<s>[INST] What is the integral of $\int\frac{dx}{\sqrt{x^3+a^3}}$? 
What is the integral of $$\int\frac{dx}{\sqrt{x^3+a^3}}?$$

I came across this integration in a physics problem. I suspect role of complex numbers here.
'$a$' is a constant
 [/INST] It's this:
$$\frac{2 \sqrt[6]{-1} \sqrt[3]{a^3} \sqrt{(-1)^{5/6} \left(\frac{\sqrt[3]{-1} x}{\sqrt[3]{a^3}}-1\right)} \sqrt{\frac{(-1)^{2/3} x^2}{\left(a^3\right)^{2/3}}+\frac{\sqrt[3]{-1} x}{\sqrt[3]{a^3}}+1} F\left(\left.\sin ^{-1}\left(\frac{\sqrt{-\frac{(-1)^{5/6} x}{\sqrt[3]{a^3}}-(-1)^{5/6}}}{\sqrt[4]{3}}\right)\right|\sqrt[3]{-1}\right)}{\sqrt[4]{3} \sqrt{a^3+x^3}}$$
where $F$ denotes the Elliptic Integral.
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