There's a MySQL driver for scout, which uses MySQL's full text indexing:
https://github.com/YABhq/laravel-scout-mysql-driver
One warning on shared hosting. Doing searches on a database can quickly become resource intensive. If it does that too much, then don't be surprised when you get a notice from your hosting provider about having to upgrade or find a new provider. Best thing to do is put some kind of protection on any public facing search so that people can't abuse it. Something like Captcha or throttling.