# Calculate inverse laplace transform by integration with four singular points?

I have a function U[s,x] to do inverse Laplace transform for analytical solution, as following

-((km (kp + la + (
E^(-((Sqrt[(-kp + KE la + KE la s)/KE] x)/Sqrt[
la])) (km - kp) (km (kp + la) + la (kp - KE la (1 + s))))/(
KE Sqrt[(-km + KE la + KE la s)/
KE] (km Sqrt[(-km + KE la + KE la s)/KE] -
kp Sqrt[(-kp + KE la + KE la s)/KE]))))/((-km + kp) (-(kp/KE) +
la + la s)))


Where

kp = 1/2 (-1 + KE la - KE s + KE la s + Sqrt[
4 la KE + (1 + KE s - KE la (1 + s))^2])

km = 1/2 (-1 + KE la - KE s + KE la s - Sqrt[
4 la KE + (1 + KE s - KE la (1 + s))^2])


KE and la are positive constants, say KE=100,la= 0.05

Since it contains nested square roots, the direct use of InverseLaplaceTransform was failed. Then I tried Apart followed by direct integration. I have difficulties when calculating residue of the following term:

KE/(2 (-1 + la + la s) Sqrt[4 KE la + (1 + KE s - KE la (1 + s))^2])


It seems have three singular points:

(1 - la)/la,

(-KE + KE la + KE^2 la - KE^2 la^2 -
2 Sqrt[-KE^3 la + 2 KE^3 la^2 - KE^3 la^3])/(KE^2 - 2 KE^2 la +
KE^2 la^2),

(-KE + KE la + KE^2 la - KE^2 la^2 +
2 Sqrt[-KE^3 la + 2 KE^3 la^2 - KE^3 la^3])/(KE^2 - 2 KE^2 la +
KE^2 la^2),


Is there any hints how to calculate the residue with nested square root? Thank you.

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