Algebraic Numbers

A number is called algebraic if there is a polynomial with rational coefficients for which the number is a root. For example, $\sqrt{2}$ is algebraic because it is a root of the polynomial $x^2 - 2$. The number $\sqrt{ 2 + \sqrt{3} + \sqrt{5} }$ is also algebraic because it is a root of a monic polynomial of degree 8, namely $x^8 + ax^7 + bx^6 + cx^5 + dx^4 + ex^3 + fx^2 + gx + h.$ Find $|a| + |b| + |c| + |d| + |e| + |f| + |g| + |h|.$

Details and assumptions:

• A monic polynomial is a polynomial whose leading coefficient is 1.
• For those who can't see the square roots clearly, the number is $(2 + 3^{\frac 1 2 } + 5^{\frac 1 2 } )^{\frac 1 2}$.