In a Hilbert spaceHgivenA:H -> 2Ha maximally monotone operator, we study the convergence properties of a general class of relaxed inertial proximal algorithms. This study aims to extend to the case of the general monotone inclusionAxCONTAINS AS MEMBER0the acceleration techniques initially introduced by Nesterov in the case of convex minimization. The relaxed form of the proximal algorithms plays a central role. It comes naturally with the regularization of the operatorAby its Yosida approximation with a variable parameter, a technique recently introduced by Attouch-Peypouquet (Math Program Ser B,2018. 10.1007/s10107-018-1252-x) for a particular class of inertial proximal algorithms. Our study provides an algorithmic version of the convergence results obtained by Attouch-Cabot (J Differ Equ 264:7138-7182,2018) in the case of continuous dynamical systems.