"-리라": 1) a sentence ending, and 2) a connective to make a quotation-like clause. * -리라 as a sentence ending: mostly used in literature and song lyrics, with two meanings. 1. "would", "is likely to" (an expectation or conjecture). Similar to -것 같은데 or 아마 ... 겠지 in speech. - 설이 지나면 물가가 오르리라: Prices would go up after the new year's day. - 그가 고의로 그렇게 한 것은 아니리라: He isn't likely to have done it on purpose. - 헛된 꿈은 아니리: It wouldn't be a futile dream. (shortened to -리, often seen in poetic language) 2. "I will" (expresses a resolve in one's mind). Similar to -ㄹ 테야 or -겠어 in speech. - 나는 돌아가리라: I am going back / I will go back. - 내 기필코 나의 꿈을 이루고야 말리라: I will realize my dream no matter what happens. * -리라고 (or -리라) as a connective inside of a sentence. It carries the same nuance of expectation and conjecture. This form is used fairly frequently in speech too. It helps to compare it with other quotation forms: 1. (past) 그가 잘했다고 생각해 (I think he did well) / 그가 잘했다고 들었어 (I heard he did well). 2. (present - fact) 그가 잘한다고 생각해 (I think he does well) / 그가 잘하고 있다고 믿어 (I believe he's doing well). 3. (present - expectation; or future) 그가 잘하리라고 믿어/생각해 (I expect he would do well; I expect him to do well). 4. (future) 그가 잘할 것이라고 믿어/생각해 (I expect he will do well; I expect him to do well). -ㄴ다고 of #2 is the present tense quotation form which has a plain nuance of restating a fact. -리라고 in #3 is softer, as either a belief/conjecture in the present tense, or about the future sometimes. So it is often used with verbs such as 믿어, 기대해, and 생각해. In all of them, "고" in -다고 or -라고 is often dropped, especially in conversation, so you get "-리라 믿어", etc.

It is a pattern of prospective assertive -(으)리라[고]: will probably be/do, usually quoting an internal thought. A fancy equivalent to -(으)ㄹ 것이다. The followed verbs are "믿다, 생각하다, 여기다" 알아서 잘하리라 믿어. I believe that you would take care of it well.