I've just been down this road, being relatively new to bass. I've spent about 18 months.
Things I've been told by people smarter than me:
Both DI and mic'd cab contribute differently to the sound. So Ideally use both.
Things I've discovered:
DI is much more controllable as far as even response is concerned. A DI "pedal" may also further shape the tone and or gain (distortion) of your signal.
I bought a bass amp and speaker box for rehersal and it turns out the XLR out of the amp gives the best sound by far. But not counting all of the expensive gear in the form of pedals and amps, Di to your interface and then ideally usinfg "Plugin" to simulate AMP/Mi/Room is the way to go. Otherwise, yes, I find just the DI sound even with EQ and compression just sounds too raw and not like anything you might hear on the radio.
Mic'ing the bass amp was difficult to tame. My "studio" is one big rectangular box, so even with all the mic placement and cabinet movement, I had to manage the build up of frequencies, big time.
So in direct answer to your question, yes go the mic'd cab befor the DI, unless you want to get something to better that DI sound.
...and like witzendoz said, this is all assuming that the strings and setup and near to good condition. This *will* make a huge difference and is kind of pushed aside because bass strings are expensive.
Hope this helps.
FM
I run one lead to the amp and I mic the amp, run the line out from the same amp for some added growl and go DI straight from bass to DAW via the splitter box. Blend the signal to one track eventually and Bob's your uncle.