\int \sqrt{x^3(ax+b)} dx =\left [ 
\frac{b}{12a}-
\frac{b^2}{8a^2x}+
\frac{x}{3}\right]\sqrt{x^3(ax+b)} + 
\frac{b^3}{8a^{5/2}}\ln \left | a\sqrt{x} + \sqrt{a(ax+b)} \right |+C