<s>[INST] Evaluate $\int_0^a \sqrt{\frac{x^3}{a^3-x^3}} dx$. Evaluate following in terms of Gamma function:
$$\int_0^a \sqrt{\frac{x^3}{a^3-x^3}} dx$$
I don't know how to proceed. So, please tell the intuition behind the solution.
 [/INST] Assuming $a>0$,
$$\int_{0}^{a}\sqrt{\frac{x^3}{a^3-x^3}}\,dx \stackrel{x \mapsto az}{=} a\int_{0}^{1}z^{3/2}(1-z^3)^{-1/2}\,dz \stackrel{z \mapsto u^{1/3}}{=}\frac{a}{3}\int_{0}^{1}u^{-1/6}(1-u)^{-1/2}\,du $$
equals
$$\tfrac{a}{3}\,B\left(\tfrac{5}{6},\tfrac{1}{2}\right)=a \cdot\frac{\Gamma\left(\tfrac{5}{6}\right)\sqrt{\pi}}{\Gamma\left(\tfrac{1}{3}\right)}=\frac{2^{4/3}\pi a}{\sqrt{3}\,\Gamma\left(\tfrac{1}{3}\right)^3}=\frac{\text{AGM}(2,\sqrt{2+\sqrt{3}})}{2\pi\cdot 3^{1/4}}\cdot a $$
by Legendre duplication formula and the reflection formula. The last equality follows from the relation between $\Gamma\left(\frac{1}{3}\right)$ and an elliptic integral and allows a very efficient numerical evaluation of the LHS through the arithmetic-geometric mean.
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