^2*f^2*(d*e + c*f) + 6*a*b*d*f*(d^2*e^2 + 3*c*d*e*f + c ^2*f^2) + b^2*(d^3*e^3 + 9*c*d^2*e^2*f + 9*c^2*d*e*f^2 + c^3*f^3))*x^11)/1 1 + (d*f*(a^2*d^2*f^2 + 6*a*b*d*f*(d*e + c*f) + 3*b^2*(d^2*e^2 + 3*c*d*e*f + c^2*f^2))*x^13)/13 + (b*d^2*f^2*(2*a*d*f + 3*b*(d*e + c*f))*x^15)/15 + (b^2*d^3*f^3*x^17)/17
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 = 0.53 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.12
| method | result | size |
| default | \(\frac {b^{2} d^{3} f^{3} x^{17}}{17}+\frac {\left (\left (2 a \,d^{3} b +3 b^{2} c \,d^{2}\right ) f^{3}+3 b^{2} d^{3} e \,f^{2}\right ) x^{15}}{15}+\frac {\left (\left (a^{2} d^{3}+6 a c \,d^{2} b +3 b^{2} c^{2} d \right ) f^{3}+3 \left (2 a \,d^{3} b +3 b^{2} c \,d^{2}\right ) e \,f^{2}+3 b^{2} d^{3} e^{2} f \right ) x^{13}}{13}+\frac {\left (\left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) f^{3}+3 \left (a^{2} d^{3}+6 a c \,d^{2} b +3 b^{2} c^{2} d \right ) e \,f^{2}+3 \left (2 a \,d^{3} b +3 b^{2} c \,d^{2}\right ) e^{2} f +b^{2} d^{3} e^{3}\right ) x^{11}}{11}+\frac {\left (\left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) f^{3}+3 \left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) e \,f^{2}+3 \left (a^{2} d^{3}+6 a c \,d^{2} b +3 b^{2} c^{2} d \right ) e^{2} f +\left (2 a \,d^{3} b +3 b^{2} c \,d^{2}\right ) e^{3}\right ) x^{9}}{9}+\frac {\left (a^{2} c^{3} f^{3}+3 \left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) e \,f^{2}+3 \left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) e^{2} f +\left (a^{2} d^{3}+6 a c \,d^{2} b +3 b^{2} c^{2} d \right ) e^{3}\right ) x^{7}}{7}+\frac {\left (3 a^{2} c^{3} e \,f^{2}+3 \left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) e^{2} f +\left (3 c \,a^{2} d^{2}+6 a b \,c^{2} d +b^{2} c^{3}\right ) e^{3}\right ) x^{5}}{5}+\frac {\left (3 a^{2} c^{3} e^{2} f +\left (3 a^{2} c^{2} d +2 a b \,c^{3}\right ) e^{3}\right ) x^{3}}{3}+a^{2} c^{3} e^{3} x\) | \(553\) |
| norman | \(a^{2} c^{3} e^{3} x +\left (a^{2} c^{3} e^{2} f +a^{2} c^{2} d \,e^{3}+\frac {2}{3} a b \,c^{3} e^{3}\right ) x^{3}+\left (\frac {3}{5} a^{2} c^{3} e \,f^{2}+\frac {9}{5} a^{2} c^{2} d \,e^{2} f +\frac {3}{5} a^{2} c \,d^{2} e^{3}+\frac {6}{5} a b \,c^{3} e^{2} f +\frac {6}{5} a b \,c^{2} d \,e^{3}+\frac {1}{5} b^{2} c^{3} e^{3}\right ) x^{5}+\left (\frac {1}{7} a^{2} c^{3} f^{3}+\frac {9}{7} a^{2} c^{2} d e \,f^{2}+\frac {9}{7} a^{2} c \,d^{2} e^{2} f +\frac {1}{7} a^{2} d^{3} e^{3}+\frac {6}{7} a b \,c^{3} e \,f^{2}+\frac {18}{7} a b \,c^{2} d \,e^{2} f +\frac {6}{7} a b c \,d^{2} e^{3}+\frac {3}{7} b^{2} c^{3} e^{2} f +\frac {3}{7} b^{2} c^{2} d \,e^{3}\right ) x^{7}+\left (\frac {1}{3} a^{2} c^{2} d \,f^{3}+a^{2} c \,d^{2} e \,f^{2}+\frac {1}{3} a^{2} d^{3} e^{2} f +\frac {2}{9} a b \,c^{3} f^{3}+2 a b \,c^{2} d e \,f^{2}+2 a b c \,d^{2} e^{2} f +\frac {2}{9} a b \,d^{3} e^{3}+\frac {1}{3} b^{2} c^{3} e \,f^{2}+b^{2} c^{2} d \,e^{2} f +\frac {1}{3} b^{2} c \,d^{2} e^{3}\right ) x^{9}+\left (\frac {3}{11} a^{2} c \,d^{2} f^{3}+\frac {3}{11} a^{2} d^{3} e \,f^{2}+\frac {6}{11} a b \,c^{2} d \,f^{3}+\frac {18}{11} a b c \,d^{2} e \,f^{2}+\frac {6}{11} a b \,d^{3} e^{2} f +\frac {1}{11} b^{2} c^{3} f^{3}+\frac {9}{11} b^{2} c^{2} d e \,f^{2}+\frac {9}{11} b^{2} c \,d^{2} e^{2} f +\frac {1}{11} b^{2} d^{3} e^{3}\right ) x^{11}+\left (\frac {1}{13} a^{2} d^{3} f^{3}+\frac {6}{13} a b c \,d^{2} f^{3}+\frac {6}{13} a b \,d^{3} e \,f^{2}+\frac {3}{13} b^{2} c^{2} d \,f^{3}+\frac {9}{13} b^{2} c \,d^{2} e \,f^{2}+\frac {3}{13} b^{2} d^{3} e^{2} f \right ) x^{13}+\left (\frac {2}{15} a b \,d^{3} f^{3}+\frac {1}{5} b^{2} c \,d^{2} f^{3}+\frac {1}{5} b^{2} d^{3} e \,f^{2}\right ) x^{15}+\frac {b^{2} d^{3} f^{3} x^{17}}{17}\) | \(596\) |
