  Let $\mathbb{C}_q$ be a non-commutative Laurent polynomial ring associated
with a $(n+1)\times (n+1)$ rational quantum matrix $q$. Let
$\mathfrak{sl}_d(\mathbb{C}_q)\oplus HC_1(\mathbb{C}_q)$ be the universal
central extension of Lie subalgebra $\mathfrak{sl}_d(\mathbb{C}_q)$ of
$\mathfrak{gl}_d(\mathbb{C}_q)$. Now let us take the Lie algebra
$\tau=\mathfrak{gl}_d(\mathbb{C}_q)\oplus HC_1(\mathbb{C}_q)$. Let
$Der(\mathbb{C}_q)$ be the Lie algebra of all derivations of $\mathbb{C}_q$.
Now we consider the Lie algebra $\tilde{\tau}=\tau\rtimes Der(\mathbb{C}_q)$,
called as full toroidal Lie algebra co-ordinated by rational quantum tori. In
this paper we get a classification of irreducible integrable modules with
finite dimensional weight spaces for $\tilde{\tau}$ with nonzero central action
on the modules.
