The Briancon-Skoda theorem is a comparison relating the integral closure of powers of a finitely generated ideal with its ordinary power. The theorem was originally proved via analytic methods for coordinate rings of smooth varieties over the complex numbers. The full algebraic version for all regular local rings was obtained by Lipman and Sathaye. Since then, there have been other proofs and generalizations to mild singularities, most notably by tight closure theory in positive characteristic and reduction mod p. In this talk, we prove a general Briançon-Skoda containment for pseudo-rational singularities in all characteristics. Our method is quite simple, and it recovers most of previously known results and extends them to mixed characteristic. It also yields some new results on F-pure and Du Bois singularities (and in fact a characteristic free version). This is based on joint work with Peter McDonald, Rebecca R.G., and Karl Schwede.