| gosper | \(\frac {9}{5} x^{5} a^{2} c^{2} d \,e^{2} f +\frac {6}{5} x^{5} a b \,c^{3} e^{2} f +\frac {6}{5} x^{5} a b \,c^{2} d \,e^{3}+\frac {9}{7} x^{7} a^{2} c^{2} d e \,f^{2}+\frac {9}{7} x^{7} a^{2} c \,d^{2} e^{2} f +\frac {1}{3} x^{9} b^{2} c^{3} e \,f^{2}+\frac {1}{11} x^{11} b^{2} d^{3} e^{3}+\frac {1}{13} x^{13} a^{2} d^{3} f^{3}+x^{3} a^{2} c^{3} e^{2} f +\frac {18}{11} x^{11} a b c \,d^{2} e \,f^{2}+\frac {3}{11} x^{11} a^{2} c \,d^{2} f^{3}+\frac {3}{11} x^{11} a^{2} d^{3} e \,f^{2}+\frac {2}{9} x^{9} a b \,d^{3} e^{3}+\frac {1}{5} x^{15} b^{2} c \,d^{2} f^{3}+\frac {1}{5} x^{15} b^{2} d^{3} e \,f^{2}+\frac {9}{13} x^{13} b^{2} c \,d^{2} e \,f^{2}+\frac {1}{5} x^{5} b^{2} c^{3} e^{3}+\frac {1}{7} x^{7} a^{2} c^{3} f^{3}+\frac {1}{7} x^{7} a^{2} d^{3} e^{3}+\frac {1}{11} x^{11} b^{2} c^{3} f^{3}+\frac {1}{3} x^{9} a^{2} c^{2} d \,f^{3}+\frac {1}{3} x^{9} a^{2} d^{3} e^{2} f +\frac {2}{9} x^{9} a b \,c^{3} f^{3}+\frac {18}{7} x^{7} a b \,c^{2} d \,e^{2} f +2 x^{9} a b \,c^{2} d e \,f^{2}+2 x^{9} a b c \,d^{2} e^{2} f +\frac {1}{3} x^{9} b^{2} c \,d^{2} e^{3}+\frac {3}{5} x^{5} a^{2} c \,d^{2} e^{3}+\frac {3}{7} x^{7} b^{2} c^{3} e^{2} f +\frac {3}{7} x^{7} b^{2} c^{2} d \,e^{3}+x^{9} a^{2} c \,d^{2} e \,f^{2}+x^{9} b^{2} c^{2} d \,e^{2} f +\frac {6}{11} x^{11} a b \,c^{2} d \,f^{3}+\frac {3}{13} x^{13} b^{2} d^{3} e^{2} f +\frac {2}{15} x^{15} a b \,d^{3} f^{3}+\frac {6}{7} x^{7} a b \,c^{3} e \,f^{2}+\frac {6}{7} x^{7} a b c \,d^{2} e^{3}+x^{3} a^{2} c^{2} d \,e^{3}+\frac {2}{3} x^{3} a b \,c^{3} e^{3}+\frac {3}{5} x^{5} a^{2} c^{3} e \,f^{2}+\frac {3}{13} x^{13} b^{2} c^{2} d \,f^{3}+\frac {6}{11} x^{11} a b \,d^{3} e^{2} f +\frac {9}{11} x^{11} b^{2} c^{2} d e \,f^{2}+\frac {9}{11} x^{11} b^{2} c \,d^{2} e^{2} f +\frac {6}{13} x^{13} a b c \,d^{2} f^{3}+\frac {6}{13} x^{13} a b \,d^{3} e \,f^{2}+a^{2} c^{3} e^{3} x +\frac {1}{17} b^{2} d^{3} f^{3} x^{17}\) | \(699\) |
| risch | \(\frac {9}{5} x^{5} a^{2} c^{2} d \,e^{2} f +\frac {6}{5} x^{5} a b \,c^{3} e^{2} f +\frac {6}{5} x^{5} a b \,c^{2} d \,e^{3}+\frac {9}{7} x^{7} a^{2} c^{2} d e \,f^{2}+\frac {9}{7} x^{7} a^{2} c \,d^{2} e^{2} f +\frac {1}{3} x^{9} b^{2} c^{3} e \,f^{2}+\frac {1}{11} x^{11} b^{2} d^{3} e^{3}+\frac {1}{13} x^{13} a^{2} d^{3} f^{3}+x^{3} a^{2} c^{3} e^{2} f +\frac {18}{11} x^{11} a b c \,d^{2} e \,f^{2}+\frac {3}{11} x^{11} a^{2} c \,d^{2} f^{3}+\frac {3}{11} x^{11} a^{2} d^{3} e \,f^{2}+\frac {2}{9} x^{9} a b \,d^{3} e^{3}+\frac {1}{5} x^{15} b^{2} c \,d^{2} f^{3}+\frac {1}{5} x^{15} b^{2} d^{3} e \,f^{2}+\frac {9}{13} x^{13} b^{2} c \,d^{2} e \,f^{2}+\frac {1}{5} x^{5} b^{2} c^{3} e^{3}+\frac {1}{7} x^{7} a^{2} c^{3} f^{3}+\frac {1}{7} x^{7} a^{2} d^{3} e^{3}+\frac {1}{11} x^{11} b^{2} c^{3} f^{3}+\frac {1}{3} x^{9} a^{2} c^{2} d \,f^{3}+\frac {1}{3} x^{9} a^{2} d^{3} e^{2} f +\frac {2}{9} x^{9} a b \,c^{3} f^{3}+\frac {18}{7} x^{7} a b \,c^{2} d \,e^{2} f +2 x^{9} a b \,c^{2} d e \,f^{2}+2 x^{9} a b c \,d^{2} e^{2} f +\frac {1}{3} x^{9} b^{2} c \,d^{2} e^{3}+\frac {3}{5} x^{5} a^{2} c \,d^{2} e^{3}+\frac {3}{7} x^{7} b^{2} c^{3} e^{2} f +\frac {3}{7} x^{7} b^{2} c^{2} d \,e^{3}+x^{9} a^{2} c \,d^{2} e \,f^{2}+x^{9} b^{2} c^{2} d \,e^{2} f +\frac {6}{11} x^{11} a b \,c^{2} d \,f^{3}+\frac {3}{13} x^{13} b^{2} d^{3} e^{2} f +\frac {2}{15} x^{15} a b \,d^{3} f^{3}+\frac {6}{7} x^{7} a b \,c^{3} e \,f^{2}+\frac {6}{7} x^{7} a b c \,d^{2} e^{3}+x^{3} a^{2} c^{2} d \,e^{3}+\frac {2}{3} x^{3} a b \,c^{3} e^{3}+\frac {3}{5} x^{5} a^{2} c^{3} e \,f^{2}+\frac {3}{13} x^{13} b^{2} c^{2} d \,f^{3}+\frac {6}{11} x^{11} a b \,d^{3} e^{2} f +\frac {9}{11} x^{11} b^{2} c^{2} d e \,f^{2}+\frac {9}{11} x^{11} b^{2} c \,d^{2} e^{2} f +\frac {6}{13} x^{13} a b c \,d^{2} f^{3}+\frac {6}{13} x^{13} a b \,d^{3} e \,f^{2}+a^{2} c^{3} e^{3} x +\frac {1}{17} b^{2} d^{3} f^{3} x^{17}\) | \(699\) |
