Find the inverse laplace transform :
$\displaystyle F(s) = \sqrt{ s + \sqrt{1+s^2}}$
If you mean Bromwich style, then look at this one first:
$\displaystyle \mathop\oint\limits_{H} e^{st}\sqrt{1+s^2}\,ds$ where $\displaystyle H$ is the contour below. It doesn't matter if this one doesn't have an inverse transform. Just first figure out how the integrand varies across the (finite) red-blue and yellow-green branch-cuts and also the remaining parts of the contour. Then try and extend that work to $\displaystyle F(s)$ above. However, this may not be the best contour to use.