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For positive integer $n$ and positive reals $x,y,a,b$

$x a^n + y b^n < (x+\frac{a}{3})\int (a+x+\frac{1+a}{b})^n da + (y+\frac{b}{3})\int (b+y+\frac{1+b}{a})^ndb+\frac{9}{4} $

Is this true ?

What is the easiest way to decide this ?

In inequalities is there a priority in proofs for calculus tools vs algebraic methods ?

I sometimes find inequalities puzzling. Some advice is welcome.

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