The $\alpha'^2$ correction from double field theory

It is known that the order $\alpha'$ correction to the tree-level effective action for the bosonic and heterotic string can be described in the framework of Double Field Theory (DFT). Here we determine the DFT action and transformations at order $\alpha'^2$ by a direct calculation. The result is vastly simpler than previous proposals. We show that this correction reproduces the known $\alpha'^2$ correction to the heterotic string effective action. The relation of our action to an (implicit) all order proposal coming from the so-called generalized Bergshoeff-de Roo identification is also discussed.