Section C.3 Integrals Involving \(\sqrt{a + bu}\)
\begin{align}
\int u^n \sqrt{a + bu} \, du \amp=\amp \frac{2}{b(2n+3)}\left[ u^n(a+bu)^{3/2} - n a \int u^{n-1} \sqrt{a + bu} \, du \right]\tag{C.3.1}\\
\int \frac{u^n}{\sqrt{a + bu}} \, du \amp=\amp \frac{2u^n \sqrt{a + bu}}{b(2n+1)} - \frac{2 n a}{b(2n+1)} \int \frac{u^{n-1}}{\sqrt{a + bu}} \, du\tag{C.3.2}\\
\int \frac{1}{u^n \sqrt{a + bu}} \, du \amp=\amp - \frac{\sqrt{a + bu}}{a(n-1)u^{n-1}} - \frac{b(2n-3)}{2 a (n-1)} \int \frac{1}{u^{n-1} \sqrt{a + bu}} \, du\tag{C.3.3}
\end{align}
