Integrand size = 26, antiderivative size = 158 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=a c^2 e^2 x+\frac {1}{3} c e (b c e+2 a (d e+c f)) x^3+\frac {1}{5} \left (2 b c e (d e+c f)+a \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (2 a d f (d e+c f)+b \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^7+\frac {1}{9} d f (a d f+2 b (d e+c f)) x^9+\frac {1}{11} b d^2 f^2 x^{11} \] Output:
a*c^2*e^2*x+1/3*c*e*(b*c*e+2*a*(c*f+d*e))*x^3+1/5*(2*b*c*e*(c*f+d*e)+a*(c^ 2*f^2+4*c*d*e*f+d^2*e^2))*x^5+1/7*(2*a*d*f*(c*f+d*e)+b*(c^2*f^2+4*c*d*e*f+ d^2*e^2))*x^7+1/9*d*f*(a*d*f+2*b*(c*f+d*e))*x^9+1/11*b*d^2*f^2*x^11
Time = 0.06 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=a c^2 e^2 x+\frac {1}{3} c e (b c e+2 a (d e+c f)) x^3+\frac {1}{5} \left (2 b c e (d e+c f)+a \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^5+\frac {1}{7} \left (2 a d f (d e+c f)+b \left (d^2 e^2+4 c d e f+c^2 f^2\right )\right ) x^7+\frac {1}{9} d f (a d f+2 b (d e+c f)) x^9+\frac {1}{11} b d^2 f^2 x^{11} \] Input:
Integrate[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^2,x]
Output:
a*c^2*e^2*x + (c*e*(b*c*e + 2*a*(d*e + c*f))*x^3)/3 + ((2*b*c*e*(d*e + c*f ) + a*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^5)/5 + ((2*a*d*f*(d*e + c*f) + b* (d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^7)/7 + (d*f*(a*d*f + 2*b*(d*e + c*f))*x ^9)/9 + (b*d^2*f^2*x^11)/11
Time = 0.37 (sec) , antiderivative size = 158, 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.
| \(\displaystyle \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 396 |
| \(\displaystyle \int \left (x^6 \left (2 a d f (c f+d e)+b \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+x^4 \left (a \left (c^2 f^2+4 c d e f+d^2 e^2\right )+2 b c e (c f+d e)\right )+d f x^8 (a d f+2 b (c f+d e))+c e x^2 (2 a (c f+d e)+b c e)+a c^2 e^2+b d^2 f^2 x^{10}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} x^7 \left (2 a d f (c f+d e)+b \left (c^2 f^2+4 c d e f+d^2 e^2\right )\right )+\frac {1}{5} x^5 \left (a \left (c^2 f^2+4 c d e f+d^2 e^2\right )+2 b c e (c f+d e)\right )+\frac {1}{9} d f x^9 (a d f+2 b (c f+d e))+\frac {1}{3} c e x^3 (2 a (c f+d e)+b c e)+a c^2 e^2 x+\frac {1}{11} b d^2 f^2 x^{11}\) |
Input:
Int[(a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^2,x]
Output:
