  Equation of state with quantum statistics corrections is derived for systems
of the Fermi and Bose particles by using their van der Waals (vdW) and
effective density-dependent Skyrme mean-field interactions. First few orders of
these corrections over the small quantum statistics parameter, $\varepsilon
\approx \hbar^3 n(mT)^{-3/2}g^{-1}$, where $n$ and $T$ are the particle number
density and temperature, $m$ and $g$ the mass and degeneracy factor of
particles, are analytically obtained. For interacting system of nucleon and
$\alpha$ - particles, a small impurity of $\alpha$ - particles to a nucleon
system at leading first order in both $\alpha $-particle and nucleon small
parameters $\varepsilon$ does not change much the basic results for the
symmetric nuclear matter in the quantum vdW consideration. Our approximate
analytical results for the quantum vdW and Skyrme mean-field approaches are in
a good agreement with accurate numerical calculations.
