Zapier comes to mind. You may already be aware of it. Integrations across applications via configurable conditional paths.
Already integrates with Slack, Github, and Jenkins, but not Jetbrains yet as of this writing.
As far as project management, it really depends on what you're using it for. This article outlines many options, categorized into:
-Core Project Management
-Chat and Communications in Projects
-Customizable Project Tools
I have some colleagues who are absolutely transforming their productivity using Trello, for example, integrated through Zapier with all their other applications. But it's by no means the only solution out there. Don't know if the Jetbrains absence is a deal breaker or not. Some businesspeople are turned off by the 'markety' feel of its whole interface/content strategy, which I can relate to. I do think that a strong indication of the power of the platform, though, is all the energy Microsoft, for example, has so rapidly put into getting Office365, Excel, and Dynamics CRM up and running with Zapier (MS Exchange is forthcoming).