A novel method is presented for evaluating definite and infinite integrals involving polyalgorithms by using Laplace, Mellin and Stieltjes transforms. The author shows that the standard technique of Laplace transform, together with the use of operations of integration, differentiation, integration under integral sign in combination with the rules and theorems of the operational calculus, can be used for evaluating a wide range of integrals. The method is illustrated by evaluating a variety of different types of integrals.