\(\int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^6} \, dx\) [307]

Optimal result

Integrand size = 37, antiderivative size = 574 \[ \int \frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {e+f x^2}}{x^6} \, dx=-\frac {a \sqrt {c+d x^2} \sqrt {e+f x^2}}{5 x^5 \sqrt {a+b x^2}}-\frac {(4 b c e+a d e+a c f) \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 c e x^3 \sqrt {a+b x^2}}+\frac {\left (b^2 c e-3 a b (d e+c f)+2 a^2 \left (\frac {d^2 e}{c}-d f+\frac {c f^2}{e}\right )\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 a c e x \sqrt {a+b x^2}}+\frac {2 \sqrt {b c-a d} \left (b^2 c^2 e^2-a b c e (d e+c f)+a^2 \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )|\frac {c (b e-a f)}{(b c-a d) e}\right )}{15 a^2 c^{5/2} e \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {b c-a d} (d e-c f) (b c e-2 a d e+a c f) \sqrt {c+d x^2} \sqrt {\frac {a \left (e+f x^2\right )}{e \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right ),\frac {c (b e-a f)}{(b c-a d) e}\right )}{15 a c^{5/2} e \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {e+f x^2}} \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 450

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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