  In this paper we derive a novel cosmological model from the $f(R,T)$ theory
of gravitation, for which $R$ is the Ricci scalar and $T$ is the trace of the
energy-momentum tensor. We consider the functional form $f(R,T)=f(R)+f(T)$,
with $f(R)$ being the Starobinksy model, named $R+\alpha R^{2}$, and
$f(T)=2\gamma T$, with $\alpha$ and $\gamma$ being constants. We show that a
hybrid expansion law form for the scale factor is a solution for the derived
Friedmann-like equations. In this way, the model is able to predict both the
decelerated and the accelerated regimes of expansion of the universe, with the
transition redshift between these stages being in accordance with recent
observations. We also apply the energy conditions to our material content
solutions. Such an application makes us able to obtain the range of
acceptability for the free parameters of the model, named $\alpha$ and
$\gamma$.
