Analytic approximants for the eigenvalues of the one-dimensional Schrödinger equation with potentials of the form V (x) = xa + λ xb are found using a multi-point quasi-rational approximation technique. This technique is based on the use of the power series and asymptotic expansion of the eigenvalues in λ, as well as the expansion at intermediate points. These expansions are found through a system of differential equations. The approximants found are valid and accurate for any values of λ > 0 (with > a). As an example, the technique is applied to the quartic anharmonic oscillator.