  An extension of the Starobinsky model is proposed. Besides the usual
Starobinsky Lagrangian, a term proportional to the derivative of the scalar
curvature, $\nabla_{\mu}R\nabla^{\mu}R$, 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.
