[tex]\displaystyle\int\frac{x+1}{\sqrt x}\,\mathrm dx=\int\frac x{x^{1/2}}+\frac1{x^{1/2}}\,\mathrm dx[/tex]
[tex]=\displaystyle\int x^{1/2}+x^{-1/2}\,\mathrm dx[/tex]
By the power rule,
[tex]=\dfrac{x^{3/2}}{\frac32}+\dfrac{x^{1/2}}{\frac12}+C[/tex]
(this seems to be the step you're not getting?)
[tex]=\dfrac23x^{3/2}+2x^{1/2}+C[/tex]
The next step is to pull out a common factor of [tex]x^{1/2}[/tex] from the antiderivative:
[tex]x^{3/2}=x^{1/2+1}=x^{1/2}\cdot x^1[/tex]
so that the final result is
[tex]\dfrac23x^{3/2}+2x^{1/2}+C=\dfrac23x^{1/2}(x+3)+C[/tex]
