What's the best 10.4 inch graphics card

Solution
I'll see what's available for about $250. Since it's for BF4 specifically you should get an AMD card (Mantle).

Points:
1) Your CPU still needs to be very good to keep the frame rate high, especially in large multi-player maps.

2) Your Power Supply must have the proper 6/8-pin PCIe power connections for the graphics card.

3) You'll need Windows 7/8 64-bit, not 32-bit.

4) 8GB DDR3 memory also recommended.

5) TWEAKING THE GAME SETTINGS is the most important thing. Monitor with VSYNC OFF while running FRAPS, and look at tweaking guides as well. Some settings add very little to the visuals but are demanding. Turn VSYNC ON again once things are tweaked. You may need to drop a few settings LATER if you can't run smoothly.

RECOMMENDED CARD: http://www.newegg.com/Product/Product.aspx?Item=N82E16814121802

SUMMARY BENCHMARK: http://www.techpowerup.com/reviews/ASUS/R9_270X_Direct_Cu_II_TOP/26.html

The GTX760 averages a little higher, but BF4 performance should receive a big boost with MANTLE in the near future, but no benchmarks yet. The R9-270X is also slightly cheaper so everything considered I think this is the best card for you.

If it wasn't for BF4 though, I'd be recommending the Asus GTX760 for about $240 after MIR which also has two games with it: http://www.newegg.com/Product/Product.aspx?Item=N82E16814121775

CARD LENGTH: 10.43 inches.

So, it's cutting it pretty close but the PCIe power connectors are on top at least. You can MEASURE your case, but I suspect it will fit fine as usually a case lists the card length supported to account for PCIe connectors on the END of the card. Again, measure.
 
Update: comparing the GTX760 to R9-270X

If we compare the same Asus cooler on both cards, the GTX760 averages 12% faster, based on 18 games so that's a reliable number.

Mantle in BF4 will boost the R9-270X ahead. Nobody knows yet, but I'd guess at a 20% difference, but then that would mean maybe only a 10% lead? Again, nobody knows yet; maybe will be really surprised... It's not just BF4 of course, but just how many games will actually support Mantle in the next two or three years will be small. By the time it succeeds (I think it will) you may want a new card anyway.

Just trying to be unbiased and provide ALL the info so you can make an informed decision.

NVIDIA BENEFITS:
1) NVidia Shadowplay (hardware recording of gameplay)
2) PhysX
3) G-Sync (requires new monitor in 2014 to work)
4) Gamestream (via Shield or upcoming device to HDTV from remote PC)
5) GTX760 averges 12% higher frame rates versus R9-270X (however VALUE is basically equal due to price difference)
6) Free games included with card.

There's no RIGHT answer for everybody. I hope you've found this pretty useful.

CARD LENGTH for Asus GTX760 DCU2:
8.6 inches !!

Power:
1x8-pin PCIe (or included 2x6-pin to 8-pin adapter)

Asus site: https://www.asus.com/Graphics_Cards/GTX760DC2OC2GD5/#overview
 
Best with the budget is a GTX 760.

A Superclocked GTX 760 from EVGA (9.5")
http://us.ncix.com/products/?usaffiliateid=1000031504&sku=85649&vpn=02G-P4-2765-KR&manufacture=eVGA&promoid=1221

Or the silent one from Asus (8.6")
http://www.microcenter.com/product/416680/GTX760-DC2OC-2GD5_NVIDIA_GeForce_GTX_760_DirectCU_II_Overclocked_2048MB_GDDR5_PCIe_30_x16_Video_Card


If you're in on the Mantle ghostwagon, you can get the R9 270X or HD 7870 which is cheaper but still good to run at high settings, as photonboy suggested.