Why not split it up equally? About 5 days in each?

I've been to Vienna and Budapest, but not Prague yet. From what I heard it's more of a wandering city and that only a few days are recommended.

Vienna has tons of sites to see. Truly beautiful city! You have the Schonbrunn, Kunsthistorische, Belvedere, Hofburg Palace, St Stephen's Cathedral, St. Charles's Church, world class Opera, to name a few things!!

Also, day trips are easier to take, such as those to Melk Abbey and Bratislava as mentioned by others.

I feel compelled to add, Vienna is perfect for classical music buffs. My husband is one - I am not. We went to the Barber of Seville last weekend in Budapest. On the tram he said "Julie, you are really gonna like this one. The overture is the music from Bugs Bunny".

Gesh, a tease. After 7 minutes of Bugs Bunny (which was very good) - there was two and a half hours of opera. And for some reason, he hates when I say "Can we at least get drinks at half time.".

If the OP is a music fan, and most particularly opera, Vienna is hard to beat.