bookshelf speakers would require an amplifier if they arent powered (if they dont have a ac wall plug and specifically say they are powered speakers)
5.25" speakers are not bookshelf size. those are likely satelite sized and likely will not be a whole lot better than what you have now.
bestbuy.com has some nice pioneer and boston acoustics bookshelf speakers for about $99/pair however that leaves zero room for an amplifier.
for clarification this is a bookshelf speaker
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
this is a satelite speaker.
bookshelves are bigger and have a little more bass output. however they still lack a subwoofer so might be a little lighter on bass than what you have now.
if you really want a subwoofer then perhaps you should look into either an upper end logitech or perhaps another 2.1 computer speaker set.
if you could raise your budget enough to afford an amplifier (think $20-50) then a bookshelf speaker set might also do well for you.
if you dont mind spending $100 PLUS that $50 gift card then i think you can use any of those options. if you have only $100 INLCUDING that gift card then i think you are limited to likely picking out another pc speaker set.