We study the discrete spectrum of the Hamiltonian H = −∆ + 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/r 2 − 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) q . 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.