  We develop a Laplace transform method for constructing universal invariants
of 3-manifolds. As an application, we recover Habiro's theory of integer
homology 3-spheres and extend it to some classes of rational homology 3-spheres
with cyclic homology. If |H_1|=2, we give explicit formulas for universal
invariants dominating the sl(2) and SO(3) Witten--Reshetikhin--Turaev
invariants, as well as their spin and cohomological refinements at all roots of
unity. New results on the Ohtsuki series and the integrality of quantum
invariants are the main applications of our construction.
