Are there any useful restrictions to the prime factors of the number
$$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}?$$
The two smallest are $6771419$ and $72153167$, which I found by trial division. The number is small enough for ECM, but this is quite slow for numbers of this magnitude.
Is there anything better than trial division or ECM ?