  We consider weakly interacting bosonic gases with local and non-local
multi-body interactions. By using the Bogoliubov approximation, we first
investigate contact interactions, studying the case in which the interparticle
potential can be written as a sum of N-body {\delta}-interactions, and then
considering general contact potentials. Results for the quasi-particle spectrum
and the stability are presented. We then examine non-local interactions,
focusing on two different cases of 3-body non-local interactions. Our results
are used for systems with 2- and 3-body {\delta}-interactions and applied for
realistic values of the trap parameters. Finally, the effect of conservative
3-body terms in dipolar systems and soft-core potentials (that can be simulated
with Rydberg dressed atoms) is also studied.
