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Differential operators and integral operators are linked together by thefirst fundamental theorem of calculus. Based on this principle, the notion of adifferential Rota-Baxter algebra was proposed by Guo and Keigher from analgebraic abstraction point of view. Recently, the subject has attracted moreattention since it is associated with many areas in mathematics, such asintegro-differential algebras. This paper considers differential algebras,Rota-Baxter algebras and differential Rota-Baxter algebras in thequasi-idempotent operator context. We establish a Gr\"obner-Shirshov basis forfree commutative quasi-idempotent differential algebras (resp. Rota-Baxteralgebras, resp. differential Rota-Baxter algebras). This provides a linearbasis of free object in each of the three corresponding categories byComposition-Diamond lemma.

Construction of free quasi-idempotent differential Rota-Baxter algebras by Gr\"obner-Shirshov bases