Amazon sell batteries for your machine...
Enter AS10D51 - hover the pointy finger over the blue description and a panel comes up showing all the models the battery fits into.
Check out the star ratings to see the relative quality of each offering - cheapest is not necessarily the best.
The figures 4.2 Ah and 5200 mAh indicate in units of current and time how much power the battery holds and should not concern you very much 5200 mAh is 5.2 Amps for an hour - or - if you like - 1 Amp for 5.2 hours.
If you run your computer without its Power Adapter then these figures are important to you - otherwise forget it.
The number of cells is somewhat immaterial - the voltage will always be 10.8V
Forget cells - forget voltage - forget Ah and mAh - just concentrate on (1) your computer's model number - (2) the cost - and (3) what people say about it - check out the number of One Star ratings on Amazon as your best guide.
Good luck...
February 2015