# Inverse Laplace Transformation of an Exponential function

I really appreciate if anyone could indicate me how to handle this inverse Laplace transformation (ILT):

L-1[Exp(-c0*Sqrt(a(s)))/Sqrt(a(s))]

where

a(s)=(s2+c1s)/(c2s+c3)

c0,c1,c2,c3 are all constants.

I searched some literatures regarding the ILT of Exp funtions but no such form. I used some general formulas of ILT and basic expressions with exponential functions, and finally the result involves an integral of the product of several functions including BesselJ_0 function and exponential function, which seems hard to integrate. Im also wondering is there any rule to judge whether the ILT of a function can be obtained analytically or not? If yes for this function, how to do it? Many thanks for any help!

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