I am trying to compute the following integral: $$\int_0^{a_1} \int_v^{a_2} \exp(-\frac{1}{2}u) \exp(-bv ) I_0(c \sqrt{ v \, u}) \,du \,dv,$$ where $a_1$, $a_2$, $b$ and $c$ are positive constants, and $I_0$ is the modified Bessel function of the first kind. Note that $u$ and $v$ are $\ge 0$.
Any idea on how to do the calculation ?