a*c^2*e^2*x + (c*e*(b*c*e + 2*a*(d*e + c*f))*x^3)/3 + ((2*b*c*e*(d*e + c*f ) + a*(d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^5)/5 + ((2*a*d*f*(d*e + c*f) + b* (d^2*e^2 + 4*c*d*e*f + c^2*f^2))*x^7)/7 + (d*f*(a*d*f + 2*b*(d*e + c*f))*x ^9)/9 + (b*d^2*f^2*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 = 0.50 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(\frac {b \,d^{2} f^{2} x^{11}}{11}+\frac {\left (\left (a \,d^{2}+2 b c d \right ) f^{2}+2 b \,d^{2} e f \right ) x^{9}}{9}+\frac {\left (\left (2 a c d +b \,c^{2}\right ) f^{2}+2 \left (a \,d^{2}+2 b c d \right ) e f +b \,d^{2} e^{2}\right ) x^{7}}{7}+\frac {\left (a \,c^{2} f^{2}+2 \left (2 a c d +b \,c^{2}\right ) e f +\left (a \,d^{2}+2 b c d \right ) e^{2}\right ) x^{5}}{5}+\frac {\left (2 a \,c^{2} e f +\left (2 a c d +b \,c^{2}\right ) e^{2}\right ) x^{3}}{3}+a \,c^{2} e^{2} x\) | \(169\) |
| norman | \(\frac {b \,d^{2} f^{2} x^{11}}{11}+\left (\frac {1}{9} a \,d^{2} f^{2}+\frac {2}{9} b c d \,f^{2}+\frac {2}{9} b \,d^{2} e f \right ) x^{9}+\left (\frac {2}{7} a c d \,f^{2}+\frac {2}{7} a \,d^{2} e f +\frac {1}{7} b \,c^{2} f^{2}+\frac {4}{7} b c d e f +\frac {1}{7} b \,d^{2} e^{2}\right ) x^{7}+\left (\frac {1}{5} a \,c^{2} f^{2}+\frac {4}{5} a c e f d +\frac {1}{5} a \,d^{2} e^{2}+\frac {2}{5} b \,c^{2} e f +\frac {2}{5} b c d \,e^{2}\right ) x^{5}+\left (\frac {2}{3} a \,c^{2} e f +\frac {2}{3} a c \,e^{2} d +\frac {1}{3} b \,c^{2} e^{2}\right ) x^{3}+a \,c^{2} e^{2} x\) | \(175\) |
| gosper | \(\frac {1}{11} b \,d^{2} f^{2} x^{11}+\frac {1}{9} x^{9} a \,d^{2} f^{2}+\frac {2}{9} x^{9} b c d \,f^{2}+\frac {2}{9} x^{9} b \,d^{2} e f +\frac {2}{7} x^{7} a c d \,f^{2}+\frac {2}{7} x^{7} a \,d^{2} e f +\frac {1}{7} x^{7} b \,c^{2} f^{2}+\frac {4}{7} x^{7} b c d e f +\frac {1}{7} x^{7} b \,d^{2} e^{2}+\frac {1}{5} x^{5} a \,c^{2} f^{2}+\frac {4}{5} x^{5} a c e f d +\frac {1}{5} x^{5} a \,d^{2} e^{2}+\frac {2}{5} x^{5} b \,c^{2} e f +\frac {2}{5} x^{5} b c d \,e^{2}+\frac {2}{3} x^{3} a \,c^{2} e f +\frac {2}{3} x^{3} a c \,e^{2} d +\frac {1}{3} x^{3} b \,c^{2} e^{2}+a \,c^{2} e^{2} x\) | \(203\) |
| risch | \(\frac {1}{11} b \,d^{2} f^{2} x^{11}+\frac {1}{9} x^{9} a \,d^{2} f^{2}+\frac {2}{9} x^{9} b c d \,f^{2}+\frac {2}{9} x^{9} b \,d^{2} e f +\frac {2}{7} x^{7} a c d \,f^{2}+\frac {2}{7} x^{7} a \,d^{2} e f +\frac {1}{7} x^{7} b \,c^{2} f^{2}+\frac {4}{7} x^{7} b c d e f +\frac {1}{7} x^{7} b \,d^{2} e^{2}+\frac {1}{5} x^{5} a \,c^{2} f^{2}+\frac {4}{5} x^{5} a c e f d +\frac {1}{5} x^{5} a \,d^{2} e^{2}+\frac {2}{5} x^{5} b \,c^{2} e f +\frac {2}{5} x^{5} b c d \,e^{2}+\frac {2}{3} x^{3} a \,c^{2} e f +\frac {2}{3} x^{3} a c \,e^{2} d +\frac {1}{3} x^{3} b \,c^{2} e^{2}+a \,c^{2} e^{2} x\) | \(203\) |
