integration - Looking for closed-form solution of the following integral

I have been trying to calculate the following triple integral:

$$ I(a,b,c) \,=\, \int_{x=0}^{a}\int_{y=0}^{b}\int_{z=0}^{c} \frac{dx\,dy\,dz}{(1+x^{2}+y^{2}+z^{2})^{3}} $$

I can find values numerically for given $a,b,c$ but, since I know that $I(a,b,c)\rightarrow\frac{\pi^{2}}{32}$ as $a,b,c\rightarrow\infty$, I wondered whether the integral has a closed-form solution for arbitrary $a,b,c$ ? I certainly haven't found one and hoped someone might be able to help.