I’m working through an Assimil book, and I found this sentence:
Ich hoffe, er bleibt nicht lang.
I thought that, in a sub-clause, the verb and the subject were reversed and thus it should be:
Ich hoffe, bleibt er nicht lang.
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Your first example ("Ich hoffe, er bleibt nicht lang") is correct. But it may be even better to say:
Ich hoffe, daß er nicht lange bleibt.
As it stands, you connect two sentences, but this does not change the word order in the individual sentences. There is simply no rule that says it should.
Ich glaube er sieht mich.
Sie denkt er kommt.
Ich hoffe, bleibt er nicht lang
might with some bending and effort be understood as the semantically strange statement
Ich hoffe, sofern er nicht lang bleibt [I hope if he doesn't stay long]
(though this is (more) usual with the subclause before the main clause - so a less but still quite strange statement would be "Bleibt er nicht lang, [dann] hoffe ich") while
Ich hoffe, er bleibt nicht lang [I hope he doesn't stay long]
is the same as the more complete
Ich hoffe, dass er nicht lange bleibt [I hope that he doesn't stay long]
Also note that in three of these four examples, the subject is still (correctly) before the verb! The verb switches with the subject in questions, while you probably think of verb and object ("Er bleibt nicht lang." -> "er nicht lang bleibt" in subclause and "Bleibt er nicht lang?" in a question).
There are two different kinds of subclauses relative to the main clause: coordinated subclauses and subordinated subclauses.
Coordinated subclauses keep the usual word order:
conjugated verb in second position, rest of the predicate at the end
Subordinated subclauses have another word order:
all verbs at the end with the conjugated verb in last position
The changed word order in subordinated subclauses might yield a reversion the order of subject and verb, but this can also be achieved in coordinated subclauses as the subject need not be in first position in German.
So the correct word order depends on the ordination form of the subclause.
etwas hoffen can be realized in different ways concerning the object
Ich hoffe, dass er nicht lang *bleibt*.
Ich hoffe, er *bleibt* nicht lang.
Ich hoffe, nicht lange zu bleiben.(Attention: change in meaning)
So the first sentence is correct. Your alternative has wrong word order.