Integrand size = 24, antiderivative size = 94 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=a c e^2 x+\frac {1}{3} e (b c e+a d e+2 a c f) x^3+\frac {1}{5} (a f (2 d e+c f)+b e (d e+2 c f)) x^5+\frac {1}{7} f (2 b d e+b c f+a d f) x^7+\frac {1}{9} b d f^2 x^9 \] Output:
a*c*e^2*x+1/3*e*(2*a*c*f+a*d*e+b*c*e)*x^3+1/5*(a*f*(c*f+2*d*e)+b*e*(2*c*f+ d*e))*x^5+1/7*f*(a*d*f+b*c*f+2*b*d*e)*x^7+1/9*b*d*f^2*x^9
Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=a c e^2 x+\frac {1}{3} e (b c e+a d e+2 a c f) x^3+\frac {1}{5} \left (b d e^2+2 b c e f+2 a d e f+a c f^2\right ) x^5+\frac {1}{7} f (2 b d e+b c f+a d f) x^7+\frac {1}{9} b d f^2 x^9 \] Input:
Integrate[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^2,x]
Output:
a*c*e^2*x + (e*(b*c*e + a*d*e + 2*a*c*f)*x^3)/3 + ((b*d*e^2 + 2*b*c*e*f + 2*a*d*e*f + a*c*f^2)*x^5)/5 + (f*(2*b*d*e + b*c*f + a*d*f)*x^7)/7 + (b*d*f ^2*x^9)/9
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
\(\displaystyle \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx\) |
\(\Big \downarrow \) 396 |
\(\displaystyle \int \left (f x^6 (a d f+b c f+2 b d e)+x^4 (a f (c f+2 d e)+b e (2 c f+d e))+e x^2 (2 a c f+a d e+b c e)+a c e^2+b d f^2 x^8\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{7} f x^7 (a d f+b c f+2 b d e)+\frac {1}{5} x^5 (a f (c f+2 d e)+b e (2 c f+d e))+\frac {1}{3} e x^3 (2 a c f+a d e+b c e)+a c e^2 x+\frac {1}{9} b d f^2 x^9\) |
Input:
Int[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^2,x]
Output:
a*c*e^2*x + (e*(b*c*e + a*d*e + 2*a*c*f)*x^3)/3 + ((a*f*(2*d*e + c*f) + b* e*(d*e + 2*c*f))*x^5)/5 + (f*(2*b*d*e + b*c*f + a*d*f)*x^7)/7 + (b*d*f^2*x ^9)/9
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 = 0.46 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {b d \,f^{2} x^{9}}{9}+\frac {\left (\left (a d +b c \right ) f^{2}+2 d b e f \right ) x^{7}}{7}+\frac {\left (a c \,f^{2}+2 \left (a d +b c \right ) e f +b d \,e^{2}\right ) x^{5}}{5}+\frac {\left (2 a c e f +\left (a d +b c \right ) e^{2}\right ) x^{3}}{3}+a c \,e^{2} x\) | \(94\) |
norman | \(\frac {b d \,f^{2} x^{9}}{9}+\left (\frac {1}{7} a d \,f^{2}+\frac {1}{7} b c \,f^{2}+\frac {2}{7} d b e f \right ) x^{7}+\left (\frac {1}{5} a c \,f^{2}+\frac {2}{5} a d e f +\frac {2}{5} b c e f +\frac {1}{5} b d \,e^{2}\right ) x^{5}+\left (\frac {2}{3} a c e f +\frac {1}{3} a d \,e^{2}+\frac {1}{3} b c \,e^{2}\right ) x^{3}+a c \,e^{2} x\) | \(100\) |
gosper | \(\frac {1}{9} b d \,f^{2} x^{9}+\frac {1}{7} x^{7} a d \,f^{2}+\frac {1}{7} x^{7} b c \,f^{2}+\frac {2}{7} x^{7} d b e f +\frac {1}{5} x^{5} a c \,f^{2}+\frac {2}{5} x^{5} a d e f +\frac {2}{5} x^{5} b c e f +\frac {1}{5} x^{5} b d \,e^{2}+\frac {2}{3} x^{3} a c e f +\frac {1}{3} x^{3} a d \,e^{2}+\frac {1}{3} x^{3} b c \,e^{2}+a c \,e^{2} x\) | \(115\) |
