Closed form solution of an improper integral

Question

The integrals $$ I_1=\int_0^{+\infty} \frac{\exp(-\sqrt{x^2 + c})}{\sqrt{x^2 + c}} \, dx,\qquad I_2 = \int_0^{+\infty} \sqrt{x^2 + c} \, \exp(-\sqrt{x^2 + c}) \, dx $$ where $c\in \mathbb{C}$ and $\Re[c] < 0, \Im[c]>0$. These integrals comes from some electromagnetic Green's function. They seem to be quite trivial, but I can't seem to find a way to construct closed curves that allow use of e.g. Jordan's lemma.