[en] We study the discrete spectrum of the Hamiltonian H = -Δ + V(r) for the Coulomb plus power-law potential V(r) = -1/r + β sgn(q)r^q, where β > 0, q > -2 and q ≠ 0. We show by envelope theory that the discrete eigenvalues E_nl of H may be approximated by the semiclassical expression E_nl(q) ∼ min_r>0{1/r² - 1/(μr) + sgn(q)β(νr)^q}. Values of μ and ν are prescribed which yield upper and lower bounds. Accurate upper bounds are also obtained by use of a trial function of the form, ψ(r) = r^l+1 e^-(xr)d. We give detailed results for V(r) = -1/r + βr^q, q = 0.5, 1, 2 for n = 1, l = 0, 1, 2, along with comparison eigenvalues found by direct numerical methods