Integrand size = 26, antiderivative size = 226 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=a c^2 e^3 x+\frac {1}{3} c e^2 (b c e+2 a d e+3 a c f) x^3+\frac {1}{5} e \left (b c e (2 d e+3 c f)+a \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (a f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+b e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^7+\frac {1}{9} f \left (a d f (3 d e+2 c f)+b \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x^9+\frac {1}{11} d f^2 (3 b d e+2 b c f+a d f) x^{11}+\frac {1}{13} b d^2 f^3 x^{13} \] Output:
a*c^2*e^3*x+1/3*c*e^2*(3*a*c*f+2*a*d*e+b*c*e)*x^3+1/5*e*(b*c*e*(3*c*f+2*d* e)+a*(3*c^2*f^2+6*c*d*e*f+d^2*e^2))*x^5+1/7*(a*f*(c^2*f^2+6*c*d*e*f+3*d^2* e^2)+b*e*(3*c^2*f^2+6*c*d*e*f+d^2*e^2))*x^7+1/9*f*(a*d*f*(2*c*f+3*d*e)+b*( c^2*f^2+6*c*d*e*f+3*d^2*e^2))*x^9+1/11*d*f^2*(a*d*f+2*b*c*f+3*b*d*e)*x^11+ 1/13*b*d^2*f^3*x^13
Time = 0.09 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=a c^2 e^3 x+\frac {1}{3} c e^2 (b c e+2 a d e+3 a c f) x^3+\frac {1}{5} e \left (b c e (2 d e+3 c f)+a \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (a f \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )+b e \left (d^2 e^2+6 c d e f+3 c^2 f^2\right )\right ) x^7+\frac {1}{9} f \left (a d f (3 d e+2 c f)+b \left (3 d^2 e^2+6 c d e f+c^2 f^2\right )\right ) x^9+\frac {1}{11} d f^2 (3 b d e+2 b c f+a d f) x^{11}+\frac {1}{13} b d^2 f^3 x^{13} \] Input:
Integrate[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^3,x]
Output:
a*c^2*e^3*x + (c*e^2*(b*c*e + 2*a*d*e + 3*a*c*f)*x^3)/3 + (e*(b*c*e*(2*d*e + 3*c*f) + a*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^5)/5 + ((a*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) + b*e*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^7)/7 + (f*(a*d*f*(3*d*e + 2*c*f) + b*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2))*x^9)/9 + (d*f^2*(3*b*d*e + 2*b*c*f + a*d*f)*x^11)/11 + (b*d^2*f^3*x^13)/13
Time = 0.45 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
| \(\displaystyle \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 396 |
| \(\displaystyle \int \left (f x^8 \left (a d f (2 c f+3 d e)+b \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )\right )+x^6 \left (a f \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+b e \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )\right )+e x^4 \left (a \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+b c e (3 c f+2 d e)\right )+c e^2 x^2 (3 a c f+2 a d e+b c e)+d f^2 x^{10} (a d f+2 b c f+3 b d e)+a c^2 e^3+b d^2 f^3 