Integrand size = 24, antiderivative size = 172 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^4 \, dx=a c e^4 x+\frac {1}{3} e^3 (b c e+a d e+4 a c f) x^3+\frac {1}{5} e^2 (2 a f (2 d e+3 c f)+b e (d e+4 c f)) x^5+\frac {2}{7} e f (a f (3 d e+2 c f)+b e (2 d e+3 c f)) x^7+\frac {1}{9} f^2 (a f (4 d e+c f)+2 b e (3 d e+2 c f)) x^9+\frac {1}{11} f^3 (4 b d e+b c f+a d f) x^{11}+\frac {1}{13} b d f^4 x^{13} \]
a*c*e^4*x+1/3*e^3*(4*a*c*f+a*d*e+b*c*e)*x^3+1/5*e^2*(2*a*f*(3*c*f+2*d*e)+b *e*(4*c*f+d*e))*x^5+2/7*e*f*(a*f*(2*c*f+3*d*e)+b*e*(3*c*f+2*d*e))*x^7+1/9* f^2*(a*f*(c*f+4*d*e)+2*b*e*(2*c*f+3*d*e))*x^9+1/11*f^3*(a*d*f+b*c*f+4*b*d* e)*x^11+1/13*b*d*f^4*x^13
Time = 0.06 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^4 \, dx=a c e^4 x+\frac {1}{3} e^3 (b c e+a d e+4 a c f) x^3+\frac {1}{5} e^2 (2 a f (2 d e+3 c f)+b e (d e+4 c f)) x^5+\frac {2}{7} e f (a f (3 d e+2 c f)+b e (2 d e+3 c f)) x^7+\frac {1}{9} f^2 (a f (4 d e+c f)+2 b e (3 d e+2 c f)) x^9+\frac {1}{11} f^3 (4 b d e+b c f+a d f) x^{11}+\frac {1}{13} b d f^4 x^{13} \]
a*c*e^4*x + (e^3*(b*c*e + a*d*e + 4*a*c*f)*x^3)/3 + (e^2*(2*a*f*(2*d*e + 3 *c*f) + b*e*(d*e + 4*c*f))*x^5)/5 + (2*e*f*(a*f*(3*d*e + 2*c*f) + b*e*(2*d *e + 3*c*f))*x^7)/7 + (f^2*(a*f*(4*d*e + c*f) + 2*b*e*(3*d*e + 2*c*f))*x^9 )/9 + (f^3*(4*b*d*e + b*c*f + a*d*f)*x^11)/11 + (b*d*f^4*x^13)/13
Time = 0.38 (sec) , antiderivative size = 172, 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.
\(\displaystyle \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^4 \, dx\) |
\(\Big \downarrow \) 396 |
\(\displaystyle \int \left (e^3 x^2 (4 a c f+a d e+b c e)+e^2 x^4 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+f^3 x^{10} (a d f+b c f+4 b d e)+f^2 x^8 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+2 e f x^6 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4+b d f^4 x^{12}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} e^3 x^3 (4 a c f+a d e+b c e)+\frac {1}{5} e^2 x^5 (2 a f (3 c f+2 d e)+b e (4 c f+d e))+\frac {1}{11} f^3 x^{11} (a d f+b c f+4 b d e)+\frac {1}{9} f^2 x^9 (a f (c f+4 d e)+2 b e (2 c f+3 d e))+\frac {2}{7} e f x^7 (a f (2 c f+3 d e)+b e (3 c f+2 d e))+a c e^4 x+\frac {1}{13} b d f^4 x^{13}\) |
