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Let $E(K)$ be the Mordell-Weil group of a rational elliptic surface and let $r$ be its rank. In this note we classify all the rational elliptic surfaces with Mordell-Weil group of rank $r=5$ over an algebraically closed field of arbitrary characteristic and using the theory of Mordell-Weil lattices, we find systems of generators for $E(K)$ in the coordinate-free situation.