Tough integral from particle physics

Question

I have been struggling to check if it's even possible to calculate the following integral $$ \iiint_{\Bbb R^3} d^3 \textbf{q} {1\over \sqrt{E_{\textbf{p}+\textbf{q}}(E_{\textbf{p}+\textbf{q}}+m)}}e^{-i E_{\textbf{p}+\textbf{q}} t+i \textbf{q}\cdot\textbf{x}} $$ where $\textbf{p}=(p_x,p_y,p_z)$ is constant and $$ E_{\textbf{p}+\textbf{q}} = \sqrt{|\textbf{p}+\textbf{q}|^2+m^2}\,. $$ Thank you very much for any help!