| parallelrisch | \(\frac {1}{11} b \,d^{2} f^{2} x^{11}+\frac {1}{9} x^{9} a \,d^{2} f^{2}+\frac {2}{9} x^{9} b c d \,f^{2}+\frac {2}{9} x^{9} b \,d^{2} e f +\frac {2}{7} x^{7} a c d \,f^{2}+\frac {2}{7} x^{7} a \,d^{2} e f +\frac {1}{7} x^{7} b \,c^{2} f^{2}+\frac {4}{7} x^{7} b c d e f +\frac {1}{7} x^{7} b \,d^{2} e^{2}+\frac {1}{5} x^{5} a \,c^{2} f^{2}+\frac {4}{5} x^{5} a c e f d +\frac {1}{5} x^{5} a \,d^{2} e^{2}+\frac {2}{5} x^{5} b \,c^{2} e f +\frac {2}{5} x^{5} b c d \,e^{2}+\frac {2}{3} x^{3} a \,c^{2} e f +\frac {2}{3} x^{3} a c \,e^{2} d +\frac {1}{3} x^{3} b \,c^{2} e^{2}+a \,c^{2} e^{2} x\) | \(203\) |
| orering | \(\frac {x \left (315 b \,d^{2} f^{2} x^{10}+385 a \,d^{2} f^{2} x^{8}+770 b c d \,f^{2} x^{8}+770 b \,d^{2} e f \,x^{8}+990 a c d \,f^{2} x^{6}+990 a \,d^{2} e f \,x^{6}+495 b \,c^{2} f^{2} x^{6}+1980 b c d e f \,x^{6}+495 b \,d^{2} e^{2} x^{6}+693 a \,c^{2} f^{2} x^{4}+2772 a c d e f \,x^{4}+693 a \,d^{2} e^{2} x^{4}+1386 b \,c^{2} e f \,x^{4}+1386 b c d \,e^{2} x^{4}+2310 a \,c^{2} e f \,x^{2}+2310 a c d \,e^{2} x^{2}+1155 b \,c^{2} e^{2} x^{2}+3465 a \,c^{2} e^{2}\right )}{3465}\) | \(206\) |
Input:
int((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e)^2,x,method=_RETURNVERBOSE)
Output:
1/11*b*d^2*f^2*x^11+1/9*((a*d^2+2*b*c*d)*f^2+2*b*d^2*e*f)*x^9+1/7*((2*a*c* d+b*c^2)*f^2+2*(a*d^2+2*b*c*d)*e*f+b*d^2*e^2)*x^7+1/5*(a*c^2*f^2+2*(2*a*c* d+b*c^2)*e*f+(a*d^2+2*b*c*d)*e^2)*x^5+1/3*(2*a*c^2*e*f+(2*a*c*d+b*c^2)*e^2 )*x^3+a*c^2*e^2*x
Time = 0.07 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {1}{11} \, b d^{2} f^{2} x^{11} + \frac {1}{9} \, {\left (2 \, b d^{2} e f + {\left (2 \, b c d + a d^{2}\right )} f^{2}\right )} x^{9} + \frac {1}{7} \, {\left (b d^{2} e^{2} + 2 \, {\left (2 \, b c d + a d^{2}\right )} e f + {\left (b c^{2} + 2 \, a c d\right )} f^{2}\right )} x^{7} + a c^{2} e^{2} x + \frac {1}{5} \, {\left (a c^{2} f^{2} + {\left (2 \, b c d + a d^{2}\right )} e^{2} + 2 \, {\left (b c^{2} + 2 \, a c d\right )} e f\right )} x^{5} + \frac {1}{3} \, {\left (2 \, a c^{2} e f + {\left (b c^{2} + 2 \, a c d\right )} e^{2}\right )} x^{3} \] Input:
integrate((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e)^2,x, algorithm="fricas")
Output:
1/11*b*d^2*f^2*x^11 + 1/9*(2*b*d^2*e*f + (2*b*c*d + a*d^2)*f^2)*x^9 + 1/7* (b*d^2*e^2 + 2*(2*b*c*d + a*d^2)*e*f + (b*c^2 + 2*a*c*d)*f^2)*x^7 + a*c^2* e^2*x + 1/5*(a*c^2*f^2 + (2*b*c*d + a*d^2)*e^2 + 2*(b*c^2 + 2*a*c*d)*e*f)* x^5 + 1/3*(2*a*c^2*e*f + (b*c^2 + 2*a*c*d)*e^2)*x^3