a*c*e^4*x + (e^3*(b*c*e + a*d*e + 4*a*c*f)*x^3)/3 + (e^2*(2*a*f*(2*d*e + 3 *c*f) + b*e*(d*e + 4*c*f))*x^5)/5 + (2*e*f*(a*f*(3*d*e + 2*c*f) + b*e*(2*d *e + 3*c*f))*x^7)/7 + (f^2*(a*f*(4*d*e + c*f) + 2*b*e*(3*d*e + 2*c*f))*x^9 )/9 + (f^3*(4*b*d*e + b*c*f + a*d*f)*x^11)/11 + (b*d*f^4*x^13)/13
3.1.1.3.1 Defintions of rubi rules used
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 = 4.08 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {b d \,f^{4} x^{13}}{13}+\frac {\left (\left (a d +b c \right ) f^{4}+4 b d e \,f^{3}\right ) x^{11}}{11}+\frac {\left (a c \,f^{4}+4 \left (a d +b c \right ) e \,f^{3}+6 b d \,e^{2} f^{2}\right ) x^{9}}{9}+\frac {\left (4 a c e \,f^{3}+6 \left (a d +b c \right ) e^{2} f^{2}+4 b d \,e^{3} f \right ) x^{7}}{7}+\frac {\left (6 a c \,e^{2} f^{2}+4 \left (a d +b c \right ) e^{3} f +b d \,e^{4}\right ) x^{5}}{5}+\frac {\left (4 a c \,e^{3} f +\left (a d +b c \right ) e^{4}\right ) x^{3}}{3}+a c \,e^{4} x\) | \(176\) |
norman | \(\frac {b d \,f^{4} x^{13}}{13}+\left (\frac {1}{11} a d \,f^{4}+\frac {1}{11} b c \,f^{4}+\frac {4}{11} b d e \,f^{3}\right ) x^{11}+\left (\frac {1}{9} a c \,f^{4}+\frac {4}{9} a d e \,f^{3}+\frac {4}{9} b c e \,f^{3}+\frac {2}{3} b d \,e^{2} f^{2}\right ) x^{9}+\left (\frac {4}{7} a c e \,f^{3}+\frac {6}{7} a d \,e^{2} f^{2}+\frac {6}{7} b c \,e^{2} f^{2}+\frac {4}{7} b d \,e^{3} f \right ) x^{7}+\left (\frac {6}{5} a c \,e^{2} f^{2}+\frac {4}{5} a d \,e^{3} f +\frac {4}{5} b c \,e^{3} f +\frac {1}{5} b d \,e^{4}\right ) x^{5}+\left (\frac {4}{3} a c \,e^{3} f +\frac {1}{3} a d \,e^{4}+\frac {1}{3} b c \,e^{4}\right ) x^{3}+a c \,e^{4} x\) | \(190\) |
gosper | \(\frac {1}{13} b d \,f^{4} x^{13}+\frac {1}{11} x^{11} a d \,f^{4}+\frac {1}{11} x^{11} b c \,f^{4}+\frac {4}{11} x^{11} b d e \,f^{3}+\frac {1}{9} x^{9} a c \,f^{4}+\frac {4}{9} x^{9} a d e \,f^{3}+\frac {4}{9} x^{9} b c e \,f^{3}+\frac {2}{3} x^{9} b d \,e^{2} f^{2}+\frac {4}{7} x^{7} a c e \,f^{3}+\frac {6}{7} x^{7} a d \,e^{2} f^{2}+\frac {6}{7} x^{7} b c \,e^{2} f^{2}+\frac {4}{7} x^{7} b d \,e^{3} f +\frac {6}{5} x^{5} a c \,e^{2} f^{2}+\frac {4}{5} x^{5} a d \,e^{3} f +\frac {4}{5} x^{5} b c \,e^{3} f +\frac {1}{5} x^{5} b d \,e^{4}+\frac {4}{3} x^{3} a c \,e^{3} f +\frac {1}{3} x^{3} a d \,e^{4}+\frac {1}{3} x^{3} b c \,e^{4}+a c \,e^{4} x\) | \(219\) |