x^{12}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{9} f x^9 \left (a d f (2 c f+3 d e)+b \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )\right )+\frac {1}{7} x^7 \left (a f \left (c^2 f^2+6 c d e f+3 d^2 e^2\right )+b e \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )\right )+\frac {1}{5} e x^5 \left (a \left (3 c^2 f^2+6 c d e f+d^2 e^2\right )+b c e (3 c f+2 d e)\right )+\frac {1}{3} c e^2 x^3 (3 a c f+2 a d e+b c e)+\frac {1}{11} d f^2 x^{11} (a d f+2 b c f+3 b d e)+a c^2 e^3 x+\frac {1}{13} b d^2 f^3 x^{13}\) |
Input:
Int[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^3,x]
Output:
a*c^2*e^3*x + (c*e^2*(b*c*e + 2*a*d*e + 3*a*c*f)*x^3)/3 + (e*(b*c*e*(2*d*e + 3*c*f) + a*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^5)/5 + ((a*f*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2) + b*e*(d^2*e^2 + 6*c*d*e*f + 3*c^2*f^2))*x^7)/7 + (f*(a*d*f*(3*d*e + 2*c*f) + b*(3*d^2*e^2 + 6*c*d*e*f + c^2*f^2))*x^9)/9 + (d*f^2*(3*b*d*e + 2*b*c*f + a*d*f)*x^11)/11 + (b*d^2*f^3*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 = 0.51 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\frac {b \,d^{2} f^{3} x^{13}}{13}+\frac {\left (\left (a \,d^{2}+2 b c d \right ) f^{3}+3 b \,d^{2} e \,f^{2}\right ) x^{11}}{11}+\frac {\left (\left (2 a c d +b \,c^{2}\right ) f^{3}+3 \left (a \,d^{2}+2 b c d \right ) e \,f^{2}+3 b \,d^{2} e^{2} f \right ) x^{9}}{9}+\frac {\left (a \,c^{2} f^{3}+3 \left (2 a c d +b \,c^{2}\right ) e \,f^{2}+3 \left (a \,d^{2}+2 b c d \right ) e^{2} f +e^{3} b \,d^{2}\right ) x^{7}}{7}+\frac {\left (3 a \,c^{2} e \,f^{2}+3 \left (2 a c d +b \,c^{2}\right ) e^{2} f +\left (a \,d^{2}+2 b c d \right ) e^{3}\right ) x^{5}}{5}+\frac {\left (3 a \,c^{2} e^{2} f +\left (2 a c d +b \,c^{2}\right ) e^{3}\right ) x^{3}}{3}+a \,c^{2} e^{3} x\) | \(237\) |
| norman | \(\frac {b \,d^{2} f^{3} x^{13}}{13}+\left (\frac {1}{11} a \,d^{2} f^{3}+\frac {2}{11} b c d \,f^{3}+\frac {3}{11} b \,d^{2} e \,f^{2}\right ) x^{11}+\left (\frac {2}{9} a c d \,f^{3}+\frac {1}{3} a \,d^{2} e \,f^{2}+\frac {1}{9} b \,c^{2} f^{3}+\frac {2}{3} b c d e \,f^{2}+\frac {1}{3} b \,d^{2} e^{2} f \right ) x^{9}+\left (\frac {1}{7} a \,c^{2} f^{3}+\frac {6}{7} a c d e \,f^{2}+\frac {3}{7} a \,d^{2} e^{2} f +\frac {3}{7} b \,c^{2} e \,f^{2}+\frac {6}{7} b c d \,e^{2} f +\frac {1}{7} e^{3} b \,d^{2}\right ) x^{7}+\left (\frac {3}{5} a \,c^{2} e \,f^{2}+\frac {6}{5} a c d \,e^{2} f +\frac {1}{5} a \,d^{2} e^{3}+\frac {3}{5} b \,c^{2} e^{2} f +\frac {2}{5} b c d \,e^{3}\right ) x^{5}+\left (a \,c^{2} e^{2} f +\frac {2}{3} a c d \,e^{3}+\frac {1}{3} b \,c^{2} e^{3}\right ) x^{3}+a \,c^{2} e^{3} x\) | \(249\) |
