# 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.

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