# Simplify and Substitute

Calculus Level 5

$\large \int_0^{\infty} \frac{dx}{\sqrt{x}\left[x^2 + \left(1 + 2\sqrt{2}\right)x + 1\right]\left[1 - x + x^2 - x^3 + \cdots + x^{50}\right]}$

The value of the above integral can be expressed in the form $\pi\left(a\sqrt{b} - c\right)$, where $a$, $b$, and $c$ are coprime positive integers and $b$ is square-free. Find $(a + 1)(b + 3)(c + 5)$.

Bonus: Generalize for integrals of the form

$\int_0^{\infty} \frac{dx}{\sqrt{x}\left[x^2 + ax + 1\right]\left[1 - x + x^2 - x^3 + \cdots + (- x)^n\right]}$

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