risch | \(\frac {1}{13} b d \,f^{4} x^{13}+\frac {1}{11} x^{11} a d \,f^{4}+\frac {1}{11} x^{11} b c \,f^{4}+\frac {4}{11} x^{11} b d e \,f^{3}+\frac {1}{9} x^{9} a c \,f^{4}+\frac {4}{9} x^{9} a d e \,f^{3}+\frac {4}{9} x^{9} b c e \,f^{3}+\frac {2}{3} x^{9} b d \,e^{2} f^{2}+\frac {4}{7} x^{7} a c e \,f^{3}+\frac {6}{7} x^{7} a d \,e^{2} f^{2}+\frac {6}{7} x^{7} b c \,e^{2} f^{2}+\frac {4}{7} x^{7} b d \,e^{3} f +\frac {6}{5} x^{5} a c \,e^{2} f^{2}+\frac {4}{5} x^{5} a d \,e^{3} f +\frac {4}{5} x^{5} b c \,e^{3} f +\frac {1}{5} x^{5} b d \,e^{4}+\frac {4}{3} x^{3} a c \,e^{3} f +\frac {1}{3} x^{3} a d \,e^{4}+\frac {1}{3} x^{3} b c \,e^{4}+a c \,e^{4} x\) | \(219\) |
parallelrisch | \(\frac {1}{13} b d \,f^{4} x^{13}+\frac {1}{11} x^{11} a d \,f^{4}+\frac {1}{11} x^{11} b c \,f^{4}+\frac {4}{11} x^{11} b d e \,f^{3}+\frac {1}{9} x^{9} a c \,f^{4}+\frac {4}{9} x^{9} a d e \,f^{3}+\frac {4}{9} x^{9} b c e \,f^{3}+\frac {2}{3} x^{9} b d \,e^{2} f^{2}+\frac {4}{7} x^{7} a c e \,f^{3}+\frac {6}{7} x^{7} a d \,e^{2} f^{2}+\frac {6}{7} x^{7} b c \,e^{2} f^{2}+\frac {4}{7} x^{7} b d \,e^{3} f +\frac {6}{5} x^{5} a c \,e^{2} f^{2}+\frac {4}{5} x^{5} a d \,e^{3} f +\frac {4}{5} x^{5} b c \,e^{3} f +\frac {1}{5} x^{5} b d \,e^{4}+\frac {4}{3} x^{3} a c \,e^{3} f +\frac {1}{3} x^{3} a d \,e^{4}+\frac {1}{3} x^{3} b c \,e^{4}+a c \,e^{4} x\) | \(219\) |
1/13*b*d*f^4*x^13+1/11*((a*d+b*c)*f^4+4*b*d*e*f^3)*x^11+1/9*(a*c*f^4+4*(a* d+b*c)*e*f^3+6*b*d*e^2*f^2)*x^9+1/7*(4*a*c*e*f^3+6*(a*d+b*c)*e^2*f^2+4*b*d *e^3*f)*x^7+1/5*(6*a*c*e^2*f^2+4*(a*d+b*c)*e^3*f+b*d*e^4)*x^5+1/3*(4*a*c*e ^3*f+(a*d+b*c)*e^4)*x^3+a*c*e^4*x
Time = 0.29 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^4 \, dx=\frac {1}{13} \, b d f^{4} x^{13} + \frac {1}{11} \, {\left (4 \, b d e f^{3} + {\left (b c + a d\right )} f^{4}\right )} x^{11} + \frac {1}{9} \, {\left (6 \, b d e^{2} f^{2} + a c f^{4} + 4 \, {\left (b c + a d\right )} e f^{3}\right )} x^{9} + \frac {2}{7} \, {\left (2 \, b d e^{3} f + 2 \, a c e f^{3} + 3 \, {\left (b c + a d\right )} e^{2} f^{2}\right )} x^{7} + a c e^{4} x + \frac {1}{5} \, {\left (b d e^{4} + 6 \, a c e^{2} f^{2} + 4 \, {\left (b c + a d\right )} e^{3} f\right )} x^{5} + \frac {1}{3} \, {\left (4 \, a c e^{3} f + {\left (b c + a d\right )} e^{4}\right )} x^{3} \]
