Consider the following.

${\displaystyle \underset{1}{\overset{4}{}}}\frac{8+x^2}{\sqrt[\phantom{2}]{x}}dx$

Use algebra to rewrite the integrand using rational exponents.

${\displaystyle \underset{1}{\overset{4}{}}}\frac{8+x^2}{\sqrt[\phantom{2}]{x}}dx={\displaystyle \underset{1}{\overset{4}{}}}(8x^{\frac{1}{2}}+x^{-\frac{1}{2}})dx$

${\displaystyle \underset{1}{\overset{4}{}}}\frac{8+x^2}{\sqrt[\phantom{2}]{x}}dx={\displaystyle \underset{1}{\overset{4}{}}}(8x^{\frac{1}{2}}+x^{\frac{5}{2}})dx$

${\displaystyle \underset{1}{\overset{4}{}}}\frac{8+x^2}{\sqrt[\phantom{2}]{x}}dx={\displaystyle \underset{1}{\overset{4}{}}}(8x^{-\frac{1}{2}}+x^{\frac{3}{2}})dx$

${\displaystyle \underset{1}{\overset{4}{}}}\frac{8+x^2}{\sqrt[\phantom{2}]{x}}dx={\displaystyle \underset{1}{\overset{4}{}}}(8x^{\frac{1}{2}}+x^{\frac{3}{2}})dx$

${\displaystyle \underset{1}{\overset{4}{}}}\frac{8+x^2}{\sqrt[\phantom{2}]{x}}dx={\displaystyle \underset{1}{\overset{4}{}}}(8x^{\frac{3}{2}}+x^{\frac{5}{2}})dx$