Best Integrated that you've heard under $5k

In my experience listening to 20+ amps of various combinations,

Luxman L-590 AXII (fabulous Class A integrated I’ve owned in this price range)

I own this piece.

I agree with MC...go with a Raven Blackhawk 3.1, tubes from Dave as well.  You won't be disappointed.  The rig is astounding, 

To say "there is no best" is absolute hogwash.
He didn’t ask "what is the best". he asked "what is the best YOU’VE heard".
Everyone has their own criteria. The other mistake is to omit the "why" you think was the best you’ve heard so as to establish personal preferences in sound to correlate with the op preferences.
The last mistake is to talk in audiophile jargon:
What the hell is "I like it because it sounds musical" supposed to mean..Huh? Or.."it brings out the emotion". or "it sounds warm". What?
What is "warm " to you mean? Your "emotions" are your own. Do you even know why that happens?

So pure and simple, what’s the best you’ve heard so far in your life and WHY?
For me, with my ProAc Studio 100’s.under 5k, it was I 30 series (w the exception of the I-32) from Primare.
I like clarity, neutrality, timbre correctness, impactful first transient notes along with no treble round off and harmonic weight which is really a bit of distortion added. These are the qualities that imbue "realism" for me and a perfect balance between clinical and muddled.
So this is the practical side of me.
I hesitate to mention what the epitome of what I’ve heard in intergrateds is for me: that being the Line Magnetic 845iA.
This thing is big (73 lbs) runs as hot as Hades and is my preference single ended configuration.
It can be lethal if touched incorrectly and will burn your skin off if you brush against the 845’s.
Not the best to have w young grandkids around the home and it being the only amp source when listening to holiday music w everyone around.
Other than that, it is truly the best I’ve heard for the above mentioned criteria.

I have a soft spot for good integrated amps, and, while the list is short compared to many audiophiles, I have enjoyed a few good ones over the years.

The list includes:

PASS Aleph (many years ago)
Bryston 4B (ditto)
Jeff Rowland Concentra I (loved it; >$5k, but I bought it used)
re-capped Accuphase e-303x (fabulous for a vintage amp)
HEGEL h190 (hard to fault)

Which leads to my answer to the OP’s question: the GATO AMP-150

This Danish-designed amp is largely under-the-radar in the U.S., but fairly well-known and received by audiophiles in Europe. The unusual aesthetics are somewhat polarizing – some love it, others, not so much. But I find the sound produced to be exceptionally good. It is very neutral/accurate, but with a touch of warmth, which I happen to like.

I used the GATO, HEGEL and Accuphase with my current Harbeth 30.1

The topology is interesting, and worth reading about (or watch the reviews below).

FWIW, but Steve Guttenberg and John Darko loved the amp, and have dedicated episodes to it. Darko compared it both directly and favorably to the more expensive Hegel H390.