\(\int \frac {1}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) [365]

Optimal result

Integrand size = 37, antiderivative size = 638 \[ \int \frac {1}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=-\frac {\left (\frac {8 b^2 c e^2}{a}+7 b e (d e+c f)+a \left (\frac {8 d^2 e^2}{c}+7 d e f+8 c f^2\right )\right ) \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 a^2 c^2 e^2 x \sqrt {e+f x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{5 a c e x^5}+\frac {4 (b c e+a d e+a c f) \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 a^2 c^2 e^2 x^3}-\frac {\sqrt {-b e+a f} \left (8 b^2 c^2 e^2+7 a b c e (d e+c f)+a^2 \left (8 d^2 e^2+7 c d e f+8 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{15 a^{7/2} c^2 e^3 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {\left (8 b^3 c^2 e^2+3 a b^2 c e (d e+c f)-4 a^3 d f (d e+c f)+a^2 b \left (4 d^2 e^2+c d e f+4 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{15 a^{7/2} c^2 e^2 \sqrt {-b e+a f} \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

-1/15*(8*b^2*c*e^2/a+7*b*e*(c*f+d*e)+a*(8*d^2*e^2/c+7*d*e*f+8*c*f^2))*(b*x 
^2+a)^(1/2)*(d*x^2+c)^(1/2)/a^2/c^2/e^2/x/(f*x^2+e)^(1/2)-1/5*(b*x^2+a)^(1 
/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/a/c/e/x^5+4/15*(a*c*f+a*d*e+b*c*e)*(b* 
x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/a^2/c^2/e^2/x^3-1/15*(a*f-b*e 
)^(1/2)*(8*b^2*c^2*e^2+7*a*b*c*e*(c*f+d*e)+a^2*(8*c^2*f^2+7*c*d*e*f+8*d^2* 
e^2))*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticE((a*f-b*e)^ 
(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),(a*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a^(7/2) 
/c^2/e^3/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)-1/15*(8*b^3*c^2*e 
^2+3*a*b^2*c*e*(c*f+d*e)-4*a^3*d*f*(c*f+d*e)+a^2*b*(4*c^2*f^2+c*d*e*f+4*d^ 
2*e^2))*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticF((a*f-b*e 
)^(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),(a*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a^(7/ 
2)/c^2/e^2/(a*f-b*e)^(1/2)/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
 

Mathematica [F]

\[ \int \frac {1}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx \] Input:

Integrate[1/(x^6*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
 

Output:

Integrate[1/(x^6*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x]
 

Rubi [F]

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.

\(\displaystyle \int \frac {1}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\)

\(\Big \downarrow \) 450

\(\displaystyle \int \frac {1}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}dx\)

Input:

Int[1/(x^6*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 450

Int[((g_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^ 
(q_.)*((e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Unintegrable[(g*x)^m*(a + 
b*x^2)^p*(c + d*x^2)^q*(e + f*x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, g, m, 
p, q, r}, x]
 

Maple [F]

\[\int \frac {1}{x^{6} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}}d x\]

Input:

int(1/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
 

Output:

int(1/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
 

Fricas [F]

\[ \int \frac {1}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e} x^{6}} \,d x } \] Input:

integrate(1/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorit 
hm="fricas")
 

Output:

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(b*d*f*x^12 + (b* 
d*e + (b*c + a*d)*f)*x^10 + a*c*e*x^6 + (a*c*f + (b*c + a*d)*e)*x^8), x)
 

Sympy [F]

\[ \int \frac {1}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{x^{6} \sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \sqrt {e + f x^{2}}}\, dx \] Input:

integrate(1/x**6/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2),x)
 

Output:

Integral(1/(x**6*sqrt(a + b*x**2)*sqrt(c + d*x**2)*sqrt(e + f*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e} x^{6}} \,d x } \] Input:

integrate(1/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorit 
hm="maxima")
 

Output:

integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)*x^6), x)
 

Giac [F]

\[ \int \frac {1}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e} x^{6}} \,d x } \] Input:

integrate(1/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, algorit 
hm="giac")
 

Output:

integrate(1/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)*x^6), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{x^6\,\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}} \,d x \] Input:

int(1/(x^6*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)),x)
 

Output:

int(1/(x^6*(a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^6 \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{x^{6} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}d x \] Input:

int(1/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
 

Output:

int(1/x^6/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)