I'm trying to find some nice expression for integrals of the form $$ I(a,b,c)=\int_{-\infty}^{\infty} dx \frac{1}{({a x^4+ b x^2+c})^{3/2}}. $$
So far I failed to find a rewriting of this in terms of special function (elliptic, hypergeometric, Bessel..) but something tells me that this should be possible. Does anyone know if this is possible?
If the denominator has a real zero, the integral diverges. So assume the denominator has no real zeros. Factor it. For example, we get this if $u,v>0$, $u \ne v$: $$ \int_{-\infty }^{\infty }\! \frac{{\rm d}x}{\left( \left( {x}^{2}+u \right) \left( { x}^{2}+v \right) \right) ^{3/2}}\, ={\frac {2}{\sqrt {v}\; \left( u-v \right) ^{2}\;u} \left[ \left( u+v \right) {\rm E} \left( \sqrt {1-{\frac {u}{v}}} \;\right) -2u\;{\rm K} \left( \sqrt {1-{\frac {u}{v}}} \;\right) \right] } $$ and in case $u=v$ we get $(3/8)\pi u^{-5/2}$. The complete elliptic integrals E and K may be found here.