Looking for an integrated amp around 4000 to 5000.

Can't go wrong with primaluna. I love my prologue bought used. Best sound ever.

Belles Virtuoso, when it becomes available.

The newly introduced NAD M-33 is worth a long look. It checks all your requirements and adds a few. MSRP is exactly $5000. Check it out. It looks to me at age 73 like a geezer's dream come true.
I'm currently using the M-10, which is a great unit in its own right. M-33 is more and better all the way around. Bluesound is very good and very convenient. 

PrimaLuna HP (14 tubes!) with my Vienna Acoustics produce excellent, tight bass. Luxman solid state was unexciting in comparison. Rotel and Parasound were okay, but did not equal PL.

if you can pick up a used ML 585 within your budget you’ll never look back. It’s a 12k list price unit that checks all the boxes. it also has a very good hi rez built in dac. It makes some of these recommendations look like toys.