risch | \(\frac {1}{9} b d \,f^{2} x^{9}+\frac {1}{7} x^{7} a d \,f^{2}+\frac {1}{7} x^{7} b c \,f^{2}+\frac {2}{7} x^{7} d b e f +\frac {1}{5} x^{5} a c \,f^{2}+\frac {2}{5} x^{5} a d e f +\frac {2}{5} x^{5} b c e f +\frac {1}{5} x^{5} b d \,e^{2}+\frac {2}{3} x^{3} a c e f +\frac {1}{3} x^{3} a d \,e^{2}+\frac {1}{3} x^{3} b c \,e^{2}+a c \,e^{2} x\) | \(115\) |
parallelrisch | \(\frac {1}{9} b d \,f^{2} x^{9}+\frac {1}{7} x^{7} a d \,f^{2}+\frac {1}{7} x^{7} b c \,f^{2}+\frac {2}{7} x^{7} d b e f +\frac {1}{5} x^{5} a c \,f^{2}+\frac {2}{5} x^{5} a d e f +\frac {2}{5} x^{5} b c e f +\frac {1}{5} x^{5} b d \,e^{2}+\frac {2}{3} x^{3} a c e f +\frac {1}{3} x^{3} a d \,e^{2}+\frac {1}{3} x^{3} b c \,e^{2}+a c \,e^{2} x\) | \(115\) |
orering | \(\frac {x \left (35 b d \,f^{2} x^{8}+45 a d \,f^{2} x^{6}+45 b c \,f^{2} x^{6}+90 b d e f \,x^{6}+63 a c \,f^{2} x^{4}+126 a d e f \,x^{4}+126 b c e f \,x^{4}+63 b d \,e^{2} x^{4}+210 a c e f \,x^{2}+105 a d \,e^{2} x^{2}+105 b c \,e^{2} x^{2}+315 a c \,e^{2}\right )}{315}\) | \(118\) |
Input:
int((b*x^2+a)*(d*x^2+c)*(f*x^2+e)^2,x,method=_RETURNVERBOSE)
Output:
1/9*b*d*f^2*x^9+1/7*((a*d+b*c)*f^2+2*d*b*e*f)*x^7+1/5*(a*c*f^2+2*(a*d+b*c) *e*f+b*d*e^2)*x^5+1/3*(2*a*c*e*f+(a*d+b*c)*e^2)*x^3+a*c*e^2*x
Time = 0.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\frac {1}{9} \, b d f^{2} x^{9} + \frac {1}{7} \, {\left (2 \, b d e f + {\left (b c + a d\right )} f^{2}\right )} x^{7} + \frac {1}{5} \, {\left (b d e^{2} + a c f^{2} + 2 \, {\left (b c + a d\right )} e f\right )} x^{5} + a c e^{2} x + \frac {1}{3} \, {\left (2 \, a c e f + {\left (b c + a d\right )} e^{2}\right )} x^{3} \] Input:
integrate((b*x^2+a)*(d*x^2+c)*(f*x^2+e)^2,x, algorithm="fricas")
Output:
1/9*b*d*f^2*x^9 + 1/7*(2*b*d*e*f + (b*c + a*d)*f^2)*x^7 + 1/5*(b*d*e^2 + a *c*f^2 + 2*(b*c + a*d)*e*f)*x^5 + a*c*e^2*x + 1/3*(2*a*c*e*f + (b*c + a*d) *e^2)*x^3
Time = 0.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.29 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=a c e^{2} x + \frac {b d f^{2} x^{9}}{9} + x^{7} \left (\frac {a d f^{2}}{7} + \frac {b c f^{2}}{7} + \frac {2 b d e f}{7}\right ) + x^{5} \left (\frac {a c f^{2}}{5} + \frac {2 a d e f}{5} + \frac {2 b c e f}{5} + \frac {b d e^{2}}{5}\right ) + x^{3} \cdot \left (\frac {2 a c e f}{3} + \frac {a d e^{2}}{3} + \frac {b c e^{2}}{3}\right ) \] Input:
integrate((b*x**2+a)*(d*x**2+c)*(f*x**2+e)**2,x)
Output:
a*c*e**2*x + b*d*f**2*x**9/9 + x**7*(a*d*f**2/7 + b*c*f**2/7 + 2*b*d*e*f/7 ) + x**5*(a*c*f**2/5 + 2*a*d*e*f/5 + 2*b*c*e*f/5 + b*d*e**2/5) + x**3*(2*a *c*e*f/3 + a*d*e**2/3 + b*c*e**2/3)
