If $$x=(a+\sqrt {a^2+b^3})^\frac {1}{3} + (a-\sqrt {a^2+b^3})^\frac {1}{3}$$ then prove that $x^3+3bx=2a$.

By observing the given question, I thought about cubing on both sides. But it becomes quiet vague and complex. 

Can anyone help me with a simpler proof?