In this paper, we investigate an intelligent reflecting surface (IRS) assisted cell-free multiple input multiple output (MIMO) communication system, where distributed multiple IRSs are dedicated deployed to assist distributed multiple base stations (BSs) for cooperative transmission. Our objective is to maximize the achievable sum-rate of the cell-free system by jointly optimizing the active transmit beamforming matrices at BSs and the passive reflecting beamforming matrices at IRSs, while the constraints on the maximum transmit power of each BS and the phase shift of each IRS element are satisfied. We propose an efficient framework to jointly design the BSs, the IRSs, and the user equipment (UEs). As a compromise approach, we first transform the non-convex problem into an equivalent form based on the fractional programming methods and then decompose the reformulated problem into two subproblems and solve them alternately. Particularly, we propose a Lagrangian dual sub-gradient based algorithm to solve the subproblem of optimizing the active transmit beamforming with nearly closed-form solutions. We reformulate the subproblem of optimizing the passive reflecting beamforming as a constant modulus constrained quadratic programming (CMC-QP) problem. We first provide two feasible solutions by proposing a pair of relaxation-based algorithms. We also develop a low-complexity alternating sequential optimization (ASO) algorithm to obtain closed-form solutions. All three algorithms are guaranteed to converge to locally optimal solutions. Simulation results demonstrate that the proposed algorithms achieve considerable performance improvements compared with the benchmark schemes.