Integrand size = 24, antiderivative size = 130 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=a c e^3 x+\frac {1}{3} e^2 (b c e+a d e+3 a c f) x^3+\frac {1}{5} e (3 a f (d e+c f)+b e (d e+3 c f)) x^5+\frac {1}{7} f (3 b e (d e+c f)+a f (3 d e+c f)) x^7+\frac {1}{9} f^2 (3 b d e+b c f+a d f) x^9+\frac {1}{11} b d f^3 x^{11} \] Output:
a*c*e^3*x+1/3*e^2*(3*a*c*f+a*d*e+b*c*e)*x^3+1/5*e*(3*a*f*(c*f+d*e)+b*e*(3* c*f+d*e))*x^5+1/7*f*(3*b*e*(c*f+d*e)+a*f*(c*f+3*d*e))*x^7+1/9*f^2*(a*d*f+b *c*f+3*b*d*e)*x^9+1/11*b*d*f^3*x^11
Time = 0.07 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=a c e^3 x+\frac {1}{3} e^2 (b c e+a d e+3 a c f) x^3+\frac {1}{5} e (3 a f (d e+c f)+b e (d e+3 c f)) x^5+\frac {1}{7} f (3 b e (d e+c f)+a f (3 d e+c f)) x^7+\frac {1}{9} f^2 (3 b d e+b c f+a d f) x^9+\frac {1}{11} b d f^3 x^{11} \] Input:
Integrate[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^3,x]
Output:
a*c*e^3*x + (e^2*(b*c*e + a*d*e + 3*a*c*f)*x^3)/3 + (e*(3*a*f*(d*e + c*f) + b*e*(d*e + 3*c*f))*x^5)/5 + (f*(3*b*e*(d*e + c*f) + a*f*(3*d*e + c*f))*x ^7)/7 + (f^2*(3*b*d*e + b*c*f + a*d*f)*x^9)/9 + (b*d*f^3*x^11)/11
Time = 0.32 (sec) , antiderivative size = 130, 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.
| \(\displaystyle \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 396 |
| \(\displaystyle \int \left (e^2 x^2 (3 a c f+a d e+b c e)+f^2 x^8 (a d f+b c f+3 b d e)+f x^6 (a f (c f+3 d e)+3 b e (c f+d e))+e x^4 (3 a f (c f+d e)+b e (3 c f+d e))+a c e^3+b d f^3 x^{10}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} e^2 x^3 (3 a c f+a d e+b c e)+\frac {1}{9} f^2 x^9 (a d f+b c f+3 b d e)+\frac {1}{7} f x^7 (a f (c f+3 d e)+3 b e (c f+d e))+\frac {1}{5} e x^5 (3 a f (c f+d e)+b e (3 c f+d e))+a c e^3 x+\frac {1}{11} b d f^3 x^{11}\) |
Input:
Int[(a + b*x^2)*(c + d*x^2)*(e + f*x^2)^3,x]
Output:
a*c*e^3*x + (e^2*(b*c*e + a*d*e + 3*a*c*f)*x^3)/3 + (e*(3*a*f*(d*e + c*f) + b*e*(d*e + 3*c*f))*x^5)/5 + (f*(3*b*e*(d*e + c*f) + a*f*(3*d*e + c*f))*x ^7)/7 + (f^2*(3*b*d*e + b*c*f + a*d*f)*x^9)/9 + (b*d*f^3*x^11)/11
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 = 0.45 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.04
| method | result | size |
| default | \(\frac {b d \,f^{3} x^{11}}{11}+\frac {\left (\left (a d +b c \right ) f^{3}+3 b d e \,f^{2}\right ) x^{9}}{9}+\frac {\left (a c \,f^{3}+3 \left (a d +b c \right ) e \,f^{2}+3 b d \,e^{2} f \right ) x^{7}}{7}+\frac {\left (3 a c e \,f^{2}+3 \left (a d +b c \right ) e^{2} f +b d \,e^{3}\right ) x^{5}}{5}+\frac {\left (3 a c \,e^{2} f +\left (a d +b c \right ) e^{3}\right ) x^{3}}{3}+a c \,e^{3} x\) | \(135\) |
