Problem from Latvian Sophomore's dream calculus olympiad

So I am preparing to go to this olympiad. I received previous years problems and the toughest problem in the definite integral section was this

$$\int_1^a \sqrt[5]{x^5-1}\ dx \ +\ \int_0^b \sqrt[5]{x^5+1}\ dx$$ $$a^5-b^5 = 1$$

I tried substituting the whole root sign in the respective integrals but that led to nowhere. I don't see how trigonometric substitution could be used, dummy variables or the DI method. I am really at a loss here.

Any ideas?

from Hot Weekly Questions - Mathematics Stack Exchange