Symbolic algorithms for the Painlevé test, special solutions, and recursion operators for nonlinear PDEs. This paper discusses the algorithms and implementations of three MATHEMATICA packages for the study of integrability and the computation of closed-form solutions of nonlinear polynomial partial differential equations (PDEs). The first package, PainleveTest.m, symbolically performs the Painlevé integrability test. The second package, PDESpecialSolutions.m, computes exact solutions expressible in hyperbolic or elliptic functions. The third package, PDERecursionOperator.m, generates and tests recursion operators.