| gosper | \(\frac {1}{13} b \,d^{2} f^{3} x^{13}+\frac {1}{11} x^{11} a \,d^{2} f^{3}+\frac {2}{11} x^{11} b c d \,f^{3}+\frac {3}{11} x^{11} b \,d^{2} e \,f^{2}+\frac {2}{9} x^{9} a c d \,f^{3}+\frac {1}{3} x^{9} a \,d^{2} e \,f^{2}+\frac {1}{9} x^{9} b \,c^{2} f^{3}+\frac {2}{3} x^{9} b c d e \,f^{2}+\frac {1}{3} x^{9} b \,d^{2} e^{2} f +\frac {1}{7} x^{7} a \,c^{2} f^{3}+\frac {6}{7} x^{7} a c d e \,f^{2}+\frac {3}{7} x^{7} a \,d^{2} e^{2} f +\frac {3}{7} x^{7} b \,c^{2} e \,f^{2}+\frac {6}{7} x^{7} b c d \,e^{2} f +\frac {1}{7} x^{7} e^{3} b \,d^{2}+\frac {3}{5} x^{5} a \,c^{2} e \,f^{2}+\frac {6}{5} x^{5} a c d \,e^{2} f +\frac {1}{5} x^{5} a \,d^{2} e^{3}+\frac {3}{5} x^{5} b \,c^{2} e^{2} f +\frac {2}{5} x^{5} b c d \,e^{3}+x^{3} a \,c^{2} e^{2} f +\frac {2}{3} x^{3} a c d \,e^{3}+\frac {1}{3} x^{3} b \,c^{2} e^{3}+a \,c^{2} e^{3} x\) | \(290\) |
| risch | \(\frac {1}{13} b \,d^{2} f^{3} x^{13}+\frac {1}{11} x^{11} a \,d^{2} f^{3}+\frac {2}{11} x^{11} b c d \,f^{3}+\frac {3}{11} x^{11} b \,d^{2} e \,f^{2}+\frac {2}{9} x^{9} a c d \,f^{3}+\frac {1}{3} x^{9} a \,d^{2} e \,f^{2}+\frac {1}{9} x^{9} b \,c^{2} f^{3}+\frac {2}{3} x^{9} b c d e \,f^{2}+\frac {1}{3} x^{9} b \,d^{2} e^{2} f +\frac {1}{7} x^{7} a \,c^{2} f^{3}+\frac {6}{7} x^{7} a c d e \,f^{2}+\frac {3}{7} x^{7} a \,d^{2} e^{2} f +\frac {3}{7} x^{7} b \,c^{2} e \,f^{2}+\frac {6}{7} x^{7} b c d \,e^{2} f +\frac {1}{7} x^{7} e^{3} b \,d^{2}+\frac {3}{5} x^{5} a \,c^{2} e \,f^{2}+\frac {6}{5} x^{5} a c d \,e^{2} f +\frac {1}{5} x^{5} a \,d^{2} e^{3}+\frac {3}{5} x^{5} b \,c^{2} e^{2} f +\frac {2}{5} x^{5} b c d \,e^{3}+x^{3} a \,c^{2} e^{2} f +\frac {2}{3} x^{3} a c d \,e^{3}+\frac {1}{3} x^{3} b \,c^{2} e^{3}+a \,c^{2} e^{3} x\) | \(290\) |
| parallelrisch | \(\frac {1}{13} b \,d^{2} f^{3} x^{13}+\frac {1}{11} x^{11} a \,d^{2} f^{3}+\frac {2}{11} x^{11} b c d \,f^{3}+\frac {3}{11} x^{11} b \,d^{2} e \,f^{2}+\frac {2}{9} x^{9} a c d \,f^{3}+\frac {1}{3} x^{9} a \,d^{2} e \,f^{2}+\frac {1}{9} x^{9} b \,c^{2} f^{3}+\frac {2}{3} x^{9} b c d e \,f^{2}+\frac {1}{3} x^{9} b \,d^{2} e^{2} f +\frac {1}{7} x^{7} a \,c^{2} f^{3}+\frac {6}{7} x^{7} a c d e \,f^{2}+\frac {3}{7} x^{7} a \,d^{2} e^{2} f +\frac {3}{7} x^{7} b \,c^{2} e \,f^{2}+\frac {6}{7} x^{7} b c d \,e^{2} f +\frac {1}{7} x^{7} e^{3} b \,d^{2}+\frac {3}{5} x^{5} a \,c^{2} e \,f^{2}+\frac {6}{5} x^{5} a c d \,e^{2} f +\frac {1}{5} x^{5} a \,d^{2} e^{3}+\frac {3}{5} x^{5} b \,c^{2} e^{2} f +\frac {2}{5} x^{5} b c d \,e^{3}+x^{3} a \,c^{2} e^{2} f +\frac {2}{3} x^{3} a c d \,e^{3}+\frac {1}{3} x^{3} b \,c^{2} e^{3}+a \,c^{2} e^{3} x\) | \(290\) |
