How to evaluate the following integral analytically? $$ \int_0^a \sqrt{x^3 + 1}\,dx $$ Here, $a = \sqrt[3]{63}$.
This integral represents the area between curves. Note that it can also be reduced to the calculation of the arc length for the function $y = y(x)$: $$ y = \frac{2}{5}x^{5/2},\quad x \in [0, 63^{1/3}]. $$
As Wolfram Alpha suggests, this integral probably can't be expressed in elementary functions. Could you describe an approach for analytical integration?
Take $y=\left(\frac{x}{a}\right)^{3} $, then $$I=\frac{a}{3}\int_{0}^{1}\sqrt{1+ya^{3}}y^{-\frac{2}{3}}dy $$ and now we can recognize the integral representation of the Hypergeometric function $$I=a\,_{2}F_{1}\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};-a^{3}\right)$$ and I don't think we can do much more.