On D-brane action at order $ \alpha'^2$

We use compatibility of D-brane action with linear T-duality, S-duality and with S-matrix elements as guiding principles to find all world volume couplings of one massless closed and two open strings at order $\alpha'^2$ in type II superstring theories. In particular, we find that the squares of second fundamental form appear only in world volume curvatures, and confirm the observation that dilaton appears in string frame action via the transformation $\hat{R}_{\mu\nu}\rightarrow \hat{R}_{\mu\nu}+\nabla_{\mu}\nabla_{\nu}\Phi$.