In order to study the critical behaviors of the half-integer mixed spin-3/2 and spin-5/2 Blume-Capel Ising ferrimagnetic system, we have used the exact recursion relations on the Bethe lattice. The system was studied for the coordination numbers with q = 3, 4, 5 and 6, and the obtained phase diagrams are illustrated on the (kT(c)/|J|, D-A/|J|) plane for constant values of D-B/|J|, the reduced crystal field of the sublattice with spin-5/2, and on the (kT(c)/|J|, D-B/|J|) plane for constant values of D-A/|J|, the reduced crystal field of the sublattice with spin-3/2, for q = 3 only, since the cases corresponding to q = 4, 5 and 6 reproduce results similar to the case for q = 3. In addition we have also presented the phase diagram with equal strengths of the crystal fields for q = 3, 4, 5 and 6. Besides the second- and first-order phase transitions, the system also exhibits compensation temperatures for appropriate values of the crystal fields. In this mixed spin system while the second-order phase transition lines never cut the reduced crystal field axes as in the single spin type spin-3/2 and spin-5/2 Ising models separately, the first-order phase transition lines never connect to the second-order phase transition lines and they end at the critical points, therefore the system does not give any tricritical points. In addition to this, this mixed-spin model exhibits one or two compensation temperatures depending on the values of the crystal fields, as a result the compensation temperature lines show reentrant behavior. (c) 2006 Elsevier B.V. All rights reserved.