Computing inverse two-sided Laplace transform symbolically

How can I compute the inverse two-sided Laplace transform symbolically?

I know MATLAB has ilaplace[1], but that's just for a one-sided transform.

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$1.$ MATLAB has not defined the functions for calculating inverse two-sided Laplace transform.
$2.$ Inverse one-sided Laplace transform is unique, but inverse two-sided Laplace transform is not unique. The reasons are explained in $\mathcal{B}^{-1}_{s\to x}\{e^{as^2+bs}\}$ and $\mathcal{L}^{-1}_{s\to x}\{e^{as^2+bs}\}$ , where $a\neq0$ .