When does the integral $\iiint_{\Bbb R^3}\frac{1}{(1+x^2+y^2+z^2)^p}\,dxdydz$ converge? Consider $$\iiint_{\Bbb R^3}\frac{1}{(1+x^2+y^2+z^2)^p}\,dxdydz$$

I believe it converges when $p&lt;\frac{1}{2}$, but I couldn't show it. Any hints?
Using spherical coordinates,

$$I=4\pi\int_{r=0}^\infty\frac{r^2}{(1+r^2)^p}dr.$$

The integrand is asymptotic to $r^{2-2p}$ so that the integral converges for $p&gt;3/2.$
Consider the spherical coordinates $r ∈ [0, \infty), \theta ∈ [0, \pi], \phi ∈ [0, 2π),$ with 
$$x = r\sin\theta \cos \phi, y = r\sin \theta \sin \phi, z= r\cos \theta,$$ 
then you get $$4 \pi \int_O ^\infty \frac{r^2}{(1+r^2)^p} dr,$$
which converges iff $2p -2 &gt;1.$