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In this paper we study a new class of inflation models which generalize theDirac-Born-Infeld (DBI) action with the addition of a nonminimal kineticcoupling (NKC) term. We dubbed this model as the {\it new DBI inflation model}.The NKC term does not bring new dynamical degree of freedom, so the equationsof motion remain of second order. However, with such a coupling, the action isno longer linear with respect to the Einstein curvature term ($R$ or$G^{\mu\nu}$), which leads to a correction term of $k^4$ in the perturbations.The new DBI inflation model can be viewed as theories beyond Horndeski. Withoutviolating nearly scale-invariance, such a correction may lead to new effects onthe inflationary spectra that could be tested by future observations.

New DBI Inflation model with Kinetic Coupling to Einstein Gravity