The mixed spin-2 and spin-5/2 Blume-Capel Ising ferrimagnetic system, i.e. negative bilinear interaction of the nearest-neighbor spins of the sublattices, with two crystal-field interactions, D-A for spin-2 and D-B for spin-5/2, is investigated on the Bethe lattice by using the exact recursion equations. The exact expressions for the order-parameters, the Curie temperature and the free energy are obtained in terms of the recursion relations. Then, the phase diagrams on the (kT/vertical bar J vertical bar,D-A/vertical bar J vertical bar) and (kT/vertical bar J vertical bar,D-B/vertical bar J vertical bar) planes are studied for various values of D-B/vertical bar J vertical bar and D-A/vertical bar J vertical bar, respectively, when the coordination number q = 3. It is observed that the system presents both second- and first-order phase transitions, therefore, tricritical points, besides the compensation temperatures for the sublattice magnetizations. In addition to this, we have also found that the sublattice quadrupolar moments also compensate each other meaning that their values are equal, whenever the sublattice magnetizations compensate, but at different temperatures, which is a type of compensation not mentioned in the literature as far as in our knowledge. (c) 2006 Elsevier B.V. All rights reserved.