  In this paper, we construct a canonical linear basis for free commutative
integro-differential algebras by applying the method of Gr\"obner-Shirshov
bases. We establish the Composition-Diamond Lemma for free commutative
differential Rota-Baxter algebras of order $n$. We also obtain a weakly
monomial order on these algebras, allowing us to obtain Gr\"{o}bner-Shirshov
bases for free commutative integro-differential algebras on a set. We finally
generalize the concept of functional derivations to free differential algebras
with arbitrary weight and generating sets from which to construct a canonical
linear basis for free commutative integro-differential algebras.
