In general, fast tests are a must-have, but I find that means figuring out how to write fast integration tests as well so that they can also be run as part of a TDD-like cycle. In my experience, integration tests can generally be written to be very quick, but maybe my definition of an integration test is different from yours?

For me, heavy tests implies end-to-end tests, because at that point you're interacting with the whole system including potentially a browser, and that's just going to be slow whichever way you look at it. But just accessing a database, or parsing and sending http requests doesn't have to be particularly slow, at least not compared to the speed at which I develop. I'd expect to be able to run hundreds of those sorts of tests in less than a second, which is fast enough for me.