A nested integral

Calculus Level 4

Let \(F(f(x)) \) denote \( \int_0 ^ x f(s) \, ds \).

If the algebraic expression of \( x^0 \cdot F(x^1 \cdots F(x^8 \cdot (F(x^9 \cdot F(x^{10} )))) \cdots ) \) is equal to \( \dfrac {x^A}{B} \), what is the value of \((A+B) \pmod {10^{10}} \).

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