  This paper explores the $L^{p}$ Lebesgue's integrability propagation,
$p\in(1,\infty]$, of a system of space homogeneous Boltzmann equations
modelling a multi-component mixture of polyatomic gases based on the continuous
internal energy. For typical collision kernels proposed in the literature,
$L^p$ moment-entropy-based estimates for the collision operator gain part and a
lower bound for the loss part are performed leading to a vector valued
inequality for the collision operator and, consequently, to a differential
inequality for the vector valued solutions of the system. This allows to prove
the propagation property of the polynomially weighted $L^p$ norms associated to
the vector valued solution of the system of Boltzmann equations. The case
$p=\infty$ is found as a limit of the case $p<\infty$.