| norman | \(\frac {b d \,f^{3} x^{11}}{11}+\left (\frac {1}{9} a d \,f^{3}+\frac {1}{9} b c \,f^{3}+\frac {1}{3} b d e \,f^{2}\right ) x^{9}+\left (\frac {1}{7} a c \,f^{3}+\frac {3}{7} a d e \,f^{2}+\frac {3}{7} b c e \,f^{2}+\frac {3}{7} b d \,e^{2} f \right ) x^{7}+\left (\frac {3}{5} a c e \,f^{2}+\frac {3}{5} a d \,e^{2} f +\frac {3}{5} b c \,e^{2} f +\frac {1}{5} b d \,e^{3}\right ) x^{5}+\left (a c \,e^{2} f +\frac {1}{3} a d \,e^{3}+\frac {1}{3} b c \,e^{3}\right ) x^{3}+a c \,e^{3} x\) | \(144\) |
| gosper | \(\frac {1}{11} b d \,f^{3} x^{11}+\frac {1}{9} x^{9} a d \,f^{3}+\frac {1}{9} x^{9} b c \,f^{3}+\frac {1}{3} x^{9} b d e \,f^{2}+\frac {1}{7} x^{7} a c \,f^{3}+\frac {3}{7} x^{7} a d e \,f^{2}+\frac {3}{7} x^{7} b c e \,f^{2}+\frac {3}{7} x^{7} b d \,e^{2} f +\frac {3}{5} x^{5} a c e \,f^{2}+\frac {3}{5} x^{5} a d \,e^{2} f +\frac {3}{5} x^{5} b c \,e^{2} f +\frac {1}{5} x^{5} b d \,e^{3}+x^{3} a c \,e^{2} f +\frac {1}{3} x^{3} a d \,e^{3}+\frac {1}{3} x^{3} b c \,e^{3}+a c \,e^{3} x\) | \(166\) |
| risch | \(\frac {1}{11} b d \,f^{3} x^{11}+\frac {1}{9} x^{9} a d \,f^{3}+\frac {1}{9} x^{9} b c \,f^{3}+\frac {1}{3} x^{9} b d e \,f^{2}+\frac {1}{7} x^{7} a c \,f^{3}+\frac {3}{7} x^{7} a d e \,f^{2}+\frac {3}{7} x^{7} b c e \,f^{2}+\frac {3}{7} x^{7} b d \,e^{2} f +\frac {3}{5} x^{5} a c e \,f^{2}+\frac {3}{5} x^{5} a d \,e^{2} f +\frac {3}{5} x^{5} b c \,e^{2} f +\frac {1}{5} x^{5} b d \,e^{3}+x^{3} a c \,e^{2} f +\frac {1}{3} x^{3} a d \,e^{3}+\frac {1}{3} x^{3} b c \,e^{3}+a c \,e^{3} x\) | \(166\) |
| parallelrisch | \(\frac {1}{11} b d \,f^{3} x^{11}+\frac {1}{9} x^{9} a d \,f^{3}+\frac {1}{9} x^{9} b c \,f^{3}+\frac {1}{3} x^{9} b d e \,f^{2}+\frac {1}{7} x^{7} a c \,f^{3}+\frac {3}{7} x^{7} a d e \,f^{2}+\frac {3}{7} x^{7} b c e \,f^{2}+\frac {3}{7} x^{7} b d \,e^{2} f +\frac {3}{5} x^{5} a c e \,f^{2}+\frac {3}{5} x^{5} a d \,e^{2} f +\frac {3}{5} x^{5} b c \,e^{2} f +\frac {1}{5} x^{5} b d \,e^{3}+x^{3} a c \,e^{2} f +\frac {1}{3} x^{3} a d \,e^{3}+\frac {1}{3} x^{3} b c \,e^{3}+a c \,e^{3} x\) | \(166\) |
| orering | \(\frac {x \left (315 b d \,f^{3} x^{10}+385 a d \,f^{3} x^{8}+385 b c \,f^{3} x^{8}+1155 b d e \,f^{2} x^{8}+495 a c \,f^{3} x^{6}+1485 a d e \,f^{2} x^{6}+1485 b c e \,f^{2} x^{6}+1485 b d \,e^{2} f \,x^{6}+2079 a c e \,f^{2} x^{4}+2079 a d \,e^{2} f \,x^{4}+2079 b c \,e^{2} f \,x^{4}+693 b d \,e^{3} x^{4}+3465 a c \,e^{2} f \,x^{2}+1155 a d \,e^{3} x^{2}+1155 b c \,e^{3} x^{2}+3465 a c \,e^{3}\right )}{3465}\) | \(170\) |
Input:
int((b*x^2+a)*(d*x^2+c)*(f*x^2+e)^3,x,method=_RETURNVERBOSE)
Output:
1/11*b*d*f^3*x^11+1/9*((a*d+b*c)*f^3+3*b*d*e*f^2)*x^9+1/7*(a*c*f^3+3*(a*d+ b*c)*e*f^2+3*b*d*e^2*f)*x^7+1/5*(3*a*c*e*f^2+3*(a*d+b*c)*e^2*f+b*d*e^3)*x^ 5+1/3*(3*a*c*e^2*f+(a*d+b*c)*e^3)*x^3+a*c*e^3*x