| orering | \(\frac {x \left (3465 b \,d^{2} f^{3} x^{12}+4095 a \,d^{2} f^{3} x^{10}+8190 b c d \,f^{3} x^{10}+12285 b \,d^{2} e \,f^{2} x^{10}+10010 a c d \,f^{3} x^{8}+15015 a \,d^{2} e \,f^{2} x^{8}+5005 b \,c^{2} f^{3} x^{8}+30030 b c d e \,f^{2} x^{8}+15015 b \,d^{2} e^{2} f \,x^{8}+6435 a \,c^{2} f^{3} x^{6}+38610 a c d e \,f^{2} x^{6}+19305 a \,d^{2} e^{2} f \,x^{6}+19305 b \,c^{2} e \,f^{2} x^{6}+38610 b c d \,e^{2} f \,x^{6}+6435 b \,d^{2} e^{3} x^{6}+27027 a \,c^{2} e \,f^{2} x^{4}+54054 a c d \,e^{2} f \,x^{4}+9009 a \,d^{2} e^{3} x^{4}+27027 b \,c^{2} e^{2} f \,x^{4}+18018 b c d \,e^{3} x^{4}+45045 a \,c^{2} e^{2} f \,x^{2}+30030 a c d \,e^{3} x^{2}+15015 b \,c^{2} e^{3} x^{2}+45045 a \,c^{2} e^{3}\right )}{45045}\) | \(294\) |
Input:
int((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e)^3,x,method=_RETURNVERBOSE)
Output:
1/13*b*d^2*f^3*x^13+1/11*((a*d^2+2*b*c*d)*f^3+3*b*d^2*e*f^2)*x^11+1/9*((2* a*c*d+b*c^2)*f^3+3*(a*d^2+2*b*c*d)*e*f^2+3*b*d^2*e^2*f)*x^9+1/7*(a*c^2*f^3 +3*(2*a*c*d+b*c^2)*e*f^2+3*(a*d^2+2*b*c*d)*e^2*f+e^3*b*d^2)*x^7+1/5*(3*a*c ^2*e*f^2+3*(2*a*c*d+b*c^2)*e^2*f+(a*d^2+2*b*c*d)*e^3)*x^5+1/3*(3*a*c^2*e^2 *f+(2*a*c*d+b*c^2)*e^3)*x^3+a*c^2*e^3*x
Time = 0.07 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.04 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\frac {1}{13} \, b d^{2} f^{3} x^{13} + \frac {1}{11} \, {\left (3 \, b d^{2} e f^{2} + {\left (2 \, b c d + a d^{2}\right )} f^{3}\right )} x^{11} + \frac {1}{9} \, {\left (3 \, b d^{2} e^{2} f + 3 \, {\left (2 \, b c d + a d^{2}\right )} e f^{2} + {\left (b c^{2} + 2 \, a c d\right )} f^{3}\right )} x^{9} + \frac {1}{7} \, {\left (b d^{2} e^{3} + a c^{2} f^{3} + 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f + 3 \, {\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )} x^{7} + a c^{2} e^{3} x + \frac {1}{5} \, {\left (3 \, a c^{2} e f^{2} + {\left (2 \, b c d + a d^{2}\right )} e^{3} + 3 \, {\left (b c^{2} + 2 \, a c d\right )} e^{2} f\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a c^{2} e^{2} f + {\left (b c^{2} + 2 \, a c d\right )} e^{3}\right )} x^{3} \] Input:
integrate((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e)^3,x, algorithm="fricas")
Output:
