Find all the integer solutions $a, b, c, d, e, f$ satisfying the equation $a^2b^3 + c^2d^3 = e^2f^3$.

Note that if we prove that there are no such solutions with the condition $\text{gcd}(ab, cd, ef) = 1$ and $|bdf|>1$ then we get simpler proof of Fermat's last theorem, because each positive integer $n>1$ has a representation $n=2k+3l$.