Time = 0.03 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.37 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=a c^{2} e^{2} x + \frac {b d^{2} f^{2} x^{11}}{11} + x^{9} \left (\frac {a d^{2} f^{2}}{9} + \frac {2 b c d f^{2}}{9} + \frac {2 b d^{2} e f}{9}\right ) + x^{7} \cdot \left (\frac {2 a c d f^{2}}{7} + \frac {2 a d^{2} e f}{7} + \frac {b c^{2} f^{2}}{7} + \frac {4 b c d e f}{7} + \frac {b d^{2} e^{2}}{7}\right ) + x^{5} \left (\frac {a c^{2} f^{2}}{5} + \frac {4 a c d e f}{5} + \frac {a d^{2} e^{2}}{5} + \frac {2 b c^{2} e f}{5} + \frac {2 b c d e^{2}}{5}\right ) + x^{3} \cdot \left (\frac {2 a c^{2} e f}{3} + \frac {2 a c d e^{2}}{3} + \frac {b c^{2} e^{2}}{3}\right ) \] Input:
integrate((b*x**2+a)*(d*x**2+c)**2*(f*x**2+e)**2,x)
Output:
a*c**2*e**2*x + b*d**2*f**2*x**11/11 + x**9*(a*d**2*f**2/9 + 2*b*c*d*f**2/ 9 + 2*b*d**2*e*f/9) + x**7*(2*a*c*d*f**2/7 + 2*a*d**2*e*f/7 + b*c**2*f**2/ 7 + 4*b*c*d*e*f/7 + b*d**2*e**2/7) + x**5*(a*c**2*f**2/5 + 4*a*c*d*e*f/5 + a*d**2*e**2/5 + 2*b*c**2*e*f/5 + 2*b*c*d*e**2/5) + x**3*(2*a*c**2*e*f/3 + 2*a*c*d*e**2/3 + b*c**2*e**2/3)
Time = 0.03 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.06 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {1}{11} \, b d^{2} f^{2} x^{11} + \frac {1}{9} \, {\left (2 \, b d^{2} e f + {\left (2 \, b c d + a d^{2}\right )} f^{2}\right )} x^{9} + \frac {1}{7} \, {\left (b d^{2} e^{2} + 2 \, {\left (2 \, b c d + a d^{2}\right )} e f + {\left (b c^{2} + 2 \, a c d\right )} f^{2}\right )} x^{7} + a c^{2} e^{2} x + \frac {1}{5} \, {\left (a c^{2} f^{2} + {\left (2 \, b c d + a d^{2}\right )} e^{2} + 2 \, {\left (b c^{2} + 2 \, a c d\right )} e f\right )} x^{5} + \frac {1}{3} \, {\left (2 \, a c^{2} e f + {\left (b c^{2} + 2 \, a c d\right )} e^{2}\right )} x^{3} \] Input:
integrate((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e)^2,x, algorithm="maxima")
Output:
1/11*b*d^2*f^2*x^11 + 1/9*(2*b*d^2*e*f + (2*b*c*d + a*d^2)*f^2)*x^9 + 1/7* (b*d^2*e^2 + 2*(2*b*c*d + a*d^2)*e*f + (b*c^2 + 2*a*c*d)*f^2)*x^7 + a*c^2* e^2*x + 1/5*(a*c^2*f^2 + (2*b*c*d + a*d^2)*e^2 + 2*(b*c^2 + 2*a*c*d)*e*f)* x^5 + 1/3*(2*a*c^2*e*f + (b*c^2 + 2*a*c*d)*e^2)*x^3
Time = 0.12 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.28 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {1}{11} \, b d^{2} f^{2} x^{11} + \frac {2}{9} \, b d^{2} e f x^{9} + \frac {2}{9} \, b c d f^{2} x^{9} + \frac {1}{9} \, a d^{2} f^{2} x^{9} + \frac {1}{7} \, b d^{2} e^{2} x^{7} + \frac {4}{7} \, b c d e f x^{7} + \frac {2}{7} \, a d^{2} e f x^{7} + \frac {1}{7} \, b c^{2} f^{2} x^{7} + \frac {2}{7} \, a c d f^{2} x^{7} + \frac {2}{5} \, b c d e^{2} x^{5} + \frac {1}{5} \, a d^{2} e^{2} x^{5} + \frac {2}{5} \, b c^{2} e f x^{5} + \frac {4}{5} \, a c d e f x^{5} + \frac {1}{5} \, a c^{2} f^{2} x^{5} + \frac {1}{3} \, b c^{2} e^{2} x^{3} + \frac {2}{3} \, a c d e^{2} x^{3} + \frac {2}{3} \, a c^{2} e f x^{3} + a c^{2} e^{2} x \] Input:
