Marginal likelihood function of a fragile survival model and the Laplace transform

[TomHsiung]

Hello, guys

AI and textbooks give me two different version of marginal likelihood functions in the estimation for parameters of a fragile survival model. I don't which one is correct. However, I prefer the answer given by Gemini since it is more logical as well as that it uses the Laplace transform. What do you think? Thanks.

Please compare them in the screenshots below.

A, the marginal likelihood function given by a textbook

B, the marginal likelihood function given by Gemini