\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx \]
Optimal antiderivative \[ -\frac {4 \left (2 c^{2} x -3 a x +3 b \right ) \sqrt {-x \left (-c x +\sqrt {a \,x^{2}-b x}\right )}}{15 b \,x^{2}}-\frac {4 c \sqrt {a \,x^{2}-b x}\, \sqrt {-x \left (-c x +\sqrt {a \,x^{2}-b x}\right )}}{15 b \,x^{2}} \]
command
Integrate[Sqrt[c*x^2 - x*Sqrt[-(b*x) + a*x^2]]/x^3,x]
Mathematica 13.1 output
\[ \frac {4 \sqrt {x \left (c x-\sqrt {x (-b+a x)}\right )} \left (-3 b+3 a x-c \left (2 c x+\sqrt {x (-b+a x)}\right )\right )}{15 b x^2} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx \]________________________________________________________________________________________