Find a Reduction Formula I'm trying to find a reduction formula for:
$${I}_n=\int \frac{x^{n}}{\sqrt{ax+b}}dx$$
This is where I have gotten to so far:
$${I}_n= \frac{2x^{n}\sqrt{ax+b}}{a}-\frac{2n}{a} \int \frac{x^{n-1}(ax+b)}{\sqrt{ax+b}}dx$$
I'd appreciate a little nudge in the right direction :)

You're almost there.  Simply note that for $I_n=\int \frac{x^n}{\sqrt{ax+b}}\,dx$, we have
$$\begin{align}
I_n&=\frac{2x^n\sqrt{ax+b}}{a}-\frac{2n}{a}\int x^{n-1}\sqrt{ax+b}\,dx\\\\
&=\frac{2x^n\sqrt{ax+b}}{a}-\frac{2n}{a}\int x^{n-1}\frac{ax+b}{\sqrt{ax+b}}\,dx\\\\
&=\frac{2x^n\sqrt{ax+b}}{a}-\frac{2n}{a}\left(a\int \frac{x^n}{\sqrt{ax+b}}\,dx+b\int \frac{x^{n-1}}{\sqrt{ax+b}}\right)\\\\
&=\frac{2x^n\sqrt{ax+b}}{a}-2nI_n-\frac{2nb}{a}I_{n-1}\\\\
I_n&=\frac{2x^n\sqrt{ax+b}}{a(1+2n)}-\frac{2nb}{a(2n+1)}I_{n-1}
\end{align}$$
