优化mysql中的三个表连接

我正在构建一个电子库,我已经编写了一个查询,该查询会扫描更多的行,这可能是由于索引不好。这是我的桌子结构。

1. book_categories (cat_id,parent_id default空,cat_name,parent_cat_name default空)
2. authors (作者ID,作者姓名,活动)
3. books (book_id,cat_id,author_id,book_title,book_contents,book_status enum(published,pending,deleted)发布日期)

这是SQL代码

  SELECT T.author_id, 
         T.author_name, 
         C.parent_id, 
         C.cat_id, 
         C.cat_name, 
         B.book_id, 
         B.book_title, 
         B.book_contents, 
         B.publish_date 
    FROM books B, 
         authors T, 
         book_categories C 
   WHERE B.author_id = T.author_id 
     AND C.cat_id = B.cat_id 
     AND book_status = 'published' 
ORDER BY published_date DESC

解释上述问题

id  select_type     table   type    possible_keys   key     key_len     ref     rows    Extra
-----------------------------------------------------------------------------------------------------
1   SIMPLE  C   ALL     PRIMARY     NULL    NULL    NULL    449     Using where; Using temporary; Using filesort
1   SIMPLE  B   ref     author_id,cat_id    cat_id  4   bookdb.C.cat_id     214     Using where
1   SIMPLE  T   eq_ref  PRIMARY     PRIMARY     4   bookdb.B.author_id  1    

索引和键

Alter table books 
add Foreign Key (author_id) references authors (author_id) on delete  restrict on update  restrict;

Alter table books 
add Foreign Key (cat_id) references book_categories (cat_id) on delete  restrict on update  restrict;

Create Index books_INX ON books(cat_id,book_status,published_date);