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.


1 Answer 1


Referring to Zacky’s comment it suffices to turn the cubic denominator into a quadratic one, then perform the Landen transform to come to an elliptic integral of the first kind equivalent to K(k11)_ fjaclot


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