Hot Integral - 8

Calculus Level 5

\[\displaystyle \int\limits_0^{\infty}\frac{x^6}{(10x^9 + 8)^7} \, dx\]

Given the above integral can be expressed as \(\displaystyle \frac{\Gamma(\frac{A}{B})\Gamma(\frac{C}{B})}{DE^{\frac{F}{G}}H^{\frac{I}{G}}}\) where \(A,B,C,D,E,F,G,H ,I \) all are positive integers.

Find the value of \(A+B+C+D+E+F+G+H+I\).

Details and Assumptions:

1)\(\gcd(A,B)=\gcd(C,B)=\gcd(F,G)=\gcd(I,G)=1\)

2) Here Gamma functions may or may not be reducable. For more clarity \(0 < \frac{A}{B} < 1\) and \(\frac{C}{B} > 1\)

3) Also \(\frac{F}{G}<1\) and \(\frac{I}{G} <1 \)

4) \(E\) and \(H\) are prime numbers.

Inspiration - Caboodle of Tricks.

This is a part of Hot Integrals
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