  We study a model of inflation with terms quadratic and logarithmic in the
Ricci scalar, where the gravitational action is $f(R)=R+\alpha R^2+\beta R^2
\ln R$. These terms are expected to arise from one loop corrections involving
matter fields in curved space-time. The spectral index $n_s$ and the tensor to
scalar ratio yield $10^{-4}\lesssim r\lesssim0.03$ and $0.94\lesssim n_s
\lesssim 0.99$. i.e. $r$ is an order of magnitude bigger or smaller than the
original Starobinsky model which predicted $r\sim 10^{-3}$. Further enhancement
of $r$ gives a scale invariant $n_s\sim 1$ or higher. Other inflationary
observables are $d n_s/d\ln k \gtrsim -5.2 \times 10^{-4},\, \mu \lesssim 2.1
\times 10^{-8} ,\, y \lesssim 2.6 \times 10^{-9}$. Despite the enhancement in
$r$, if the recent BICEP2 measurement stands, this model is disfavoured.
