$\int {\frac{{\sqrt x }}{{\sqrt {{a^3} - {x^3}} }}} dx$

Let $I = \int {\frac{{\sqrt x }}{{\sqrt {{a^3} - {x^3}} }}} dx = \int {\frac{{\sqrt x }}{{\sqrt {{{\left( {{a^{3/2}}} \right)}^2} - {{\left( {{x^{3/2}}} \right)}^2}} }}}$

Let's put ${x^{3/2}} = t \Rightarrow \frac{3}{2}{x^{1/2}}dx = dt$

therefore,$I = \frac{2}{3}\int {\frac{{dt}}{{\sqrt {{{\left( {{a^{3/2}}} \right)}^2} - {t^2}} }}} = \frac{2}{3}{\sin ^{ - 1}}\frac{t}{{{a^{3/2}}}} + C$

$= \frac{2}{3}{\sin ^{ - 1}}\frac{{{x^{3/2}}}}{{{a^{3/2}}}} + C = \frac{2}{3}{\sin ^{ - 1}}\sqrt {\frac{{{x^3}}}{{{a^3}}}} + C$