how to integrate $\mathrm{arcsin}\left(x^{15}\right)$? Integral by parts:
$$
I = x\sin^{-1}\left(x^{15}\right) - \int\frac{15x^{15}}{\sqrt{1-x^{30}}}dx
$$
then what?
The answer by wolfram gives an answer contains hypergeometric ${}_2F_1$ function,because it has no elementary answer. The question I want to know is, how can we find the integral of 
$$
\frac{15x^{15}}{\sqrt{1-x^{30}}}
$$
in terms of hypergeometric function?

$\int\dfrac{15x^{15}}{\sqrt{1-x^{30}}}dx$
$=\int_0^x15t^{15}(1-t^{30})^{-\frac{1}{2}}~dt+C$
$=\int_0^{x^{30}}15t^\frac{1}{2}(1-t)^{-\frac{1}{2}}~d(t^\frac{1}{30})+C$
$=\dfrac{1}{2}\int_0^{x^{30}}t^{-\frac{7}{15}}(1-t)^{-\frac{1}{2}}~dt+C$
$=\dfrac{1}{2}\int_0^1(x^{30}t)^{-\frac{7}{15}}(1-x^{30}t)^{-\frac{1}{2}}~d(x^{30}t)+C$
$=\dfrac{x^{16}}{2}\int_0^1t^{-\frac{7}{15}}(1-x^{30}t)^{-\frac{1}{2}}~dt+C$
$=\dfrac{15x^{16}}{16}~_2F_1\left(\dfrac{1}{2},\dfrac{8}{15};\dfrac{23}{15};x^{30}\right)+C$
