In this paper we study the critical behaviors of the half-integer mixed spin-3/2 and spin-5/2 Blume-Capel Ising ferrimagnetic system by using the exact recursion relations on the Bethe lattice for q = 4 and 6, whose real lattice correspondences are square and simple cubic lattices, respectively. We have obtained the phase diagrams in the (kT(c)/\J\, D-B/\J\) planes for constant values of D-A/\J\, the reduced crystal field of the sublattice with spin-5/2, and in the (kT(c)/\J\,D-B/\J\) planes for constant values of D-A/\J\, the reduced crystal field of the sublattice with spin-3/2. Even if the system presents both second- and first-order phase transitions, their lines never connect to each other and end at critical points; thus, no tricritical points are observed. We have also found the existence of one or two compensation temperatures for appropriate values of the crystal fields, therefore observing reentrant behavior for some of the compensation lines. (C) 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.