Time = 0.07 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.03 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=\frac {1}{11} \, b d f^{3} x^{11} + \frac {1}{9} \, {\left (3 \, b d e f^{2} + {\left (b c + a d\right )} f^{3}\right )} x^{9} + \frac {1}{7} \, {\left (3 \, b d e^{2} f + a c f^{3} + 3 \, {\left (b c + a d\right )} e f^{2}\right )} x^{7} + a c e^{3} x + \frac {1}{5} \, {\left (b d e^{3} + 3 \, a c e f^{2} + 3 \, {\left (b c + a d\right )} e^{2} f\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a c e^{2} f + {\left (b c + a d\right )} e^{3}\right )} x^{3} \] Input:
integrate((b*x^2+a)*(d*x^2+c)*(f*x^2+e)^3,x, algorithm="fricas")
Output:
1/11*b*d*f^3*x^11 + 1/9*(3*b*d*e*f^2 + (b*c + a*d)*f^3)*x^9 + 1/7*(3*b*d*e ^2*f + a*c*f^3 + 3*(b*c + a*d)*e*f^2)*x^7 + a*c*e^3*x + 1/5*(b*d*e^3 + 3*a *c*e*f^2 + 3*(b*c + a*d)*e^2*f)*x^5 + 1/3*(3*a*c*e^2*f + (b*c + a*d)*e^3)* x^3
Time = 0.03 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.33 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=a c e^{3} x + \frac {b d f^{3} x^{11}}{11} + x^{9} \left (\frac {a d f^{3}}{9} + \frac {b c f^{3}}{9} + \frac {b d e f^{2}}{3}\right ) + x^{7} \left (\frac {a c f^{3}}{7} + \frac {3 a d e f^{2}}{7} + \frac {3 b c e f^{2}}{7} + \frac {3 b d e^{2} f}{7}\right ) + x^{5} \cdot \left (\frac {3 a c e f^{2}}{5} + \frac {3 a d e^{2} f}{5} + \frac {3 b c e^{2} f}{5} + \frac {b d e^{3}}{5}\right ) + x^{3} \left (a c e^{2} f + \frac {a d e^{3}}{3} + \frac {b c e^{3}}{3}\right ) \] Input:
integrate((b*x**2+a)*(d*x**2+c)*(f*x**2+e)**3,x)
Output:
a*c*e**3*x + b*d*f**3*x**11/11 + x**9*(a*d*f**3/9 + b*c*f**3/9 + b*d*e*f** 2/3) + x**7*(a*c*f**3/7 + 3*a*d*e*f**2/7 + 3*b*c*e*f**2/7 + 3*b*d*e**2*f/7 ) + x**5*(3*a*c*e*f**2/5 + 3*a*d*e**2*f/5 + 3*b*c*e**2*f/5 + b*d*e**3/5) + x**3*(a*c*e**2*f + a*d*e**3/3 + b*c*e**3/3)
Time = 0.03 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.03 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=\frac {1}{11} \, b d f^{3} x^{11} + \frac {1}{9} \, {\left (3 \, b d e f^{2} + {\left (b c + a d\right )} f^{3}\right )} x^{9} + \frac {1}{7} \, {\left (3 \, b d e^{2} f + a c f^{3} + 3 \, {\left (b c + a d\right )} e f^{2}\right )} x^{7} + a c e^{3} x + \frac {1}{5} \, {\left (b d e^{3} + 3 \, a c e f^{2} + 3 \, {\left (b c + a d\right )} e^{2} f\right )} x^{5} + \frac {1}{3} \, {\left (3 \, a c e^{2} f + {\left (b c + a d\right )} e^{3}\right )} x^{3} \] Input:
integrate((b*x^2+a)*(d*x^2+c)*(f*x^2+e)^3,x, algorithm="maxima")
Output:
1/11*b*d*f^3*x^11 + 1/9*(3*b*d*e*f^2 + (b*c + a*d)*f^3)*x^9 + 1/7*(3*b*d*e ^2*f + a*c*f^3 + 3*(b*c + a*d)*e*f^2)*x^7 + a*c*e^3*x + 1/5*(b*d*e^3 + 3*a *c*e*f^2 + 3*(b*c + a*d)*e^2*f)*x^5 + 1/3*(3*a*c*e^2*f + (b*c + a*d)*e^3)* x^3
