Best Integrated that you've heard under $5k

Naim products are consistently rated better than Hegel $4$.

Kind of a hard price point for me. At dealers and shows I often hear more expensive or way cheaper integrateds.

One that stood out is Bryston B135 cubed. I just saw one on hifishark.com for $7500. Maybe you can find a cheaper one.

For cheaper my favorite is the Schiit Ragnarok 2 I own right now. It sounds way better than I thought and is a great buy in the $2k - $3k range. 

I am curious about the Lejonklou that someone mentioned above. Maybe next time I upgrade (if that happens).

I really liked my rogue audio Sphinx at the time. 
I’m basically in love with the Tubelab SSE that I built myself.

The term "integrated" traditionally has been short for "integrated amplifier". However, in the world of digital streaming systems, the part that really matter to SQ and functionality isn't so much the power amp but the pre-amp, along with the DAC. Therefore, it may make sense to focus the integration farther up the chain.  For about $5K or less, there are many good products that integrate streaming, D-A conversion, and pre-amp capabilities (volume control and source-switching).  The Cambridge Audio Edge NQ, for example. Others include the NAD Masters M33, Matrix Audio Element series, iFi Pro iDSD, and Bryston BDA 3.14.  You'll need to pair one of these with a good power amp, but the resulting total cost (and box-count) could wind up in about the same territory as you'd get with an integrated amp and source (e.g. turntable, CD player, or dedicated streamer). 

At some point (say, the next 5-10 years) the integrated amp as we know it may be going the way of the stereo receiver. If you're already marrying source selection and volume control to your streamer/DAC, it is redundant to integrate these functions with the power amp as well.  Separating the power amp then creates opportunities for mono-block or multi-channel configurations, along with the smaller footprint of a modern power amp like the Benchmark AHB2. Or use active speakers, for no visible amplifier footprint at all.

Of course, none of the above applies if you are into vinyl, tubes, and low-wattage SET amps. 

I just bought a Hegel H390 intergraded and am very happy with it.
But like others have said, a lot depends on the speakers you are driving.

Check it out.

JD