Left Join takes very long time on 150 000 rows - mysql

Question

I am having some difficulties to accomplish a task.
Here is some data from orders table:
+----+---------+
| id | bill_id |
+----+---------+
| 3 | 1 |
| 9 | 3 |
| 10 | 4 |
| 15 | 6 |
+----+---------+
And here is some data from a bills table:
+----+
| id |
+----+
| 1 |
| 2 |
| 3 |
| 4 |
| 5 |
| 6 |
+----+
I want to list all the bills that have no order associated with.
In order to achieve that, I thought that the use of LEFT JOIN was appropriated so I wrote this request:
SELECT * FROM bills
LEFT JOIN orders
ON bills.id = orders.bill_id
WHERE orders.bill_id IS NULL;
I thought that I would have the following result:
+----------+-----------+----------------+
| bills.id | orders.id | orders.bill_id |
+----------+-----------+----------------+
| 2 | NULL | NULL |
| 5 | NULL | NULL |
+----------+-----------+----------------+
But I can't reach the end of the request, it has run more than 5 minutes without result, I stopped the request because this can't be a production time anyway.
My real dataset has more than 150 000 orders and 100 000 bills. Is the dataset too big?
Is my request wrong somewhere?
Thank you very much for your tips!
EDIT: side note, the tables have no foreign keys defined... *flies away*

Answer 1

Your query is fine. I would use table aliases in writing it:
SELECT b.*
FROM bills b LEFT JOIN
orders o
ON b.id = o.bill_id
WHERE o.bill_id IS NULL;
You don't need the NULL columns from orders, probably.
You need an index on orders(bill_id):
create index idx_orders_billid on orders(bill_id);

Answer 2

By your where statement, I assume your looking for orders that have no bills.
If that's the case you don't need to do a join to the bills table as they would by definition not exist.
You will find
SELECT * FROM orders
WHERE orders.bill_id IS NULL;
A much better performing query.
Edit:
Sorry I missed your "I want to list all the bills that have no order associated with." when reading the question. As #gordon pointed out an index would certainly help. However if changing the scheme is feasible I would rather have a nullable bill.order_id column instead of a order.bill_id because you won't need a left join, an inner join would suffice to get order bills as it would be a quicker query for your other assumed requirements.