Assuming you're equipped with the error function,
[tex]\mathrm{erf}(x)=\displaystyle\frac2{\sqrt\pi}\int_0^x e^{-u^2}\,\mathrm du[/tex]
whose derivative is
[tex]\dfrac{\mathrm d}{\mathrm dx}\mathrm{erf}(x)=\dfrac2{\sqrt\pi}e^{-x^2}[/tex]
by substituting x = √y, so that x ² = y and 2x dx = dy, we have
[tex]\displastyle\int\sqrt y e^{-y^3}\,\mathrm dy=\int 2x^2 e^{-x^6}\,\mathrm dx[/tex]
Then if u = x ³ and du = 3x ² dx, we have
[tex]\displaystyle\int\sqrt y e^{-y^3}\,\mathrm dy=\int\frac23 e^{-u^2}\,\mathrm du[/tex]
[tex]\displaystyle\int\sqrt y e^{-y^3}\,\mathrm dy=\frac{\sqrt\pi}3\mathrm{erf}(u)+C[/tex]
[tex]\displaystyle\int\sqrt y e^{-y^3}\,\mathrm dy=\frac{\sqrt\pi}3\mathrm{erf}(x^3)+C[/tex]
[tex]\displaystyle\int\sqrt y e^{-y^3}\,\mathrm dy=\boxed{\frac{\sqrt\pi}3\mathrm{erf}\left(y^{\frac32}\right)+C}[/tex]
If you're not familiar with the error function, unfortunately there is no elementary antiderivative...
