Simplify and Substitute

Calculus Level 5

0dxx[x2+(1+22)x+1][1x+x2x3++x50]\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 π(abc)\pi\left(a\sqrt{b} - c\right), where aa, bb, and cc are coprime positive integers and bb is square-free. Find (a+1)(b+3)(c+5)(a + 1)(b + 3)(c + 5).

Bonus: Generalize for integrals of the form

0dxx[x2+ax+1][1x+x2x3++(x)n]\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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