Find the value of the integral of the real root of an equation

For $x\ge0$ let $f(x)$ represents the real root of the equation $$y^3+26xy=27$$ Find the value of $$\int_0^1 f^2(x) \ln(f(x)) dx$$

Whenever I see such kind of questions, I instantly think of Lisant's formula but I don't think I can apply that here.

I also tried to do by parts but that too in vain.

I saw that I can factorise the expression as $$26xy=(3-y)(y^2+3y+9)$$Although idk if it can help me in any way.

Any help is greatly appreciated.