Ab initio calculations of the orbital and the ground state energies of some open-and closed-shell atoms over Slater type orbitals with quantum numbers integer and Slater type orbitals with quantum numbers noninteger have been performed. In order to increase the efficiency of these calculations the atomic two-electron integrals were expressed in terms of incomplete beta function. Results were observed to be in good agreement with the literature.Keywords integer and noninteger Slater type orbital, ground state energy of open-and closed-shell atoms, incomplete beta function.