| parallelrisch | \(\frac {9}{5} x^{5} a^{2} c^{2} d \,e^{2} f +\frac {6}{5} x^{5} a b \,c^{3} e^{2} f +\frac {6}{5} x^{5} a b \,c^{2} d \,e^{3}+\frac {9}{7} x^{7} a^{2} c^{2} d e \,f^{2}+\frac {9}{7} x^{7} a^{2} c \,d^{2} e^{2} f +\frac {1}{3} x^{9} b^{2} c^{3} e \,f^{2}+\frac {1}{11} x^{11} b^{2} d^{3} e^{3}+\frac {1}{13} x^{13} a^{2} d^{3} f^{3}+x^{3} a^{2} c^{3} e^{2} f +\frac {18}{11} x^{11} a b c \,d^{2} e \,f^{2}+\frac {3}{11} x^{11} a^{2} c \,d^{2} f^{3}+\frac {3}{11} x^{11} a^{2} d^{3} e \,f^{2}+\frac {2}{9} x^{9} a b \,d^{3} e^{3}+\frac {1}{5} x^{15} b^{2} c \,d^{2} f^{3}+\frac {1}{5} x^{15} b^{2} d^{3} e \,f^{2}+\frac {9}{13} x^{13} b^{2} c \,d^{2} e \,f^{2}+\frac {1}{5} x^{5} b^{2} c^{3} e^{3}+\frac {1}{7} x^{7} a^{2} c^{3} f^{3}+\frac {1}{7} x^{7} a^{2} d^{3} e^{3}+\frac {1}{11} x^{11} b^{2} c^{3} f^{3}+\frac {1}{3} x^{9} a^{2} c^{2} d \,f^{3}+\frac {1}{3} x^{9} a^{2} d^{3} e^{2} f +\frac {2}{9} x^{9} a b \,c^{3} f^{3}+\frac {18}{7} x^{7} a b \,c^{2} d \,e^{2} f +2 x^{9} a b \,c^{2} d e \,f^{2}+2 x^{9} a b c \,d^{2} e^{2} f +\frac {1}{3} x^{9} b^{2} c \,d^{2} e^{3}+\frac {3}{5} x^{5} a^{2} c \,d^{2} e^{3}+\frac {3}{7} x^{7} b^{2} c^{3} e^{2} f +\frac {3}{7} x^{7} b^{2} c^{2} d \,e^{3}+x^{9} a^{2} c \,d^{2} e \,f^{2}+x^{9} b^{2} c^{2} d \,e^{2} f +\frac {6}{11} x^{11} a b \,c^{2} d \,f^{3}+\frac {3}{13} x^{13} b^{2} d^{3} e^{2} f +\frac {2}{15} x^{15} a b \,d^{3} f^{3}+\frac {6}{7} x^{7} a b \,c^{3} e \,f^{2}+\frac {6}{7} x^{7} a b c \,d^{2} e^{3}+x^{3} a^{2} c^{2} d \,e^{3}+\frac {2}{3} x^{3} a b \,c^{3} e^{3}+\frac {3}{5} x^{5} a^{2} c^{3} e \,f^{2}+\frac {3}{13} x^{13} b^{2} c^{2} d \,f^{3}+\frac {6}{11} x^{11} a b \,d^{3} e^{2} f +\frac {9}{11} x^{11} b^{2} c^{2} d e \,f^{2}+\frac {9}{11} x^{11} b^{2} c \,d^{2} e^{2} f +\frac {6}{13} x^{13} a b c \,d^{2} f^{3}+\frac {6}{13} x^{13} a b \,d^{3} e \,f^{2}+a^{2} c^{3} e^{3} x +\frac {1}{17} b^{2} d^{3} f^{3} x^{17}\) | \(699\) |
| orering | \(\frac {x \left (45045 b^{2} d^{3} f^{3} x^{16}+102102 a b \,d^{3} f^{3} x^{14}+153153 b^{2} c \,d^{2} f^{3} x^{14}+153153 b^{2} d^{3} e \,f^{2} x^{14}+58905 a^{2} d^{3} f^{3} x^{12}+353430 a b c \,d^{2} f^{3} x^{12}+353430 a b \,d^{3} e \,f^{2} x^{12}+176715 b^{2} c^{2} d \,f^{3} x^{12}+530145 b^{2} c \,d^{2} e \,f^{2} x^{12}+176715 b^{2} d^{3} e^{2} f \,x^{12}+208845 a^{2} c \,d^{2} f^{3} x^{10}+208845 a^{2} d^{3} e \,f^{2} x^{10}+417690 a b \,c^{2} d \,f^{3} x^{10}+1253070 a b c \,d^{2} e \,f^{2} x^{10}+417690 a b \,d^{3} e^{2} f \,x^{10}+69615 b^{2} c^{3} f^{3} x^{10}+626535 b^{2} c^{2} d e \,f^{2} x^{10}+626535 b^{2} c \,d^{2} e^{2} f \,x^{10}+69615 b^{2} d^{3} e^{3} x^{10}+255255 a^{2} c^{2} d \,f^{3} x^{8}+765765 a^{2} c \,d^{2} e \,f^{2} x^{8}+255255 a^{2} d^{3} e^{2} f \,x^{8}+170170 a b \,c^{3} f^{3} x^{8}+1531530 a b \,c^{2} d e \,f^{2} x^{8}+1531530 a b c \,d^{2} e^{2} f \,x^{8}+170170 a b \,d^{3} e^{3} x^{8}+255255 b^{2} c^{3} e \,f^{2} x^{8}+765765 b^{2} c^{2} d \,e^{2} f \,x^{8}+255255 b^{2} c \,d^{2} e^{3} x^{8}+109395 a^{2} c^{3} f^{3} x^{6}+984555 a^{2} c^{2} d e \,f^{2} x^{6}+984555 a^{2} c \,d^{2} e^{2} f \,x^{6}+109395 a^{2} d^{3} e^{3} x^{6}+656370 a b \,c^{3} e \,f^{2} x^{6}+1969110 a b \,c^{2} d \,e^{2} f \,x^{6}+656370 a b c \,d^{2} e^{3} x^{6}+328185 b^{2} c^{3} e^{2} f \,x^{6}+328185 b^{2} c^{2} d \,e^{3} x^{6}+459459 a^{2} c^{3} e \,f^{2} x^{4}+1378377 a^{2} c^{2} d \,e^{2} f \,x^{4}+459459 a^{2} c \,d^{2} e^{3} x^{4}+918918 a b \,c^{3} e^{2} f \,x^{4}+918918 a b \,c^{2} d \,e^{3} x^{4}+153153 b^{2} c^{3} e^{3} x^{4}+765765 a^{2} c^{3} e^{2} f \,x^{2}+765765 a^{2} c^{2} d \,e^{3} x^{2}+510510 a b \,c^{3} e^{3} x^{2}+765765 a^{2} c^{3} e^{3}\right )}{765765}\) | \(706\) |
Input:
int((b*x^2+a)^2*(d*x^2+c)^3*(f*x^2+e)^3,x,method=_RETURNVERBOSE)
Output:
