Subwoofer insight.

If you want to get the highest technologies ,and performance plus a great app for phone or tablet and 2019 sub of the year 
SVS 3000 SB for $995 delivered , 45 day audition 
don’t like it they will even pay the freight back .
i have a $1600 JL Audio,and rel sub that we’re both over 1500
this SVS sealer sub is so much more advanced
and even borrows from the flagship all technologies and is a 13 inch Aluminum driver and over 25 lb driver 50 or 52 bit processor 
the 2 part pole piece uses the small inner section for low volume,and outer as demand requests it ,meaning very tunefull
tight Bass even st low volume ,unheard of and only a 14 x16  52 lbs , and the app all adjustments and volume from your listening position , if you want to tune the room ,buy the $90 Mike and rew
on your computer , and truly tunes your room ,a learn8ng curve there though, and SVS can assist on that. Thunderous Bass to 
earthquake 18 HZ. One is plenty ,but for movies and just to balance the room two is best . Highly recommended !!

Make sure your room acoustics are dealt with first. They can make speakers sound larger than they did before, by tightening up the bass and controlling the mid/treble.  Talk to GIK acoustics.

That's first, next, subwoofers are great, but be prepared to take some effort to integrate them well.

With that large of room, Audio Kinesis Swarm is worth checking out.  Mcarbon used the same Audio Kinesis Dayton amp and built his own swarm from 4 unassembled box speakers from Dayton.  I used car audio sealed sub boxes and Kicker drivers.  Now trying find space for a 4th sub box, my room is much smaller than yours.  As gochurchgo mentioned 6 are even better for a large space.  I have a very good friend that sells car audio, hence my choice, and he insisted on letting me pay cost only.  2 years ago we built a Howe truss bridge across a creek on property he owns ten miles out of town in the woods.  

@ghosthouse
It sounds like you are using powered subs, but feeding them with a "speaker level signal from the amp" -- doesn't that cause overpowering of the sub?  I would think it also halves the impedance of each channel, which can be a challenge for the power amp.  Do the cables from each sub go directly to a pair (R or L) of power amp outputs, or to a pair of (R or L) speaker terminals? I wouldn't think it makes a difference, but I've learned not to trust my assumptions in this hobby.  

There are separate right and left 8 ohm taps on the integrated so seems okay for connection purposes.  Main speakers hooked to the 4 ohm taps. Super nice guy I dealt with today.  He gave me a couple of weeks to see if they work for me.  Now I just need to figure out how to get these jokers dialed in properly.  Thank you Eric for the room treatment advice. I recognize that the room size is somewhat of a problem.  I think the best I am going to be able to accomplish is maybe some local treatments around the equipment.  My wife is never going to allow anything hanging on wall.  I don't blame her.  It is as much as her house as mine.  I also appreciate Millercarbon advice on more subs but no way she is going to let cables run across her main living area.  Anyways, I got them hooked up without dialing in or proper positioning.  First impression is good.  Somewhat of guttural feeling or sensation.  Never listened through subs before so it is different indeed.  Need a little more time to get it as right as situation allows, watch some videos, read some discussions, etc on how to do it properly recognizing that there are limitations without professional room treatments and the like.  Any pointers (either personal or recommended videos or articles) from those who have similar experience as I am going though on how to properly dial them in would be appreciated.