Time = 0.03 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.99 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\frac {1}{9} \, b d f^{2} x^{9} + \frac {1}{7} \, {\left (2 \, b d e f + {\left (b c + a d\right )} f^{2}\right )} x^{7} + \frac {1}{5} \, {\left (b d e^{2} + a c f^{2} + 2 \, {\left (b c + a d\right )} e f\right )} x^{5} + a c e^{2} x + \frac {1}{3} \, {\left (2 \, a c e f + {\left (b c + a d\right )} e^{2}\right )} x^{3} \] Input:
integrate((b*x^2+a)*(d*x^2+c)*(f*x^2+e)^2,x, algorithm="maxima")
Output:
1/9*b*d*f^2*x^9 + 1/7*(2*b*d*e*f + (b*c + a*d)*f^2)*x^7 + 1/5*(b*d*e^2 + a *c*f^2 + 2*(b*c + a*d)*e*f)*x^5 + a*c*e^2*x + 1/3*(2*a*c*e*f + (b*c + a*d) *e^2)*x^3
Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.21 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\frac {1}{9} \, b d f^{2} x^{9} + \frac {2}{7} \, b d e f x^{7} + \frac {1}{7} \, b c f^{2} x^{7} + \frac {1}{7} \, a d f^{2} x^{7} + \frac {1}{5} \, b d e^{2} x^{5} + \frac {2}{5} \, b c e f x^{5} + \frac {2}{5} \, a d e f x^{5} + \frac {1}{5} \, a c f^{2} x^{5} + \frac {1}{3} \, b c e^{2} x^{3} + \frac {1}{3} \, a d e^{2} x^{3} + \frac {2}{3} \, a c e f x^{3} + a c e^{2} x \] Input:
integrate((b*x^2+a)*(d*x^2+c)*(f*x^2+e)^2,x, algorithm="giac")
Output:
1/9*b*d*f^2*x^9 + 2/7*b*d*e*f*x^7 + 1/7*b*c*f^2*x^7 + 1/7*a*d*f^2*x^7 + 1/ 5*b*d*e^2*x^5 + 2/5*b*c*e*f*x^5 + 2/5*a*d*e*f*x^5 + 1/5*a*c*f^2*x^5 + 1/3* b*c*e^2*x^3 + 1/3*a*d*e^2*x^3 + 2/3*a*c*e*f*x^3 + a*c*e^2*x
Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=x^5\,\left (\frac {a\,c\,f^2}{5}+\frac {b\,d\,e^2}{5}+\frac {2\,a\,d\,e\,f}{5}+\frac {2\,b\,c\,e\,f}{5}\right )+x^3\,\left (\frac {a\,d\,e^2}{3}+\frac {b\,c\,e^2}{3}+\frac {2\,a\,c\,e\,f}{3}\right )+x^7\,\left (\frac {a\,d\,f^2}{7}+\frac {b\,c\,f^2}{7}+\frac {2\,b\,d\,e\,f}{7}\right )+a\,c\,e^2\,x+\frac {b\,d\,f^2\,x^9}{9} \] Input:
int((a + b*x^2)*(c + d*x^2)*(e + f*x^2)^2,x)
Output:
x^5*((a*c*f^2)/5 + (b*d*e^2)/5 + (2*a*d*e*f)/5 + (2*b*c*e*f)/5) + x^3*((a* d*e^2)/3 + (b*c*e^2)/3 + (2*a*c*e*f)/3) + x^7*((a*d*f^2)/7 + (b*c*f^2)/7 + (2*b*d*e*f)/7) + a*c*e^2*x + (b*d*f^2*x^9)/9
Time = 0.15 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.24 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^2 \, dx=\frac {x \left (35 b d \,f^{2} x^{8}+45 a d \,f^{2} x^{6}+45 b c \,f^{2} x^{6}+90 b d e f \,x^{6}+63 a c \,f^{2} x^{4}+126 a d e f \,x^{4}+126 b c e f \,x^{4}+63 b d \,e^{2} x^{4}+210 a c e f \,x^{2}+105 a d \,e^{2} x^{2}+105 b c \,e^{2} x^{2}+315 a c \,e^{2}\right )}{315} \] Input:
int((b*x^2+a)*(d*x^2+c)*(f*x^2+e)^2,x)
Output:
(x*(315*a*c*e**2 + 210*a*c*e*f*x**2 + 63*a*c*f**2*x**4 + 105*a*d*e**2*x**2 + 126*a*d*e*f*x**4 + 45*a*d*f**2*x**6 + 105*b*c*e**2*x**2 + 126*b*c*e*f*x **4 + 45*b*c*f**2*x**6 + 63*b*d*e**2*x**4 + 90*b*d*e*f*x**6 + 35*b*d*f**2* x**8))/315