Time = 0.12 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.27 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=\frac {1}{11} \, b d f^{3} x^{11} + \frac {1}{3} \, b d e f^{2} x^{9} + \frac {1}{9} \, b c f^{3} x^{9} + \frac {1}{9} \, a d f^{3} x^{9} + \frac {3}{7} \, b d e^{2} f x^{7} + \frac {3}{7} \, b c e f^{2} x^{7} + \frac {3}{7} \, a d e f^{2} x^{7} + \frac {1}{7} \, a c f^{3} x^{7} + \frac {1}{5} \, b d e^{3} x^{5} + \frac {3}{5} \, b c e^{2} f x^{5} + \frac {3}{5} \, a d e^{2} f x^{5} + \frac {3}{5} \, a c e f^{2} x^{5} + \frac {1}{3} \, b c e^{3} x^{3} + \frac {1}{3} \, a d e^{3} x^{3} + a c e^{2} f x^{3} + a c e^{3} x \] Input:
integrate((b*x^2+a)*(d*x^2+c)*(f*x^2+e)^3,x, algorithm="giac")
Output:
1/11*b*d*f^3*x^11 + 1/3*b*d*e*f^2*x^9 + 1/9*b*c*f^3*x^9 + 1/9*a*d*f^3*x^9 + 3/7*b*d*e^2*f*x^7 + 3/7*b*c*e*f^2*x^7 + 3/7*a*d*e*f^2*x^7 + 1/7*a*c*f^3* x^7 + 1/5*b*d*e^3*x^5 + 3/5*b*c*e^2*f*x^5 + 3/5*a*d*e^2*f*x^5 + 3/5*a*c*e* f^2*x^5 + 1/3*b*c*e^3*x^3 + 1/3*a*d*e^3*x^3 + a*c*e^2*f*x^3 + a*c*e^3*x
Time = 1.66 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.10 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=x^5\,\left (\frac {b\,d\,e^3}{5}+\frac {3\,a\,c\,e\,f^2}{5}+\frac {3\,a\,d\,e^2\,f}{5}+\frac {3\,b\,c\,e^2\,f}{5}\right )+x^7\,\left (\frac {a\,c\,f^3}{7}+\frac {3\,a\,d\,e\,f^2}{7}+\frac {3\,b\,c\,e\,f^2}{7}+\frac {3\,b\,d\,e^2\,f}{7}\right )+x^3\,\left (\frac {a\,d\,e^3}{3}+\frac {b\,c\,e^3}{3}+a\,c\,e^2\,f\right )+x^9\,\left (\frac {a\,d\,f^3}{9}+\frac {b\,c\,f^3}{9}+\frac {b\,d\,e\,f^2}{3}\right )+a\,c\,e^3\,x+\frac {b\,d\,f^3\,x^{11}}{11} \] Input:
int((a + b*x^2)*(c + d*x^2)*(e + f*x^2)^3,x)
Output:
x^5*((b*d*e^3)/5 + (3*a*c*e*f^2)/5 + (3*a*d*e^2*f)/5 + (3*b*c*e^2*f)/5) + x^7*((a*c*f^3)/7 + (3*a*d*e*f^2)/7 + (3*b*c*e*f^2)/7 + (3*b*d*e^2*f)/7) + x^3*((a*d*e^3)/3 + (b*c*e^3)/3 + a*c*e^2*f) + x^9*((a*d*f^3)/9 + (b*c*f^3) /9 + (b*d*e*f^2)/3) + a*c*e^3*x + (b*d*f^3*x^11)/11
Time = 0.16 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.30 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^3 \, dx=\frac {x \left (315 b d \,f^{3} x^{10}+385 a d \,f^{3} x^{8}+385 b c \,f^{3} x^{8}+1155 b d e \,f^{2} x^{8}+495 a c \,f^{3} x^{6}+1485 a d e \,f^{2} x^{6}+1485 b c e \,f^{2} x^{6}+1485 b d \,e^{2} f \,x^{6}+2079 a c e \,f^{2} x^{4}+2079 a d \,e^{2} f \,x^{4}+2079 b c \,e^{2} f \,x^{4}+693 b d \,e^{3} x^{4}+3465 a c \,e^{2} f \,x^{2}+1155 a d \,e^{3} x^{2}+1155 b c \,e^{3} x^{2}+3465 a c \,e^{3}\right )}{3465} \] Input:
int((b*x^2+a)*(d*x^2+c)*(f*x^2+e)^3,x)
Output:
(x*(3465*a*c*e**3 + 3465*a*c*e**2*f*x**2 + 2079*a*c*e*f**2*x**4 + 495*a*c* f**3*x**6 + 1155*a*d*e**3*x**2 + 2079*a*d*e**2*f*x**4 + 1485*a*d*e*f**2*x* *6 + 385*a*d*f**3*x**8 + 1155*b*c*e**3*x**2 + 2079*b*c*e**2*f*x**4 + 1485* b*c*e*f**2*x**6 + 385*b*c*f**3*x**8 + 693*b*d*e**3*x**4 + 1485*b*d*e**2*f* x**6 + 1155*b*d*e*f**2*x**8 + 315*b*d*f**3*x**10))/3465