I am trying to solve the contour integral
$$\frac{1}{2\pi i}\int_C\frac{e^{t}}{t(2at+x^2)}{\rm{exp}}\left(\frac{ax^2}{2(2at+x^2)}\right)\,{\rm d}t $$
where the path of integration $C$ starts at $-\infty-i0$ on the real axis, goes to $-\varepsilon-i0$, circles the origin in the counterclockwise direction with radius $\varepsilon$ to the point $-\varepsilon+i0$ and returns to the point $-\infty+i0$ (I got such path from Hankel's contour integral of reciprocal Gamma function $1/\Gamma(z)$).
I concluded that a given function $$\frac{e^{t}}{t(2at+x^2)}{\rm{exp}}\left(\frac{ax^2}{2(2at+x^2)}\right)$$ has two poles: First one is $t=0$ and I also calculated the residual in that pole and it is equal to $\frac{e^{a/2}}{x^2}$; second one is at $t=-\frac{x^2}{2a}$ and I am not sure is it a pole of first or second order (because of the denominator of exponential part). Also, when I computed residual at such pole (I tried to compute residuals in both cases -when it is a pole of the first kind and also the second kind) in each case I got $+\infty$ and I do not know what that means.
Please help me with some ideas to calculate such integral. Thank you in advance!