The coupled Schrodinger equation for three-body atomic systems such as Ps(-)(e(+)e(-)e(-)), Mu(mu(+)e(-)e(-)), p(+)mu(-)mu(-), d(+)mu(+)mu(-), and t(+)mu(-)mu(-) is solved by using the hyperspherical harmonic-generalized Laguerre polynomial expansion method (HHGLP). Ground-state eigenenergies are calculated for the certain number of basis functions. The eigenenergies of the first excited states are obtained for both pure Coulomb, V(r, Omega) = (Z) over cap(Omega)/r, and Fues-Kratzer-type (FK-type) potential, V(r, Omega) = (Z) over cap(Omega)/r + (A) over cap(Omega)/r(2). As an example, to illustrate the effect of FK-type interaction on the lowest excited states, the eigenenergies of the Mu(-) ion are given together as a function of the certain number of basis functions for both potentials. Our results were compared with the other theoretical calculations. Ground-state eigenenergies of atoms such as X+Y-Y- as a function of the reduced mass were investigated. The potential curves of the ground and first excited states of Ps(-)(e(+)e(-)e(-)), Mu(-)(mu(+)e(-)e(-)), p(+)mu(-)mu(-) systems were obtained by a diagonalization process. It is pointed out that the FK-type potential makes the HH method more useful and more extended for atomic calculations. (C) 2004 Wiley Periodicals, Inc.