Will this GPU fit in my computer?

As recently as January I've bought a new GPU for my HP Pavilion Slimline (2009). It works great (AMD Radeon 7750) however, my PC is coming up with problems here and there. If I however choose to change computers, I saw this one that has decent performance, but the GPU is terrible. It's a low profile but doesn't do well on BF3. On my current computer and GPU I'm able to run BF3 on Medium graphics perfectly no lag at all. I would guess maybe 40FPS. So my question is, if I were to buy the HPE H8 1360T desktop pc, would my AMD Radon HD 7750 GPU fit in the HPE H81360T? The Radeon came with 2 different holders for the back and I used the smaller one, but would the larger one fit? I measured it as about 4.75 inches. If it's too small, are there any custom fittings to make it fit? Would my GPU be compatible with that PC? Thanks! (Im buying it from shopping.hp.com too for about $720)

The are only 2 standard sized for the PCI slot back plates for which your card has both as an option. So it will fit. The question is whether the PSU is capable of supporting the extra load the GPU upgrade adds to the system.
+1 A good point I didn’t think of in my post
If your PC is coming up with intermittent problems since your GPU install my guess is that slimline case is the root of your problems. OEM cases like yours are a gamers nightmare, with an enthusiast GPU inside the close confines of a Slimline case, all that extra heat generated has no way to escape(you could probably fry an egg in there). If there are extra vents that will allow the installation of extra exhaust fans that is what you need to do to get the internal case temp down. If not then buy a modest gaming case that does and transfer the guts out of the OEM case and into that one and add all the fans that case will allow.
Good Luck