<p>This paper introduces a new splitting algorithm to solve a three-operator monotone inclusion problem that comprises of the sum of a maximal monotone operator, Lipschitz continuous monotone operator, and a cocoercive operator in real Hilbert spaces. The new splitting algorithm features the following (i) three different inertial extrapolation steps; (ii) one forward evaluation of the Lipschitz continuous monotone operator, one forward evaluation of the cocoercive operator and one backward evaluation of the maximal monotone operator at each iteration. The more interesting feature of the proposed algorithm is that each of the involved operators is evaluated at different inertial step. We establish weak, strong and linear convergence of the sequence of iterates under standard assumptions, respectively. Several known splitting algorithms for the monotone inclusion problems of three-operator sum that have appeared in the literature are considered as special cases of our algorithm. Numerical tests confirm the superiority of our algorithm over related ones in the literature.</p>