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Recently, some transportation service providers attempt to integrate the ride services offered by multiple independent ride-sourcing platforms, and passengers are able to request ride through such third-party integrators or connectors and receive service from any one of the platforms. This novel business model, termed as third-party platform-integration in this paper, has potentials to alleviate the cost of market fragmentation due to the demand splitting among multiple platforms. While most existing studies focus on the operation strategies for one single monopolist platform, much less is known about the competition and platform-integration as well as the implications on operation strategy and system efficiency. In this paper, we propose mathematical models to describe the ride-sourcing market with multiple competing platforms and compare system performance metrics between two market scenarios, i.e., with and without platform-integration, at Nash equilibrium as well as social optimum. We find that platform-integration can increase total realized demand and social welfare at both Nash equilibrium and social optimum, but may not necessarily generate a greater profit when vehicle supply is sufficiently large or/and market is too fragmented. We show that the market with platform-integration generally achieves greater social welfare. On one hand, the integrator in platform-integration is able to generate a thicker market and reduce matching frictions; on the other hand, multiple platforms are still competing by independently setting their prices, which help to mitigate monopoly mark-up as in the monopoly market.