The spinless Bethe-Salpeter equation is solved for three attractive static quark-antiquark potentials of the form V(r) = -ar(-beta)+br(beta)+c, 0 < beta less-than-or-equal-to 1, and the effective non-Coulombic power-law potential of the. form V(r)=ar 0.1+c to obtain the spin-averaged energy levels in bottomonium UPSILON(bbBAR) and charmonium psi(ccBAR) families. The shifted 1/N expansion technique is used. Calculations of the energy eigenvalues are carried out up to third order and parameters of each potential are adjusted to obtain the best agreement with the experimental spin-averaged data (SAD). Flavor-dependent and flavor-independent cases are considered in this work.