Evaluating $\int \sqrt{\frac{x^a}{(x+b)^c}} dx$ for $a,b,c\in\mathbb N$ and $a \lt c$ For $a,b,c\in\mathbb N$ are they any procedures how to calculate the following indefinite integral?
$$\int \sqrt{\frac{x^a}{(x+b)^c}} dx$$
it is also possible to assume that $a \lt c$ but it would be great if it works more general.

It seems in any particular case, Maple knows how to do this.
$$
\int \frac{x^{\frac{3}{2}}}{(x + 5)^{\frac{5}{2}}} d x = \frac{-2\sqrt{5} x^{\frac{5}{2}}
\Biggl(4 x^{\frac{7}{2}} + 15 x^{\frac{5}{2}} - 15 \sqrt{5} \operatorname{arcsinh} \biggl(\frac{\sqrt{5} \sqrt{x}}{5}\biggr) x^{2} \biggl(\frac{x}{5} + 1\biggr)^{\frac{3}{2}}\Biggr)}{
75 x^{\frac{9}{2}} \biggl(\frac{x}{5} + 1\biggr)^{\frac{3}{2}}}
$$
So I assume you can get some reduction formulas through integration by parts, then do a few beginning cases, like $a=1,2$, $c=1,2$
added substitution $y=\mathrm{arcsinh}\sqrt{x/b}$ leads to
$$
\int \frac{x^{\bigl(\frac{a}{2}\bigr)}}{(x + b)^{\bigl(\frac{c}{2}\bigr)}} d x = 2 b^{\Bigl(1 + \frac{a}{2} - \frac{c}{2}\Bigr)} \int {\operatorname{sinh} (y)^{(1 + a)}}{\operatorname{cosh} (y)^{(1-c)}} d y
$$
and the method for this trig-type integral is known.
