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"For example, if the time required by an algorithm on all inputs of size n is at most 5n^3 + 3n for any n (bigger than some n_0), the asymptotic time complexity is O(n^3)."
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for, for would be O(n^2) but we have for, for and for O(2n^2)
With time complexity we drop out the constant term.
https://en.wikipedia.org/wiki/Time_complexity
"For example, if the time required by an algorithm on all inputs of size n is at most 5n^3 + 3n for any n (bigger than some n_0), the asymptotic time complexity is O(n^3)."
... you're right^^