Integrar desarrollando la función que conforma el integrando

Integrar
$$\int\dfrac{(x^m+x^n)^2}{\sqrt{x}}dx$$
Donde $m,n\in\mathbb{Z}$.

Solución.

Desarrollamos la función a integrar
$$\begin{align}
\dfrac{(x^m+x^n)^2}{\sqrt{x}}&=\dfrac{x^{2m}+2x^{m+n}+x^{2n}}{x^{1/2}}\\
&=x^{2m-1/2}+2x^{m+n-1/2}+x^{2n-1/2}\\
&=x^{(4m-1)/2}+2x^{(2m+2n-1)/2}+x^{(4n-1)/2}
\end{align}$$
Integramos
$$\begin{align}
\int\dfrac{(x^m+x^n)^2}{\sqrt{x}}dx&=\int\left(x^{(4m-1)/2}+2x^{(2m+2n-1)/2}+x^{(4n-1)/2}\right)dx\\
&=\dfrac{x^{(4m-1)/2+1}}{(4m-1)/2+1}+\dfrac{2x^{(2m+2n-1)/2+1}}{(2m+2n-1)/2+1}+\dfrac{x^{(4n-1)/2+1}}{(4n-1)/2+1}+C\\
&=\dfrac{2x^{(4m+1)/2}}{4m+1}+\dfrac{4x^{(2m+2n+1)/2}}{2m+2n+1}+\dfrac{2x^{(4n+1)/2}}{4n+1}+C
\end{align}$$
Donde $C$ es la constante de integración.