integrate((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e)^2,x, algorithm="giac")
Output:
1/11*b*d^2*f^2*x^11 + 2/9*b*d^2*e*f*x^9 + 2/9*b*c*d*f^2*x^9 + 1/9*a*d^2*f^ 2*x^9 + 1/7*b*d^2*e^2*x^7 + 4/7*b*c*d*e*f*x^7 + 2/7*a*d^2*e*f*x^7 + 1/7*b* c^2*f^2*x^7 + 2/7*a*c*d*f^2*x^7 + 2/5*b*c*d*e^2*x^5 + 1/5*a*d^2*e^2*x^5 + 2/5*b*c^2*e*f*x^5 + 4/5*a*c*d*e*f*x^5 + 1/5*a*c^2*f^2*x^5 + 1/3*b*c^2*e^2* x^3 + 2/3*a*c*d*e^2*x^3 + 2/3*a*c^2*e*f*x^3 + a*c^2*e^2*x
Time = 0.07 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=x^5\,\left (\frac {2\,b\,c^2\,e\,f}{5}+\frac {a\,c^2\,f^2}{5}+\frac {2\,b\,c\,d\,e^2}{5}+\frac {4\,a\,c\,d\,e\,f}{5}+\frac {a\,d^2\,e^2}{5}\right )+x^7\,\left (\frac {b\,c^2\,f^2}{7}+\frac {4\,b\,c\,d\,e\,f}{7}+\frac {2\,a\,c\,d\,f^2}{7}+\frac {b\,d^2\,e^2}{7}+\frac {2\,a\,d^2\,e\,f}{7}\right )+\frac {b\,d^2\,f^2\,x^{11}}{11}+a\,c^2\,e^2\,x+\frac {c\,e\,x^3\,\left (2\,a\,c\,f+2\,a\,d\,e+b\,c\,e\right )}{3}+\frac {d\,f\,x^9\,\left (a\,d\,f+2\,b\,c\,f+2\,b\,d\,e\right )}{9} \] Input:
int((a + b*x^2)*(c + d*x^2)^2*(e + f*x^2)^2,x)
Output:
x^5*((a*c^2*f^2)/5 + (a*d^2*e^2)/5 + (2*b*c*d*e^2)/5 + (2*b*c^2*e*f)/5 + ( 4*a*c*d*e*f)/5) + x^7*((b*c^2*f^2)/7 + (b*d^2*e^2)/7 + (2*a*c*d*f^2)/7 + ( 2*a*d^2*e*f)/7 + (4*b*c*d*e*f)/7) + (b*d^2*f^2*x^11)/11 + a*c^2*e^2*x + (c *e*x^3*(2*a*c*f + 2*a*d*e + b*c*e))/3 + (d*f*x^9*(a*d*f + 2*b*c*f + 2*b*d* e))/9
Time = 0.17 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.30 \[ \int \left (a+b x^2\right ) \left (c+d x^2\right )^2 \left (e+f x^2\right )^2 \, dx=\frac {x \left (315 b \,d^{2} f^{2} x^{10}+385 a \,d^{2} f^{2} x^{8}+770 b c d \,f^{2} x^{8}+770 b \,d^{2} e f \,x^{8}+990 a c d \,f^{2} x^{6}+990 a \,d^{2} e f \,x^{6}+495 b \,c^{2} f^{2} x^{6}+1980 b c d e f \,x^{6}+495 b \,d^{2} e^{2} x^{6}+693 a \,c^{2} f^{2} x^{4}+2772 a c d e f \,x^{4}+693 a \,d^{2} e^{2} x^{4}+1386 b \,c^{2} e f \,x^{4}+1386 b c d \,e^{2} x^{4}+2310 a \,c^{2} e f \,x^{2}+2310 a c d \,e^{2} x^{2}+1155 b \,c^{2} e^{2} x^{2}+3465 a \,c^{2} e^{2}\right )}{3465} \] Input:
int((b*x^2+a)*(d*x^2+c)^2*(f*x^2+e)^2,x)
Output:
(x*(3465*a*c**2*e**2 + 2310*a*c**2*e*f*x**2 + 693*a*c**2*f**2*x**4 + 2310* a*c*d*e**2*x**2 + 2772*a*c*d*e*f*x**4 + 990*a*c*d*f**2*x**6 + 693*a*d**2*e **2*x**4 + 990*a*d**2*e*f*x**6 + 385*a*d**2*f**2*x**8 + 1155*b*c**2*e**2*x **2 + 1386*b*c**2*e*f*x**4 + 495*b*c**2*f**2*x**6 + 1386*b*c*d*e**2*x**4 + 1980*b*c*d*e*f*x**6 + 770*b*c*d*f**2*x**8 + 495*b*d**2*e**2*x**6 + 770*b* d**2*e*f*x**8 + 315*b*d**2*f**2*x**10))/3465