How to evaluate this integral $I=\int^{b}_{0}\sqrt{a-x^6}dx$ How to evaluate this integral
$$I=\int^{b}_{0}\sqrt{a-x^6}dx$$
Initially I tried substituting $x^3=\sin(t)$, but the integral becomes messy when finding $dx$.  so is there any trick to evaluate this integral?

As said in comments, the antiderivative is quite messy (involving elliptic integral of the first kind) plus many nasty terms.
For the definite integral, as Raffaele commented,
$$I=\int^{b}_{0}\sqrt{a-x^6}\,dx=b\,\sqrt{a} \, _2F_1\left(-\frac{1}{2},\frac{1}{6};\frac{7}{6};\frac{b^6}{a}\right)$$ provided that $b\geq 0\land \left((\Re(a)>0\land \Im(a)\neq 0)\lor \Re(a)>b^6\right)$.
Setting $b^6=k a$ gives $$a^{-2/3}\,I= k^{1/6} \, _2F_1\left(-\frac{1}{2},\frac{1}{6};\frac{7}{6};k\right)\qquad (0 \leq k \leq 1)$$ where the rhs has a nice looking shape.
