無理関数を分母に含む関数の不定積分

$$ \begin{aligned} \hspace{10pt}& \int_1^2 \frac{x^4 +x }{\sqrt{x} } \hspace{1pt}dx \\[1em] =& \int_1^2 \frac{x^4 +x }{x^{\frac{1}{2}} } \hspace{1pt}dx\\[1em] =& \int_1^2 \left( x^{\frac{7}{2} }+ x^{\frac{1}{2}}\right) \hspace{1pt}dx\\[1em] =& \left[\frac{1}{\frac{7}{2}+1}x^{\frac{7}{2}+1} + \frac{1}{\frac{1}{2}+1}x^{\frac{1}{2}+1} \right]_1^2\\[1em] =& \left[\frac{2}{9}x^{\frac{9}{2}} + \frac{2}{3}x^{\frac{3}{2}} \right]_1^2\\[1em] =& \frac{2}{9} (2^\frac{9}{2} - 1) + \frac{2}{3} (2^\frac{3}{2} - 1)\\[1em] =& \frac{2}{9} (16 \sqrt{2}- 1) + \frac{2}{3} (2 \sqrt{2} - 1)\\[1em] =& \frac{44}{9}\sqrt{2}-\frac{8}{9}\\[1em] \end{aligned} $$