Answer
$$\frac{2}{{b\left( {1 + 2n} \right)}}\left( {{u^n}\sqrt {a + bu} - an\int {\frac{{{u^{n - 1}}}}{{\sqrt {a + bu} }}du} } \right)$$
Work Step by Step
$$\eqalign{
& \int {\frac{{{u^n}}}{{\sqrt {a + bu} }}} du \cr
& {\text{Integrate by parts}} \cr
& {\text{Let }}t = {u^n},{\text{ }}dt = n{u^{n - 1}}du \cr
& dv = \sqrt {a + bu} ,{\text{ }}v = \int {{{\left( {a + bu} \right)}^{ - 1/2}}} du = \frac{1}{b}\frac{{{{\left( {a + bu} \right)}^{1/2}}}}{{1/2}} + C \cr
& v = \frac{2}{b}\sqrt {a + bu} \cr
& {\text{By the integration by parts formula}} \cr
& \int {tdv} = tv - \int {vdt} \cr
& \int {\frac{{{u^n}}}{{\sqrt {a + bu} }}} du = \frac{2}{b}{u^n}\sqrt {a + bu} - \int {\frac{2}{b}\sqrt {a + bu} \left( {n{u^{n - 1}}} \right)du} \cr
& \int {\frac{{{u^n}}}{{\sqrt {a + bu} }}} du = \frac{2}{b}{u^n}\sqrt {a + bu} - \frac{{2n}}{b}\int {{u^{n - 1}}\sqrt {a + bu} du} \cr
& {\text{Rationalizing}} \cr
& \int {\frac{{{u^n}}}{{\sqrt {a + bu} }}} du = \frac{2}{b}{u^n}\sqrt {a + bu} - \frac{{2n}}{b}\int {{u^{n - 1}}\sqrt {a + bu} \times \frac{{\sqrt {a + bu} }}{{\sqrt {a + bu} }}du} \cr
& \int {\frac{{{u^n}}}{{\sqrt {a + bu} }}} du = \frac{2}{b}{u^n}\sqrt {a + bu} - \frac{{2n}}{b}\int {\frac{{\left( {a + bu} \right){u^{n - 1}}}}{{\sqrt {a + bu} }}du} \cr
& {\text{Distribute}} \cr
& \int {\frac{{{u^n}}}{{\sqrt {a + bu} }}} du = \frac{2}{b}{u^n}\sqrt {a + bu} - \frac{{2n}}{b}\int {\frac{{\left( {a + bu} \right){u^{n - 1}}}}{{\sqrt {a + bu} }}du} \cr
& = \frac{2}{b}{u^n}\sqrt {a + bu} - \frac{{2an}}{b}\int {\frac{{{u^{n - 1}}}}{{\sqrt {a + bu} }}du} - \frac{{2n}}{b}\int {\frac{{b{u^n}}}{{\sqrt {a + bu} }}du} \cr
& = \frac{2}{b}{u^n}\sqrt {a + bu} - \frac{{2an}}{b}\int {\frac{{{u^{n - 1}}}}{{\sqrt {a + bu} }}du} - 2n\int {\frac{{{u^n}}}{{\sqrt {a + bu} }}du} \cr
& {\text{Add }} - 2n\int {\frac{{{u^n}}}{{\sqrt {a + bu} }}du} {\text{ to both sides}} \cr
& \int {\frac{{{u^n}}}{{\sqrt {a + bu} }}} du + 2n\int {\frac{{{u^n}}}{{\sqrt {a + bu} }}du} = \frac{2}{b}{u^n}\sqrt {a + bu} - \frac{{2an}}{b}\int {\frac{{{u^{n - 1}}}}{{\sqrt {a + bu} }}du} \cr
& \int {\frac{{{u^n}}}{{\sqrt {a + bu} }}du} \left( {1 + 2n} \right) = \frac{2}{b}{u^n}\sqrt {a + bu} - \frac{{2an}}{b}\int {\frac{{{u^{n - 1}}}}{{\sqrt {a + bu} }}du} \cr
& {\text{Divide both sides by }}1 + 2n \cr
& \int {\frac{{{u^n}}}{{\sqrt {a + bu} }}du} = \frac{2}{{b\left( {1 + 2n} \right)}}{u^n}\sqrt {a + bu} - \frac{{2an}}{{b\left( {1 + 2n} \right)}}\int {\frac{{{u^{n - 1}}}}{{\sqrt {a + bu} }}du} \cr
& {\text{Factoring}} \cr
& \int {\frac{{{u^n}}}{{\sqrt {a + bu} }}du} = \frac{2}{{b\left( {1 + 2n} \right)}}\left( {{u^n}\sqrt {a + bu} - an\int {\frac{{{u^{n - 1}}}}{{\sqrt {a + bu} }}du} } \right) \cr} $$
