Modify category permalink to use post type slug

I have multiple custom post types and default post type that all need categories. Right now, each post type is using the default category taxonomy.

I would like the archive category taxonomy URL to be post-type-name/category-name for custom post types and /blog/category-name/ for default posts. Then each one of those URLs will just pull in posts of that post type within that category.

The single post views would just be post-type-name/post-name while the default post will be /blog/post-name/

Is using the default categories the best way of going about this or creating custom taxonomies for each post type?