1/13*b*d^2*f^3*x^13 + 1/11*(3*b*d^2*e*f^2 + (2*b*c*d + a*d^2)*f^3)*x^11 + 1/9*(3*b*d^2*e^2*f + 3*(2*b*c*d + a*d^2)*e*f^2 + (b*c^2 + 2*a*c*d)*f^3)*x^ 9 + 1/7*(b*d^2*e^3 + a*c^2*f^3 + 3*(2*b*c*d + a*d^2)*e^2*f + 3*(b*c^2 + 2* a*c*d)*e*f^2)*x^7 + a*c^2*e^3*x + 1/5*(3*a*c^2*e*f^2 + (2*b*c*d + a*d^2)*e ^3 + 3*(b*c^2 + 2*a*c*d)*e^2*f)*x^5 + 1/3*(3*a*c^2*e^2*f + (b*c^2 + 2*a*c* d)*e^3)*x^3
Time = 0.04 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.35 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=a c^{2} e^{3} x + \frac {b d^{2} f^{3} x^{13}}{13} + x^{11} \left (\frac {a d^{2} f^{3}}{11} + \frac {2 b c d f^{3}}{11} + \frac {3 b d^{2} e f^{2}}{11}\right ) + x^{9} \cdot \left (\frac {2 a c d f^{3}}{9} + \frac {a d^{2} e f^{2}}{3} + \frac {b c^{2} f^{3}}{9} + \frac {2 b c d e f^{2}}{3} + \frac {b d^{2} e^{2} f}{3}\right ) + x^{7} \left (\frac {a c^{2} f^{3}}{7} + \frac {6 a c d e f^{2}}{7} + \frac {3 a d^{2} e^{2} f}{7} + \frac {3 b c^{2} e f^{2}}{7} + \frac {6 b c d e^{2} f}{7} + \frac {b d^{2} e^{3}}{7}\right ) + x^{5} \cdot \left (\frac {3 a c^{2} e f^{2}}{5} + \frac {6 a c d e^{2} f}{5} + \frac {a d^{2} e^{3}}{5} + \frac {3 b c^{2} e^{2} f}{5} + \frac {2 b c d e^{3}}{5}\right ) + x^{3} \left (a c^{2} e^{2} f + \frac {2 a c d e^{3}}{3} + \frac {b c^{2} e^{3}}{3}\right ) \] Input:
integrate((b*x**2+a)*(d*x**2+c)**2*(f*x**2+e)**3,x)
Output:
a*c**2*e**3*x + b*d**2*f**3*x**13/13 + x**11*(a*d**2*f**3/11 + 2*b*c*d*f** 3/11 + 3*b*d**2*e*f**2/11) + x**9*(2*a*c*d*f**3/9 + a*d**2*e*f**2/3 + b*c* *2*f**3/9 + 2*b*c*d*e*f**2/3 + b*d**2*e**2*f/3) + x**7*(a*c**2*f**3/7 + 6* a*c*d*e*f**2/7 + 3*a*d**2*e**2*f/7 + 3*b*c**2*e*f**2/7 + 6*b*c*d*e**2*f/7 + b*d**2*e**3/7) + x**5*(3*a*c**2*e*f**2/5 + 6*a*c*d*e**2*f/5 + a*d**2*e** 3/5 + 3*b*c**2*e**2*f/5 + 2*b*c*d*e**3/5) + x**3*(a*c**2*e**2*f + 2*a*c*d* e**3/3 + b*c**2*e**3/3)
Time = 0.03 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.04 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\frac {1}{13} \, b d^{2} f^{3} x^{13} + \frac {1}{11} \, {\left (3 \, b d^{2} e f^{2} + {\left (2 \, b c d + a d^{2}\right )} f^{3}\right )} x^{11} + \frac {1}{9} \, {\left (3 \, b d^{2} e^{2} f + 3 \, {\left (2 \, b c d + a d^{2}\right )} e f^{2} + {\left (b c^{2} + 2 \, a c d\right )} f^{3}\right )} x^{9} + \frac {1}{7} \, {\left (b d^{2} e^{3} + a c^{2} f^{3} + 3 \, {\left (2 \, b c d + a d^{2}\right )} e^{2} f + 3 \, {\left (b c^{2} + 2 \, a c d\right )} e f^{2}\right )} x^{7} + a c^{2} e^{3} x + \frac {1}{5} \, {\left (3 \, a c^{2} e f^{2} + {\left (2 \, b c d + a d^{2}\right )} e^{3} + 3 \, {\left (b c^{2} + 2 \, a c d\right )} e^{2} f\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a c^{2} e^{2} f + {\left (b c^{2} + 2 \, a c d\right )} e^{3}\right )} x^{3} \] Input:
integrate((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e)^3,x, algorithm="maxima")
Output:
1/13*b*d^2*f^3*x^13 + 1/11*(3*b*d^2*e*f^2 + (2*b*c*d + a*d^2)*f^3)*x^11 + 1/9*(3*b*d^2*e^2*f + 3*(2*b*c*d + a*d^2)*e*f^2 + (b*c^2 + 2*a*c*d)*f^3)*x^ 9 + 1/7*(b*d^2*e^3 + a*c^2*f^3 + 3*(2*b*c*d + a*d^2)*e^2*f + 3*(b*c^2 + 2* a*c*d)*e*f^2)*x^7 + a*c^2*e^3*x + 1/5*(3*a*c^2*e*f^2 + (2*b*c*d + a*d^2)*e ^3 + 3*(b*c^2 + 2*a*c*d)*e^2*f)*x^5 + 1/3*(3*a*c^2*e^2*f + (b*c^2 + 2*a*c* d)*e^3)*x^3
Time = 0.13 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.28 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\frac {1}{13} \, b d^{2} f^{3} x^{13} + \frac {3}{11} \, b d^{2} e f^{2} x^{11} + \frac {2}{11} \, b c d f^{3} x^{11} + \frac {1}{11} \, a d^{2} f^{3} x^{11} + \frac {1}{3} \, b d^{2} e^{2} f x^{9} + \frac {2}{3} \, b c d e f^{2} x^{9} + \frac {1}{3} \, a d^{2} e f^{2} x^{9} + \frac {1}{9} \, b c^{2} f^{3} x^{9} + \frac {2}{9} \, a c d f^{3} x^{9} + \frac {1}{7} \, b d^{2} e^{3} x^{7} + \frac {6}{7} \, b c d e^{2} f x^{7} + \frac {3}{7} \, a d^{2} e^{2} f x^{7} + \frac {3}{7} \, b c^{2} e f^{2} x^{7} + \frac {6}{7} \, a c d e f^{2} x^{7} + \frac {1}{7} \, a c^{2} f^{3} x^{7} + \frac {2}{5} \, b c d e^{3} x^{5} + \frac {1}{5} \, a d^{2} e^{3} x^{5} + \frac {3}{5} \, b c^{2} e^{2} f x^{5} + \frac {6}{5} \, a c d e^{2} f x^{5} + \frac {3}{5} \, a c^{2} e f^{2} x^{5} + \frac {1}{3} \, b c^{2} e^{3} x^{3} + \frac {2}{3} \, a c d e^{3} x^{3} + a c^{2} e^{2} f x^{3} + a c^{2} e^{3} x \] Input:
integrate((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e)^3,x, algorithm="giac")
Output:
1/13*b*d^2*f^3*x^13 + 3/11*b*d^2*e*f^2*x^11 + 2/11*b*c*d*f^3*x^11 + 1/11*a *d^2*f^3*x^11 + 1/3*b*d^2*e^2*f*x^9 + 2/3*b*c*d*e*f^2*x^9 + 1/3*a*d^2*e*f^ 2*x^9 + 1/9*b*c^2*f^3*x^9 + 2/9*a*c*d*f^3*x^9 + 1/7*b*d^2*e^3*x^7 + 6/7*b* c*d*e^2*f*x^7 + 3/7*a*d^2*e^2*f*x^7 + 3/7*b*c^2*e*f^2*x^7 + 6/7*a*c*d*e*f^ 2*x^7 + 1/7*a*c^2*f^3*x^7 + 2/5*b*c*d*e^3*x^5 + 1/5*a*d^2*e^3*x^5 + 3/5*b* c^2*e^2*f*x^5 + 6/5*a*c*d*e^2*f*x^5 + 3/5*a*c^2*e*f^2*x^5 + 1/3*b*c^2*e^3* x^3 + 2/3*a*c*d*e^3*x^3 + a*c^2*e^2*f*x^3 + a*c^2*e^3*x