1/13*b*d*f^4*x^13 + 1/11*(4*b*d*e*f^3 + (b*c + a*d)*f^4)*x^11 + 1/9*(6*b*d *e^2*f^2 + a*c*f^4 + 4*(b*c + a*d)*e*f^3)*x^9 + 2/7*(2*b*d*e^3*f + 2*a*c*e *f^3 + 3*(b*c + a*d)*e^2*f^2)*x^7 + a*c*e^4*x + 1/5*(b*d*e^4 + 6*a*c*e^2*f ^2 + 4*(b*c + a*d)*e^3*f)*x^5 + 1/3*(4*a*c*e^3*f + (b*c + a*d)*e^4)*x^3
Time = 0.03 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.37 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^4 \, dx=a c e^{4} x + \frac {b d f^{4} x^{13}}{13} + x^{11} \left (\frac {a d f^{4}}{11} + \frac {b c f^{4}}{11} + \frac {4 b d e f^{3}}{11}\right ) + x^{9} \left (\frac {a c f^{4}}{9} + \frac {4 a d e f^{3}}{9} + \frac {4 b c e f^{3}}{9} + \frac {2 b d e^{2} f^{2}}{3}\right ) + x^{7} \cdot \left (\frac {4 a c e f^{3}}{7} + \frac {6 a d e^{2} f^{2}}{7} + \frac {6 b c e^{2} f^{2}}{7} + \frac {4 b d e^{3} f}{7}\right ) + x^{5} \cdot \left (\frac {6 a c e^{2} f^{2}}{5} + \frac {4 a d e^{3} f}{5} + \frac {4 b c e^{3} f}{5} + \frac {b d e^{4}}{5}\right ) + x^{3} \cdot \left (\frac {4 a c e^{3} f}{3} + \frac {a d e^{4}}{3} + \frac {b c e^{4}}{3}\right ) \]
a*c*e**4*x + b*d*f**4*x**13/13 + x**11*(a*d*f**4/11 + b*c*f**4/11 + 4*b*d* e*f**3/11) + x**9*(a*c*f**4/9 + 4*a*d*e*f**3/9 + 4*b*c*e*f**3/9 + 2*b*d*e* *2*f**2/3) + x**7*(4*a*c*e*f**3/7 + 6*a*d*e**2*f**2/7 + 6*b*c*e**2*f**2/7 + 4*b*d*e**3*f/7) + x**5*(6*a*c*e**2*f**2/5 + 4*a*d*e**3*f/5 + 4*b*c*e**3* f/5 + b*d*e**4/5) + x**3*(4*a*c*e**3*f/3 + a*d*e**4/3 + b*c*e**4/3)
Time = 0.23 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.02 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^4 \, dx=\frac {1}{13} \, b d f^{4} x^{13} + \frac {1}{11} \, {\left (4 \, b d e f^{3} + {\left (b c + a d\right )} f^{4}\right )} x^{11} + \frac {1}{9} \, {\left (6 \, b d e^{2} f^{2} + a c f^{4} + 4 \, {\left (b c + a d\right )} e f^{3}\right )} x^{9} + \frac {2}{7} \, {\left (2 \, b d e^{3} f + 2 \, a c e f^{3} + 3 \, {\left (b c + a d\right )} e^{2} f^{2}\right )} x^{7} + a c e^{4} x + \frac {1}{5} \, {\left (b d e^{4} + 6 \, a c e^{2} f^{2} + 4 \, {\left (b c + a d\right )} e^{3} f\right )} x^{5} + \frac {1}{3} \, {\left (4 \, a c e^{3} f + {\left (b c + a d\right )} e^{4}\right )} x^{3} \]
1/13*b*d*f^4*x^13 + 1/11*(4*b*d*e*f^3 + (b*c + a*d)*f^4)*x^11 + 1/9*(6*b*d *e^2*f^2 + a*c*f^4 + 4*(b*c + a*d)*e*f^3)*x^9 + 2/7*(2*b*d*e^3*f + 2*a*c*e *f^3 + 3*(b*c + a*d)*e^2*f^2)*x^7 + a*c*e^4*x + 1/5*(b*d*e^4 + 6*a*c*e^2*f ^2 + 4*(b*c + a*d)*e^3*f)*x^5 + 1/3*(4*a*c*e^3*f + (b*c + a*d)*e^4)*x^3
