Suppose that $a$ and $b$ satisfy $a^2b|a^3+b^3$. Prove that $a=b$. 
Suppose that $a$ and $b \in \mathbb{Z}^+$ satisfy $a^2b|a^3+b^3$. Prove that $a=b$.

I have reduced the above formulation to these two cases. Assuming $b = a + k$. Proving that any of the below two implies that $a=b$ will be enough.
$$a^2b|(a+b)^3 - 3ab^2$$
$$a^2b|2a^3+3a(a+k)+k^3$$
I can't proceed from here. How should I proceed from here?
Thanks.

Hint: Here, $a,b$ can be negative.  It suffices to prove that $|a|=|b|$ when $\gcd(a,b)=1$ and $a^2b\mid a^3+b^3$.
