The best recorded bass tone I have ever heard & seen happen on a recording session (I was co-producing, not playing) was Ken Filiano's German double bass into an Acoustic Image pumped through an Ampeg 8x10 miked with a bass drum mike if I remember correctly.
If I am in a bind (no time to setup, room acoustics suck, limited number of available tracks, the amp around happens to be cardboard sounding Markbass combo) I usually take the DI just to be safe. But if I have the time, with a good cab on hand, recording by myself, and a room with nice acoustics, I use DI, 1 mike on cab and 1 mike in the space. The miked cab is the main source for the tone. My go-to mike cab is an old Stedman N-90, which is large diaphram dynamic mic that fattens the sound a bit without coloring what comes out of the cab. In the mix it's is panned slightly with the DI, which adds definition, while the room mike is turned up just a little just so you hardly notice it, for beefing up and adding a bit room resonance. Big tone without compression or eq-ing.