Integrated platforms are an emerging model of business that integrates ride-sourcing services provided by multiple companies on a single platform. It reduces market fragmentation and brings benefits to both travelers and participating independent transport service providers. To ensure the sustainability of such platforms, the main challenge lies in efficient matching between travelers and ride-sourcing companies, so that both sides obtain non-negative utility. In this study, we propose a double auction based framework to solve the matching and pricing problems for on-demand integrated ride-sourcing platforms. To solve the real-time matching problem, we develop two online mechanisms with a critical price dynamic updating algorithm and a greedy algorithm. The proposed mechanisms are guaranteed to achieve the desired economic properties, and their efficiency is evaluated through numerical studies.