  Quite much recent studies has been attracted to the operated algebra since it
unifies various notions such as the differential algebra and the Rota-Baxter
algebra. An $\Omega$-operated algebra is a an (associative) algebra equipped
with a set $\Omega$ of linear operators which might satisfy certain operator
identities such as the Leibniz rule. A free $\Omega$-operated algebra $B$ can
be generated on an algebra $A$ similar to a free algebra generated on a set. If
$A$ has a Gr\"{o}bner-Shirshov basis $G$ and if the linear operators $\Omega$
satisfy a set $\Phi$ of operator identities, it is natural to ask when the
union $G\cup \Phi$ is a Gr\"{o}bner-Shirshov basis of $B$. A previous work
answers this question affirmatively under a mild condition, and thereby obtains
a canonical linear basis of $B$.
  In this paper, we answer this question in the general case of multiple linear
operators. As applications we get operated Gr\"{o}bner-Shirshov bases for free
differential Rota-Baxter algebras and free integro-differential algebras over
algebras as well as their linear bases. One of the key technical difficulties
is to introduce new monomial orders for the case of two operators, which might
be of independent interest.
