integrate 1/(square root x + x^3/2)
Then please make the effort to use appropriate grouping symbols.
$\displaystyle \int\frac{dx}{\sqrt{x + x^{3/2}}}$
$\displaystyle =\int\frac{dx}{\sqrt{x(1 + x^{1/2})}}$
$\displaystyle =\int\frac{dx}{\sqrt x\sqrt{1 + x^{1/2}}}$
$\displaystyle =\int\frac{x^{-1/2}\,dx}{\sqrt{1 + x^{1/2}}}$
$\displaystyle =2\int\frac{(1/2)x^{-1/2}\,dx}{\sqrt{1 + x^{1/2}}}$
You should be able to get it from here. Use $\displaystyle u=1+x^{1/2}.$