In this paper we consider the problem of computing the capacity of multi-user Gaussian MIMO systems under multiple linear transmit covariance constraints (LTCCs). These LTCCs are general enough to include many transmit power constraints such as sum power constraint (SPC) or per-antenna power constraint (PAPC) as special cases. For the considered MIMO systems with multiple LTCCs, existing solutions are based on subgradient or gradient descent methods, which are known to have slow convergence in general and are therefore not applicable to massive MIMO systems. In contrast, we propose a low-complexity semi-closed-form approach to computing the MIMO capacity for the system of interest. To this end, the considered problem in the broadcast channel is transformed into an equivalent minimax problem in the multiple access channel. The special structure of the minimax problem allows us to derive water-filling-like algorithms based on a novel combination of alternating optimization and concave-convex procedure. For the important case of joint SPC and PAPC, we also propose analytical expressions to find the optimal covariance matrix. Extensive analytical and numerical results are provided to demonstrate the effectiveness of our approach under various massive MIMO system settings.