This paper introduces a mixed-integer conic programming approach to solve the optimal capacitor placement problem in radial distribution networks. The problem is formulated to allow optimally placing fixed and switched-type capacitors; its objective is to minimize the peak power losses, the energy losses, and the costs associated with the required capacitor banks while satisfying the physical and technical constraints on the network. The proposed solution is based on the conic quadratic format of the power flow equations. As a result of using the conic format, the relaxation becomes convex and therefore a global solution to the capacitor planning problem can be obtained using a branch-and-cut algorithm. The method is tested on two radial test systems with 34 and 83 nodes having up to 12 load levels and is validated by comparison with previously published solution results.