An extension of the Starobinsky model is proposed. Besides the usual Starobinsky Lagrangian, a term proportional to the derivative of the scalar curvature, ∇μR∇μR, is considered. The analyzis is done in the Einstein frame with the introduction of a scalar field and a vector field. We show that inflation is attainable in our model, allowing for a graceful exit. We also build the cosmological perturbations and obtain the leading-order curvature power spectrum, scalar and tensor tilts and tensor-to-scalar ratio. The tensor and curvature power spectrums are compared to the most recent observations from BICEP2/Keck collaboration. We verify that the scalar-to-tensor rate r can be expected to be up to three times the values predicted by Starobinsky model