We consider the electron-positron pair production in relativistic heavy ion collisions, in which the produced electron is captured by one of the nuclei resulting, thus, in the formation of a hydrogen-like ion. These ions emerge from the collision point and hit the vacuum chamber wall inside superconducting magnets. Therefore, this process may be important for the problems of beam life time and for the quenching the irradiated magnet. A theoretical investigation for such a bound-free pair production (BFPP) at the colliders from NICA to HE LHC is presented. We obtain an approximate universal formula for the total cross section of the process. We compare it with the results of available numerical calculations and estimate that an accuracy of our calculations is better than 30% at the energies of the NICA collider and becomes of the order of a few percent for the RHIC and HE LHC colliders. Based on the obtained result, the detailed calculations are performed for future experiments at the NICA collider. We find that the expected BFPP cross sections for the Au 79 +–Au 79 + and Bi 83 +–Bi 83 + collisions are in the range from 10 to 70 barn, while for the p–Au 79 + and p–Bi 83 + collisions they are in the range of a few mbarn.