It's seems Mathematica can't find Inverse Laplace Transform.It is therefore best to use some mathematical programs like Maple or Matlab.
Maple only find symbolic solution in special caseses if
$a>0$
InverseLaplaceTransform[(Sqrt[s]*Sqrt[a + s])/(a + s), s, t] == a/2*(-BesselI[0,a*t/2] + BesselI[1, a*t/2])*Exp[-a*t/2]+DiracDelta[t]
InverseLaplaceTransform[(Sqrt[s]*Sqrt[2 a + s])/(a + s), s, t] == 1/2 a E^(-a t) (BesselI[1, a t] (2 + a \[Pi] t StruveL[0, a t]) -
a t BesselI[0, a t] (2 + \[Pi] StruveL[1, a t]))
and for:
$a<0$
InverseLaplaceTransform[(Sqrt[s]*Sqrt[a + s])/(a + s), s, t] == a/2*(BesselI[0, a*t/2] + BesselI[1,a*t/2])*Exp[a*t/2]+DiracDelta[t]
InverseLaplaceTransform[(Sqrt[s]*Sqrt[2 a + s])/(a + s), s, t] == 1/2 a E^(a t) (BesselI[1, a t] (2 + a \[Pi] t StruveL[0, a t]) -
a t BesselI[0, a t] (2 + \[Pi] StruveL[1, a t]))
If you want to solve numericall see notebook.
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