1/17*b^2*d^3*f^3*x^17+1/15*((2*a*b*d^3+3*b^2*c*d^2)*f^3+3*b^2*d^3*e*f^2)*x ^15+1/13*((a^2*d^3+6*a*b*c*d^2+3*b^2*c^2*d)*f^3+3*(2*a*b*d^3+3*b^2*c*d^2)* e*f^2+3*b^2*d^3*e^2*f)*x^13+1/11*((3*a^2*c*d^2+6*a*b*c^2*d+b^2*c^3)*f^3+3* (a^2*d^3+6*a*b*c*d^2+3*b^2*c^2*d)*e*f^2+3*(2*a*b*d^3+3*b^2*c*d^2)*e^2*f+b^ 2*d^3*e^3)*x^11+1/9*((3*a^2*c^2*d+2*a*b*c^3)*f^3+3*(3*a^2*c*d^2+6*a*b*c^2* d+b^2*c^3)*e*f^2+3*(a^2*d^3+6*a*b*c*d^2+3*b^2*c^2*d)*e^2*f+(2*a*b*d^3+3*b^ 2*c*d^2)*e^3)*x^9+1/7*(a^2*c^3*f^3+3*(3*a^2*c^2*d+2*a*b*c^3)*e*f^2+3*(3*a^ 2*c*d^2+6*a*b*c^2*d+b^2*c^3)*e^2*f+(a^2*d^3+6*a*b*c*d^2+3*b^2*c^2*d)*e^3)* x^7+1/5*(3*a^2*c^3*e*f^2+3*(3*a^2*c^2*d+2*a*b*c^3)*e^2*f+(3*a^2*c*d^2+6*a* b*c^2*d+b^2*c^3)*e^3)*x^5+1/3*(3*a^2*c^3*e^2*f+(3*a^2*c^2*d+2*a*b*c^3)*e^3 )*x^3+a^2*c^3*e^3*x
Time = 0.07 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.12 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=\frac {1}{17} \, b^{2} d^{3} f^{3} x^{17} + \frac {1}{15} \, {\left (3 \, b^{2} d^{3} e f^{2} + {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} f^{3}\right )} x^{15} + \frac {1}{13} \, {\left (3 \, b^{2} d^{3} e^{2} f + 3 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} e f^{2} + {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} f^{3}\right )} x^{13} + \frac {1}{11} \, {\left (b^{2} d^{3} e^{3} + 3 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} e^{2} f + 3 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} + {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} f^{3}\right )} x^{11} + \frac {1}{9} \, {\left ({\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} e^{2} f + 3 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} e f^{2} + {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} f^{3}\right )} x^{9} + a^{2} c^{3} e^{3} x + \frac {1}{7} \, {\left (a^{2} c^{3} f^{3} + {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} e^{3} + 3 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} e^{2} f + 3 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} e f^{2}\right )} x^{7} + \frac {1}{5} \, {\left (3 \, a^{2} c^{3} e f^{2} + {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} e^{3} + 3 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} e^{2} f\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a^{2} c^{3} e^{2} f + {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} e^{3}\right )} x^{3} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^3*(f*x^2+e)^3,x, algorithm="fricas")
Output:
1/17*b^2*d^3*f^3*x^17 + 1/15*(3*b^2*d^3*e*f^2 + (3*b^2*c*d^2 + 2*a*b*d^3)* f^3)*x^15 + 1/13*(3*b^2*d^3*e^2*f + 3*(3*b^2*c*d^2 + 2*a*b*d^3)*e*f^2 + (3 *b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*f^3)*x^13 + 1/11*(b^2*d^3*e^3 + 3*(3*b ^2*c*d^2 + 2*a*b*d^3)*e^2*f + 3*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e*f^ 2 + (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*f^3)*x^11 + 1/9*((3*b^2*c*d^2 + 2*a*b*d^3)*e^3 + 3*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^2*f + 3*(b^2*c^ 3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e*f^2 + (2*a*b*c^3 + 3*a^2*c^2*d)*f^3)*x^9 + a^2*c^3*e^3*x + 1/7*(a^2*c^3*f^3 + (3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3) *e^3 + 3*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^2*f + 3*(2*a*b*c^3 + 3*a^ 2*c^2*d)*e*f^2)*x^7 + 1/5*(3*a^2*c^3*e*f^2 + (b^2*c^3 + 6*a*b*c^2*d + 3*a^ 2*c*d^2)*e^3 + 3*(2*a*b*c^3 + 3*a^2*c^2*d)*e^2*f)*x^5 + 1/3*(3*a^2*c^3*e^2 *f + (2*a*b*c^3 + 3*a^2*c^2*d)*e^3)*x^3