Time = 0.29 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.27 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^4 \, dx=\frac {1}{13} \, b d f^{4} x^{13} + \frac {4}{11} \, b d e f^{3} x^{11} + \frac {1}{11} \, b c f^{4} x^{11} + \frac {1}{11} \, a d f^{4} x^{11} + \frac {2}{3} \, b d e^{2} f^{2} x^{9} + \frac {4}{9} \, b c e f^{3} x^{9} + \frac {4}{9} \, a d e f^{3} x^{9} + \frac {1}{9} \, a c f^{4} x^{9} + \frac {4}{7} \, b d e^{3} f x^{7} + \frac {6}{7} \, b c e^{2} f^{2} x^{7} + \frac {6}{7} \, a d e^{2} f^{2} x^{7} + \frac {4}{7} \, a c e f^{3} x^{7} + \frac {1}{5} \, b d e^{4} x^{5} + \frac {4}{5} \, b c e^{3} f x^{5} + \frac {4}{5} \, a d e^{3} f x^{5} + \frac {6}{5} \, a c e^{2} f^{2} x^{5} + \frac {1}{3} \, b c e^{4} x^{3} + \frac {1}{3} \, a d e^{4} x^{3} + \frac {4}{3} \, a c e^{3} f x^{3} + a c e^{4} x \]
1/13*b*d*f^4*x^13 + 4/11*b*d*e*f^3*x^11 + 1/11*b*c*f^4*x^11 + 1/11*a*d*f^4 *x^11 + 2/3*b*d*e^2*f^2*x^9 + 4/9*b*c*e*f^3*x^9 + 4/9*a*d*e*f^3*x^9 + 1/9* a*c*f^4*x^9 + 4/7*b*d*e^3*f*x^7 + 6/7*b*c*e^2*f^2*x^7 + 6/7*a*d*e^2*f^2*x^ 7 + 4/7*a*c*e*f^3*x^7 + 1/5*b*d*e^4*x^5 + 4/5*b*c*e^3*f*x^5 + 4/5*a*d*e^3* f*x^5 + 6/5*a*c*e^2*f^2*x^5 + 1/3*b*c*e^4*x^3 + 1/3*a*d*e^4*x^3 + 4/3*a*c* e^3*f*x^3 + a*c*e^4*x
Time = 0.10 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right ) \left (e+f x^2\right )^4 \, dx=x^3\,\left (\frac {a\,d\,e^4}{3}+\frac {b\,c\,e^4}{3}+\frac {4\,a\,c\,e^3\,f}{3}\right )+x^{11}\,\left (\frac {a\,d\,f^4}{11}+\frac {b\,c\,f^4}{11}+\frac {4\,b\,d\,e\,f^3}{11}\right )+x^5\,\left (\frac {b\,d\,e^4}{5}+\frac {4\,a\,d\,e^3\,f}{5}+\frac {4\,b\,c\,e^3\,f}{5}+\frac {6\,a\,c\,e^2\,f^2}{5}\right )+x^9\,\left (\frac {a\,c\,f^4}{9}+\frac {4\,a\,d\,e\,f^3}{9}+\frac {4\,b\,c\,e\,f^3}{9}+\frac {2\,b\,d\,e^2\,f^2}{3}\right )+\frac {2\,e\,f\,x^7\,\left (2\,a\,c\,f^2+2\,b\,d\,e^2+3\,a\,d\,e\,f+3\,b\,c\,e\,f\right )}{7}+a\,c\,e^4\,x+\frac {b\,d\,f^4\,x^{13}}{13} \]
x^3*((a*d*e^4)/3 + (b*c*e^4)/3 + (4*a*c*e^3*f)/3) + x^11*((a*d*f^4)/11 + ( b*c*f^4)/11 + (4*b*d*e*f^3)/11) + x^5*((b*d*e^4)/5 + (4*a*d*e^3*f)/5 + (4* b*c*e^3*f)/5 + (6*a*c*e^2*f^2)/5) + x^9*((a*c*f^4)/9 + (4*a*d*e*f^3)/9 + ( 4*b*c*e*f^3)/9 + (2*b*d*e^2*f^2)/3) + (2*e*f*x^7*(2*a*c*f^2 + 2*b*d*e^2 + 3*a*d*e*f + 3*b*c*e*f))/7 + a*c*e^4*x + (b*d*f^4*x^13)/13