# Integral $\int_{\sqrt{33}}^\infty\frac{dx}{\sqrt{x^3-11x^2+11x+121}}$

How can we prove $$I:=\int_{\sqrt{33}}^\infty\frac{dx}{\sqrt{x^3-11x^2+11x+121}}\\=\frac1{6\sqrt2\pi^2}\Gamma(1/11)\Gamma(3/11)\Gamma(4/11)\Gamma(5/11)\Gamma(9/11)?$$

Thoughts of this integral
This integral is in the form $$\int\frac{1}{\sqrt{P(x)}}dx,$$where $$\deg P=3$$. Therefore, this integral is an elliptic integral.
Also, I believe this integral is strongly related to Weierstrass elliptic function $$\wp(u)$$. In order to find $$g_2$$ and $$g_3$$, substitute $$x=t+11/3$$ to get $$I=2\int_{\sqrt{33}-11/3}^\infty\frac{dt}{\sqrt{4t^3-352/3t+6776/27}}$$ The question boils down to finding $$\wp(I;352/3,-6776/27)$$ but I seem to be on the wrong track.