Time = 0.11 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.03 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=x^5\,\left (\frac {3\,b\,c^2\,e^2\,f}{5}+\frac {3\,a\,c^2\,e\,f^2}{5}+\frac {2\,b\,c\,d\,e^3}{5}+\frac {6\,a\,c\,d\,e^2\,f}{5}+\frac {a\,d^2\,e^3}{5}\right )+x^9\,\left (\frac {b\,c^2\,f^3}{9}+\frac {2\,b\,c\,d\,e\,f^2}{3}+\frac {2\,a\,c\,d\,f^3}{9}+\frac {b\,d^2\,e^2\,f}{3}+\frac {a\,d^2\,e\,f^2}{3}\right )+x^7\,\left (\frac {3\,b\,c^2\,e\,f^2}{7}+\frac {a\,c^2\,f^3}{7}+\frac {6\,b\,c\,d\,e^2\,f}{7}+\frac {6\,a\,c\,d\,e\,f^2}{7}+\frac {b\,d^2\,e^3}{7}+\frac {3\,a\,d^2\,e^2\,f}{7}\right )+\frac {b\,d^2\,f^3\,x^{13}}{13}+\frac {c\,e^2\,x^3\,\left (3\,a\,c\,f+2\,a\,d\,e+b\,c\,e\right )}{3}+\frac {d\,f^2\,x^{11}\,\left (a\,d\,f+2\,b\,c\,f+3\,b\,d\,e\right )}{11}+a\,c^2\,e^3\,x \] Input:
int((a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^3,x)
Output:
x^5*((a*d^2*e^3)/5 + (2*b*c*d*e^3)/5 + (3*a*c^2*e*f^2)/5 + (3*b*c^2*e^2*f) /5 + (6*a*c*d*e^2*f)/5) + x^9*((b*c^2*f^3)/9 + (2*a*c*d*f^3)/9 + (a*d^2*e* f^2)/3 + (b*d^2*e^2*f)/3 + (2*b*c*d*e*f^2)/3) + x^7*((a*c^2*f^3)/7 + (b*d^ 2*e^3)/7 + (3*a*d^2*e^2*f)/7 + (3*b*c^2*e*f^2)/7 + (6*a*c*d*e*f^2)/7 + (6* b*c*d*e^2*f)/7) + (b*d^2*f^3*x^13)/13 + (c*e^2*x^3*(3*a*c*f + 2*a*d*e + b* c*e))/3 + (d*f^2*x^11*(a*d*f + 2*b*c*f + 3*b*d*e))/11 + a*c^2*e^3*x
Time = 0.15 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.30 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^3 \, dx=\frac {x \left (3465 b \,d^{2} f^{3} x^{12}+4095 a \,d^{2} f^{3} x^{10}+8190 b c d \,f^{3} x^{10}+12285 b \,d^{2} e \,f^{2} x^{10}+10010 a c d \,f^{3} x^{8}+15015 a \,d^{2} e \,f^{2} x^{8}+5005 b \,c^{2} f^{3} x^{8}+30030 b c d e \,f^{2} x^{8}+15015 b \,d^{2} e^{2} f \,x^{8}+6435 a \,c^{2} f^{3} x^{6}+38610 a c d e \,f^{2} x^{6}+19305 a \,d^{2} e^{2} f \,x^{6}+19305 b \,c^{2} e \,f^{2} x^{6}+38610 b c d \,e^{2} f \,x^{6}+6435 b \,d^{2} e^{3} x^{6}+27027 a \,c^{2} e \,f^{2} x^{4}+54054 a c d \,e^{2} f \,x^{4}+9009 a \,d^{2} e^{3} x^{4}+27027 b \,c^{2} e^{2} f \,x^{4}+18018 b c d \,e^{3} x^{4}+45045 a \,c^{2} e^{2} f \,x^{2}+30030 a c d \,e^{3} x^{2}+15015 b \,c^{2} e^{3} x^{2}+45045 a \,c^{2} e^{3}\right )}{45045} \] Input:
int((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e)^3,x)
Output:
(x*(45045*a*c**2*e**3 + 45045*a*c**2*e**2*f*x**2 + 27027*a*c**2*e*f**2*x** 4 + 6435*a*c**2*f**3*x**6 + 30030*a*c*d*e**3*x**2 + 54054*a*c*d*e**2*f*x** 4 + 38610*a*c*d*e*f**2*x**6 + 10010*a*c*d*f**3*x**8 + 9009*a*d**2*e**3*x** 4 + 19305*a*d**2*e**2*f*x**6 + 15015*a*d**2*e*f**2*x**8 + 4095*a*d**2*f**3 *x**10 + 15015*b*c**2*e**3*x**2 + 27027*b*c**2*e**2*f*x**4 + 19305*b*c**2* e*f**2*x**6 + 5005*b*c**2*f**3*x**8 + 18018*b*c*d*e**3*x**4 + 38610*b*c*d* e**2*f*x**6 + 30030*b*c*d*e*f**2*x**8 + 8190*b*c*d*f**3*x**10 + 6435*b*d** 2*e**3*x**6 + 15015*b*d**2*e**2*f*x**8 + 12285*b*d**2*e*f**2*x**10 + 3465* b*d**2*f**3*x**12))/45045