  This article presents cosmological models that arise in a subclass of
$f(R,T)=f(R)+f(T)$ gravity models, with different $f(R)$ functions and fixed
$T$-dependence. That is, the gravitational lagrangian is considered as
$f(R,T)=f(R)+\lambda T$, with constant $\lambda$. Here $R$ and $T$ represent
the Ricci scalar and trace of the stress-energy tensor, respectively. The
modified gravitational field equations are obtained through the metric
formalism for the Friedmann-Lema\^itre-Robertson-Walker metric with signature
$(+,-,-,-)$. We work with $f(R)=R+\alpha R^2-\frac{\mu^4}{R}$,
$f(R)=R+k\ln(\gamma R)$ and $f(R)=R+me^{[-nR]}$, with $\alpha, \mu, k, \gamma,
m$ and $n$ all free parameters, which lead to three different cosmological
models for our Universe. For the choice of $\lambda=0$, this reduces to widely
discussed $f(R)$ gravity models. This manuscript clearly describes the effects
of adding the trace of the energy-momentum tensor in the $f(R)$ lagrangian. The
exact solution of the modified field equations are obtained under the hybrid
expansion law. Also we present the Om diagnostic analysis for the discussed
models.