Time = 0.08 (sec) , antiderivative size = 719, normalized size of antiderivative = 1.46 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=a^{2} c^{3} e^{3} x + \frac {b^{2} d^{3} f^{3} x^{17}}{17} + x^{15} \cdot \left (\frac {2 a b d^{3} f^{3}}{15} + \frac {b^{2} c d^{2} f^{3}}{5} + \frac {b^{2} d^{3} e f^{2}}{5}\right ) + x^{13} \left (\frac {a^{2} d^{3} f^{3}}{13} + \frac {6 a b c d^{2} f^{3}}{13} + \frac {6 a b d^{3} e f^{2}}{13} + \frac {3 b^{2} c^{2} d f^{3}}{13} + \frac {9 b^{2} c d^{2} e f^{2}}{13} + \frac {3 b^{2} d^{3} e^{2} f}{13}\right ) + x^{11} \cdot \left (\frac {3 a^{2} c d^{2} f^{3}}{11} + \frac {3 a^{2} d^{3} e f^{2}}{11} + \frac {6 a b c^{2} d f^{3}}{11} + \frac {18 a b c d^{2} e f^{2}}{11} + \frac {6 a b d^{3} e^{2} f}{11} + \frac {b^{2} c^{3} f^{3}}{11} + \frac {9 b^{2} c^{2} d e f^{2}}{11} + \frac {9 b^{2} c d^{2} e^{2} f}{11} + \frac {b^{2} d^{3} e^{3}}{11}\right ) + x^{9} \left (\frac {a^{2} c^{2} d f^{3}}{3} + a^{2} c d^{2} e f^{2} + \frac {a^{2} d^{3} e^{2} f}{3} + \frac {2 a b c^{3} f^{3}}{9} + 2 a b c^{2} d e f^{2} + 2 a b c d^{2} e^{2} f + \frac {2 a b d^{3} e^{3}}{9} + \frac {b^{2} c^{3} e f^{2}}{3} + b^{2} c^{2} d e^{2} f + \frac {b^{2} c d^{2} e^{3}}{3}\right ) + x^{7} \left (\frac {a^{2} c^{3} f^{3}}{7} + \frac {9 a^{2} c^{2} d e f^{2}}{7} + \frac {9 a^{2} c d^{2} e^{2} f}{7} + \frac {a^{2} d^{3} e^{3}}{7} + \frac {6 a b c^{3} e f^{2}}{7} + \frac {18 a b c^{2} d e^{2} f}{7} + \frac {6 a b c d^{2} e^{3}}{7} + \frac {3 b^{2} c^{3} e^{2} f}{7} + \frac {3 b^{2} c^{2} d e^{3}}{7}\right ) + x^{5} \cdot \left (\frac {3 a^{2} c^{3} e f^{2}}{5} + \frac {9 a^{2} c^{2} d e^{2} f}{5} + \frac {3 a^{2} c d^{2} e^{3}}{5} + \frac {6 a b c^{3} e^{2} f}{5} + \frac {6 a b c^{2} d e^{3}}{5} + \frac {b^{2} c^{3} e^{3}}{5}\right ) + x^{3} \left (a^{2} c^{3} e^{2} f + a^{2} c^{2} d e^{3} + \frac {2 a b c^{3} e^{3}}{3}\right ) \] Input:
integrate((b*x**2+a)**2*(d*x**2+c)**3*(f*x**2+e)**3,x)
Output:
a**2*c**3*e**3*x + b**2*d**3*f**3*x**17/17 + x**15*(2*a*b*d**3*f**3/15 + b **2*c*d**2*f**3/5 + b**2*d**3*e*f**2/5) + x**13*(a**2*d**3*f**3/13 + 6*a*b *c*d**2*f**3/13 + 6*a*b*d**3*e*f**2/13 + 3*b**2*c**2*d*f**3/13 + 9*b**2*c* d**2*e*f**2/13 + 3*b**2*d**3*e**2*f/13) + x**11*(3*a**2*c*d**2*f**3/11 + 3 *a**2*d**3*e*f**2/11 + 6*a*b*c**2*d*f**3/11 + 18*a*b*c*d**2*e*f**2/11 + 6* a*b*d**3*e**2*f/11 + b**2*c**3*f**3/11 + 9*b**2*c**2*d*e*f**2/11 + 9*b**2* c*d**2*e**2*f/11 + b**2*d**3*e**3/11) + x**9*(a**2*c**2*d*f**3/3 + a**2*c* d**2*e*f**2 + a**2*d**3*e**2*f/3 + 2*a*b*c**3*f**3/9 + 2*a*b*c**2*d*e*f**2 + 2*a*b*c*d**2*e**2*f + 2*a*b*d**3*e**3/9 + b**2*c**3*e*f**2/3 + b**2*c** 2*d*e**2*f + b**2*c*d**2*e**3/3) + x**7*(a**2*c**3*f**3/7 + 9*a**2*c**2*d* e*f**2/7 + 9*a**2*c*d**2*e**2*f/7 + a**2*d**3*e**3/7 + 6*a*b*c**3*e*f**2/7 + 18*a*b*c**2*d*e**2*f/7 + 6*a*b*c*d**2*e**3/7 + 3*b**2*c**3*e**2*f/7 + 3 *b**2*c**2*d*e**3/7) + x**5*(3*a**2*c**3*e*f**2/5 + 9*a**2*c**2*d*e**2*f/5 + 3*a**2*c*d**2*e**3/5 + 6*a*b*c**3*e**2*f/5 + 6*a*b*c**2*d*e**3/5 + b**2 *c**3*e**3/5) + x**3*(a**2*c**3*e**2*f + a**2*c**2*d*e**3 + 2*a*b*c**3*e** 3/3)
Time = 0.04 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.12 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=\frac {1}{17} \, b^{2} d^{3} f^{3} x^{17} + \frac {1}{15} \, {\left (3 \, b^{2} d^{3} e f^{2} + {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} f^{3}\right )} x^{15} + \frac {1}{13} \, {\left (3 \, b^{2} d^{3} e^{2} f + 3 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} e f^{2} + {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} f^{3}\right )} x^{13} + \frac {1}{11} \, {\left (b^{2} d^{3} e^{3} + 3 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} e^{2} f + 3 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} + {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} f^{3}\right )} x^{11} + \frac {1}{9} \, {\left ({\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} e^{3} + 3 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} e^{2} f + 3 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} e f^{2} + {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} f^{3}\right )} x^{9} + a^{2} c^{3} e^{3} x + \frac {1}{7} \, {\left (a^{2} c^{3} f^{3} + {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} e^{3} + 3 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} e^{2} f + 3 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} e f^{2}\right )} x^{7} + \frac {1}{5} \, {\left (3 \, a^{2} c^{3} e f^{2} + {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} e^{3} + 3 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} e^{2} f\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a^{2} c^{3} e^{2} f + {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} e^{3}\right )} x^{3} \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^3*(f*x^2+e)^3,x, algorithm="maxima")
Output:
1/17*b^2*d^3*f^3*x^17 + 1/15*(3*b^2*d^3*e*f^2 + (3*b^2*c*d^2 + 2*a*b*d^3)* f^3)*x^15 + 1/13*(3*b^2*d^3*e^2*f + 3*(3*b^2*c*d^2 + 2*a*b*d^3)*e*f^2 + (3 *b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*f^3)*x^13 + 1/11*(b^2*d^3*e^3 + 3*(3*b ^2*c*d^2 + 2*a*b*d^3)*e^2*f + 3*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e*f^ 2 + (b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*f^3)*x^11 + 1/9*((3*b^2*c*d^2 + 2*a*b*d^3)*e^3 + 3*(3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3)*e^2*f + 3*(b^2*c^ 3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e*f^2 + (2*a*b*c^3 + 3*a^2*c^2*d)*f^3)*x^9 + a^2*c^3*e^3*x + 1/7*(a^2*c^3*f^3 + (3*b^2*c^2*d + 6*a*b*c*d^2 + a^2*d^3) *e^3 + 3*(b^2*c^3 + 6*a*b*c^2*d + 3*a^2*c*d^2)*e^2*f + 3*(2*a*b*c^3 + 3*a^ 2*c^2*d)*e*f^2)*x^7 + 1/5*(3*a^2*c^3*e*f^2 + (b^2*c^3 + 6*a*b*c^2*d + 3*a^ 2*c*d^2)*e^3 + 3*(2*a*b*c^3 + 3*a^2*c^2*d)*e^2*f)*x^5 + 1/3*(3*a^2*c^3*e^2 *f + (2*a*b*c^3 + 3*a^2*c^2*d)*e^3)*x^3
Time = 0.12 (sec) , antiderivative size = 698, normalized size of antiderivative = 1.42 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=\frac {1}{17} \, b^{2} d^{3} f^{3} x^{17} + \frac {1}{5} \, b^{2} d^{3} e f^{2} x^{15} + \frac {1}{5} \, b^{2} c d^{2} f^{3} x^{15} + \frac {2}{15} \, a b d^{3} f^{3} x^{15} + \frac {3}{13} \, b^{2} d^{3} e^{2} f x^{13} + \frac {9}{13} \, b^{2} c d^{2} e f^{2} x^{13} + \frac {6}{13} \, a b d^{3} e f^{2} x^{13} + \frac {3}{13} \, b^{2} c^{2} d f^{3} x^{13} + \frac {6}{13} \, a b c d^{2} f^{3} x^{13} + \frac {1}{13} \, a^{2} d^{3} f^{3} x^{13} + \frac {1}{11} \, b^{2} d^{3} e^{3} x^{11} + \frac {9}{11} \, b^{2} c d^{2} e^{2} f x^{11} + \frac {6}{11} \, a b d^{3} e^{2} f x^{11} + \frac {9}{11} \, b^{2} c^{2} d e f^{2} x^{11} + \frac {18}{11} \, a b c d^{2} e f^{2} x^{11} + \frac {3}{11} \, a^{2} d^{3} e f^{2} x^{11} + \frac {1}{11} \, b^{2} c^{3} f^{3} x^{11} + \frac {6}{11} \, a b c^{2} d f^{3} x^{11} + \frac {3}{11} \, a^{2} c d^{2} f^{3} x^{11} + \frac {1}{3} \, b^{2} c d^{2} e^{3} x^{9} + \frac {2}{9} \, a b d^{3} e^{3} x^{9} + b^{2} c^{2} d e^{2} f x^{9} + 2 \, a b c d^{2} e^{2} f x^{9} + \frac {1}{3} \, a^{2} d^{3} e^{2} f x^{9} + \frac {1}{3} \, b^{2} c^{3} e f^{2} x^{9} + 2 \, a b c^{2} d e f^{2} x^{9} + a^{2} c d^{2} e f^{2} x^{9} + \frac {2}{9} \, a b c^{3} f^{3} x^{9} + \frac {1}{3} \, a^{2} c^{2} d f^{3} x^{9} + \frac {3}{7} \, b^{2} c^{2} d e^{3} x^{7} + \frac {6}{7} \, a b c d^{2} e^{3} x^{7} + \frac {1}{7} \, a^{2} d^{3} e^{3} x^{7} + \frac {3}{7} \, b^{2} c^{3} e^{2} f x^{7} + \frac {18}{7} \, a b c^{2} d e^{2} f x^{7} + \frac {9}{7} \, a^{2} c d^{2} e^{2} f x^{7} + \frac {6}{7} \, a b c^{3} e f^{2} x^{7} + \frac {9}{7} \, a^{2} c^{2} d e f^{2} x^{7} + \frac {1}{7} \, a^{2} c^{3} f^{3} x^{7} + \frac {1}{5} \, b^{2} c^{3} e^{3} x^{5} + \frac {6}{5} \, a b c^{2} d e^{3} x^{5} + \frac {3}{5} \, a^{2} c d^{2} e^{3} x^{5} + \frac {6}{5} \, a b c^{3} e^{2} f x^{5} + \frac {9}{5} \, a^{2} c^{2} d e^{2} f x^{5} + \frac {3}{5} \, a^{2} c^{3} e f^{2} x^{5} + \frac {2}{3} \, a b c^{3} e^{3} x^{3} + a^{2} c^{2} d e^{3} x^{3} + a^{2} c^{3} e^{2} f x^{3} + a^{2} c^{3} e^{3} x \] Input:
integrate((b*x^2+a)^2*(d*x^2+c)^3*(f*x^2+e)^3,x, algorithm="giac")
Output:
1/17*b^2*d^3*f^3*x^17 + 1/5*b^2*d^3*e*f^2*x^15 + 1/5*b^2*c*d^2*f^3*x^15 + 2/15*a*b*d^3*f^3*x^15 + 3/13*b^2*d^3*e^2*f*x^13 + 9/13*b^2*c*d^2*e*f^2*x^1 3 + 6/13*a*b*d^3*e*f^2*x^13 + 3/13*b^2*c^2*d*f^3*x^13 + 6/13*a*b*c*d^2*f^3 *x^13 + 1/13*a^2*d^3*f^3*x^13 + 1/11*b^2*d^3*e^3*x^11 + 9/11*b^2*c*d^2*e^2 *f*x^11 + 6/11*a*b*d^3*e^2*f*x^11 + 9/11*b^2*c^2*d*e*f^2*x^11 + 18/11*a*b* c*d^2*e*f^2*x^11 + 3/11*a^2*d^3*e*f^2*x^11 + 1/11*b^2*c^3*f^3*x^11 + 6/11* a*b*c^2*d*f^3*x^11 + 3/11*a^2*c*d^2*f^3*x^11 + 1/3*b^2*c*d^2*e^3*x^9 + 2/9 *a*b*d^3*e^3*x^9 + b^2*c^2*d*e^2*f*x^9 + 2*a*b*c*d^2*e^2*f*x^9 + 1/3*a^2*d ^3*e^2*f*x^9 + 1/3*b^2*c^3*e*f^2*x^9 + 2*a*b*c^2*d*e*f^2*x^9 + a^2*c*d^2*e *f^2*x^9 + 2/9*a*b*c^3*f^3*x^9 + 1/3*a^2*c^2*d*f^3*x^9 + 3/7*b^2*c^2*d*e^3 *x^7 + 6/7*a*b*c*d^2*e^3*x^7 + 1/7*a^2*d^3*e^3*x^7 + 3/7*b^2*c^3*e^2*f*x^7 + 18/7*a*b*c^2*d*e^2*f*x^7 + 9/7*a^2*c*d^2*e^2*f*x^7 + 6/7*a*b*c^3*e*f^2* x^7 + 9/7*a^2*c^2*d*e*f^2*x^7 + 1/7*a^2*c^3*f^3*x^7 + 1/5*b^2*c^3*e^3*x^5 + 6/5*a*b*c^2*d*e^3*x^5 + 3/5*a^2*c*d^2*e^3*x^5 + 6/5*a*b*c^3*e^2*f*x^5 + 9/5*a^2*c^2*d*e^2*f*x^5 + 3/5*a^2*c^3*e*f^2*x^5 + 2/3*a*b*c^3*e^3*x^3 + a^ 2*c^2*d*e^3*x^3 + a^2*c^3*e^2*f*x^3 + a^2*c^3*e^3*x
Time = 2.11 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.17 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=x^7\,\left (\frac {a^2\,c^3\,f^3}{7}+\frac {9\,a^2\,c^2\,d\,e\,f^2}{7}+\frac {9\,a^2\,c\,d^2\,e^2\,f}{7}+\frac {a^2\,d^3\,e^3}{7}+\frac {6\,a\,b\,c^3\,e\,f^2}{7}+\frac {18\,a\,b\,c^2\,d\,e^2\,f}{7}+\frac {6\,a\,b\,c\,d^2\,e^3}{7}+\frac {3\,b^2\,c^3\,e^2\,f}{7}+\frac {3\,b^2\,c^2\,d\,e^3}{7}\right )+x^{11}\,\left (\frac {3\,a^2\,c\,d^2\,f^3}{11}+\frac {3\,a^2\,d^3\,e\,f^2}{11}+\frac {6\,a\,b\,c^2\,d\,f^3}{11}+\frac {18\,a\,b\,c\,d^2\,e\,f^2}{11}+\frac {6\,a\,b\,d^3\,e^2\,f}{11}+\frac {b^2\,c^3\,f^3}{11}+\frac {9\,b^2\,c^2\,d\,e\,f^2}{11}+\frac {9\,b^2\,c\,d^2\,e^2\,f}{11}+\frac {b^2\,d^3\,e^3}{11}\right )+x^5\,\left (\frac {3\,a^2\,c^3\,e\,f^2}{5}+\frac {9\,a^2\,c^2\,d\,e^2\,f}{5}+\frac {3\,a^2\,c\,d^2\,e^3}{5}+\frac {6\,a\,b\,c^3\,e^2\,f}{5}+\frac {6\,a\,b\,c^2\,d\,e^3}{5}+\frac {b^2\,c^3\,e^3}{5}\right )+x^9\,\left (\frac {a^2\,c^2\,d\,f^3}{3}+a^2\,c\,d^2\,e\,f^2+\frac {a^2\,d^3\,e^2\,f}{3}+\frac {2\,a\,b\,c^3\,f^3}{9}+2\,a\,b\,c^2\,d\,e\,f^2+2\,a\,b\,c\,d^2\,e^2\,f+\frac {2\,a\,b\,d^3\,e^3}{9}+\frac {b^2\,c^3\,e\,f^2}{3}+b^2\,c^2\,d\,e^2\,f+\frac {b^2\,c\,d^2\,e^3}{3}\right )+x^{13}\,\left (\frac {a^2\,d^3\,f^3}{13}+\frac {6\,a\,b\,c\,d^2\,f^3}{13}+\frac {6\,a\,b\,d^3\,e\,f^2}{13}+\frac {3\,b^2\,c^2\,d\,f^3}{13}+\frac {9\,b^2\,c\,d^2\,e\,f^2}{13}+\frac {3\,b^2\,d^3\,e^2\,f}{13}\right )+a^2\,c^3\,e^3\,x+\frac {b^2\,d^3\,f^3\,x^{17}}{17}+\frac {a\,c^2\,e^2\,x^3\,\left (3\,a\,c\,f+3\,a\,d\,e+2\,b\,c\,e\right )}{3}+\frac {b\,d^2\,f^2\,x^{15}\,\left (2\,a\,d\,f+3\,b\,c\,f+3\,b\,d\,e\right )}{15} \] Input:
int((a + b*x^2)^2*(c + d*x^2)^3*(e + f*x^2)^3,x)
Output:
x^7*((a^2*c^3*f^3)/7 + (a^2*d^3*e^3)/7 + (3*b^2*c^2*d*e^3)/7 + (3*b^2*c^3* e^2*f)/7 + (6*a*b*c*d^2*e^3)/7 + (6*a*b*c^3*e*f^2)/7 + (9*a^2*c*d^2*e^2*f) /7 + (9*a^2*c^2*d*e*f^2)/7 + (18*a*b*c^2*d*e^2*f)/7) + x^11*((b^2*c^3*f^3) /11 + (b^2*d^3*e^3)/11 + (3*a^2*c*d^2*f^3)/11 + (3*a^2*d^3*e*f^2)/11 + (6* a*b*c^2*d*f^3)/11 + (6*a*b*d^3*e^2*f)/11 + (9*b^2*c*d^2*e^2*f)/11 + (9*b^2 *c^2*d*e*f^2)/11 + (18*a*b*c*d^2*e*f^2)/11) + x^5*((b^2*c^3*e^3)/5 + (3*a^ 2*c*d^2*e^3)/5 + (3*a^2*c^3*e*f^2)/5 + (6*a*b*c^2*d*e^3)/5 + (6*a*b*c^3*e^ 2*f)/5 + (9*a^2*c^2*d*e^2*f)/5) + x^9*((a^2*c^2*d*f^3)/3 + (b^2*c*d^2*e^3) /3 + (a^2*d^3*e^2*f)/3 + (b^2*c^3*e*f^2)/3 + (2*a*b*c^3*f^3)/9 + (2*a*b*d^ 3*e^3)/9 + a^2*c*d^2*e*f^2 + b^2*c^2*d*e^2*f + 2*a*b*c*d^2*e^2*f + 2*a*b*c ^2*d*e*f^2) + x^13*((a^2*d^3*f^3)/13 + (3*b^2*c^2*d*f^3)/13 + (3*b^2*d^3*e ^2*f)/13 + (6*a*b*c*d^2*f^3)/13 + (6*a*b*d^3*e*f^2)/13 + (9*b^2*c*d^2*e*f^ 2)/13) + a^2*c^3*e^3*x + (b^2*d^3*f^3*x^17)/17 + (a*c^2*e^2*x^3*(3*a*c*f + 3*a*d*e + 2*b*c*e))/3 + (b*d^2*f^2*x^15*(2*a*d*f + 3*b*c*f + 3*b*d*e))/15
