# Integrate $\int \sqrt{a + x^3}dx$

How does one go about integrating:

$$\int\sqrt{a + x^3}dx$$

Wolframalpha seems to be giving a result that seems slightly too elaborate.

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For $a\ne 0$, there probably is no antiderivative expressible in terms of "elementary" functions. Presumably Alpha used elliptic functions in its answer. –  André Nicolas Dec 3 '12 at 2:32

This does not suffice, since we do not have a promising way of rewriting the integral in terms of $u$. (Edited for technical accuracy due to DonAntonio's comment.) –  000 Dec 3 '12 at 2:45
Well, it doesn't look very promising, does it? Just take into account that $$dx=\frac{1}{3x^2}du=\frac{du}{3(u-a)^{2/3}}\,...$$ –  DonAntonio Dec 3 '12 at 2:46