It sounds very complicated - have you considered a stochastic simulation (Monte Carlo)?
I think it is impossible to solve with more than two particles (?)
Is the power 1.5 correct (since I haven't studied your problem in details, I'm not sure)?
I also know how to use Google - but you do not answer my question,
maybe I should define "Large" as more than, say, 150 transforms.
I do not need to practice integration
I need a large table of Laplace transforms, do you know any good ones? Format HTML, Pdf, Ps, Latex or Word (?)
Extra question: what is the Laplace transform of a Gaussian (normal distribution), with mean m and standard deviation s (assume equal to zero for t<0)?