I am trying to find a closed form for the following function $F(s,x)$, where $s$ is a complex variable, $x$ is a real one, and $a$ is a nonzero real parameter:
$$F(s,x)=\int_0^{\infty} \frac{e^{-xt}\: t^{s-1}}{\sqrt{1-4at^2}}\: dt.$$
It can be proved that, when $a<0$, the integral defines a holomorphic function of $s$ when $\Re(s)>0$ and it can be continued to an entire function of $s$. However, I am interested in manipulating $F(s,x)$ further, and a closed form for the function would be ideal.
I have already tried using Mathematica, but the expression it gives, using hypergeometric functions, is not really helpful. I was hoping to find a simpler one, since the formula in the integrand does not seem too complicated.
I understand that the case $a>0$ might be significantly harder, but I was hoping too that maybe the hurdle of the branching point at $t=\frac{1}{2\sqrt{a}}$ could be overcome integrating on some appropriate contour, instead of directly on the real half-line. I do not have much experience with this and any help or hint would be grealty appreciated!