In the course of my research I have come across the following integral:
$\int_{0}^{\infty} e^{- \Lambda \sqrt{(z^2+a)^2+b^2}}\mathrm{d}z$.
This initially looks like it should be solvable by some suitable change of variable which will allow you to get it into a gaussian form. Unfortunately after trying for awhile I cannot find one. The constants $a$ and $b$ are combinations of parameters $s \in (0,\infty)$, $x \in [0,\infty)$:
$a = s^2-x^2$ and $b = 2sx$.
So the integral can be rewritten as:
$\int_{0}^{\infty} e^{- \Lambda \sqrt{z^4 +Az^2 +B}}\mathrm{d}z$,
with $A = 2(s^2-x^2)$ and $B = s^4 + x^4 + 2s^2x^2$.
Any help with a solution would be much appreciated.
Edit: I forgot to mention that the $s = kL$ where $L$ is a fixed value and I will eventually take a limit in which $k \rightarrow 0$, so there are opportunities for series expansions. I have tried the obvious by expanding the square root in powers of $k$, but there are then convergence issues in the region $|z-x| < k$.
A closed form solution is looking less and less likely as I try all the tricks I know and scour Gradshteyn, so a first term in $k$ (Edit: I originally said in $a$, that was a mistake) would also be much appreciated.