Time = 0.15 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.43 \[ \int \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \left (e+f x^2\right )^3 \, dx=\frac {x \left (45045 b^{2} d^{3} f^{3} x^{16}+102102 a b \,d^{3} f^{3} x^{14}+153153 b^{2} c \,d^{2} f^{3} x^{14}+153153 b^{2} d^{3} e \,f^{2} x^{14}+58905 a^{2} d^{3} f^{3} x^{12}+353430 a b c \,d^{2} f^{3} x^{12}+353430 a b \,d^{3} e \,f^{2} x^{12}+176715 b^{2} c^{2} d \,f^{3} x^{12}+530145 b^{2} c \,d^{2} e \,f^{2} x^{12}+176715 b^{2} d^{3} e^{2} f \,x^{12}+208845 a^{2} c \,d^{2} f^{3} x^{10}+208845 a^{2} d^{3} e \,f^{2} x^{10}+417690 a b \,c^{2} d \,f^{3} x^{10}+1253070 a b c \,d^{2} e \,f^{2} x^{10}+417690 a b \,d^{3} e^{2} f \,x^{10}+69615 b^{2} c^{3} f^{3} x^{10}+626535 b^{2} c^{2} d e \,f^{2} x^{10}+626535 b^{2} c \,d^{2} e^{2} f \,x^{10}+69615 b^{2} d^{3} e^{3} x^{10}+255255 a^{2} c^{2} d \,f^{3} x^{8}+765765 a^{2} c \,d^{2} e \,f^{2} x^{8}+255255 a^{2} d^{3} e^{2} f \,x^{8}+170170 a b \,c^{3} f^{3} x^{8}+1531530 a b \,c^{2} d e \,f^{2} x^{8}+1531530 a b c \,d^{2} e^{2} f \,x^{8}+170170 a b \,d^{3} e^{3} x^{8}+255255 b^{2} c^{3} e \,f^{2} x^{8}+765765 b^{2} c^{2} d \,e^{2} f \,x^{8}+255255 b^{2} c \,d^{2} e^{3} x^{8}+109395 a^{2} c^{3} f^{3} x^{6}+984555 a^{2} c^{2} d e \,f^{2} x^{6}+984555 a^{2} c \,d^{2} e^{2} f \,x^{6}+109395 a^{2} d^{3} e^{3} x^{6}+656370 a b \,c^{3} e \,f^{2} x^{6}+1969110 a b \,c^{2} d \,e^{2} f \,x^{6}+656370 a b c \,d^{2} e^{3} x^{6}+328185 b^{2} c^{3} e^{2} f \,x^{6}+328185 b^{2} c^{2} d \,e^{3} x^{6}+459459 a^{2} c^{3} e \,f^{2} x^{4}+1378377 a^{2} c^{2} d \,e^{2} f \,x^{4}+459459 a^{2} c \,d^{2} e^{3} x^{4}+918918 a b \,c^{3} e^{2} f \,x^{4}+918918 a b \,c^{2} d \,e^{3} x^{4}+153153 b^{2} c^{3} e^{3} x^{4}+765765 a^{2} c^{3} e^{2} f \,x^{2}+765765 a^{2} c^{2} d \,e^{3} x^{2}+510510 a b \,c^{3} e^{3} x^{2}+765765 a^{2} c^{3} e^{3}\right )}{765765} \] Input:
int((b*x^2+a)^2*(d*x^2+c)^3*(f*x^2+e)^3,x)
Output:
(x*(765765*a**2*c**3*e**3 + 765765*a**2*c**3*e**2*f*x**2 + 459459*a**2*c** 3*e*f**2*x**4 + 109395*a**2*c**3*f**3*x**6 + 765765*a**2*c**2*d*e**3*x**2 + 1378377*a**2*c**2*d*e**2*f*x**4 + 984555*a**2*c**2*d*e*f**2*x**6 + 25525 5*a**2*c**2*d*f**3*x**8 + 459459*a**2*c*d**2*e**3*x**4 + 984555*a**2*c*d** 2*e**2*f*x**6 + 765765*a**2*c*d**2*e*f**2*x**8 + 208845*a**2*c*d**2*f**3*x **10 + 109395*a**2*d**3*e**3*x**6 + 255255*a**2*d**3*e**2*f*x**8 + 208845* a**2*d**3*e*f**2*x**10 + 58905*a**2*d**3*f**3*x**12 + 510510*a*b*c**3*e**3 *x**2 + 918918*a*b*c**3*e**2*f*x**4 + 656370*a*b*c**3*e*f**2*x**6 + 170170 *a*b*c**3*f**3*x**8 + 918918*a*b*c**2*d*e**3*x**4 + 1969110*a*b*c**2*d*e** 2*f*x**6 + 1531530*a*b*c**2*d*e*f**2*x**8 + 417690*a*b*c**2*d*f**3*x**10 + 656370*a*b*c*d**2*e**3*x**6 + 1531530*a*b*c*d**2*e**2*f*x**8 + 1253070*a* b*c*d**2*e*f**2*x**10 + 353430*a*b*c*d**2*f**3*x**12 + 170170*a*b*d**3*e** 3*x**8 + 417690*a*b*d**3*e**2*f*x**10 + 353430*a*b*d**3*e*f**2*x**12 + 102 102*a*b*d**3*f**3*x**14 + 153153*b**2*c**3*e**3*x**4 + 328185*b**2*c**3*e* *2*f*x**6 + 255255*b**2*c**3*e*f**2*x**8 + 69615*b**2*c**3*f**3*x**10 + 32 8185*b**2*c**2*d*e**3*x**6 + 765765*b**2*c**2*d*e**2*f*x**8 + 626535*b**2* c**2*d*e*f**2*x**10 + 176715*b**2*c**2*d*f**3*x**12 + 255255*b**2*c*d**2*e **3*x**8 + 626535*b**2*c*d**2*e**2*f*x**10 + 530145*b**2*c*d**2*e*f**2*x** 12 + 153153*b**2*c*d**2*f**3*x**14 + 69615*b**2*d**3*e**3*x**10 + 176715*b **2*d**3*e**2*f*x**12 + 153153*b**2*d**3*e*f**2*x**14 + 45045*b**2*d**3...