\[ \int \frac {\sqrt {-b x+a^2 x^2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {4 \left (3 a^{3} x +5 a b \right ) \sqrt {x \left (a x +\sqrt {a^{2} x^{2}-b x}\right )}}{3 b^{2} x}+\frac {4 \left (3 a^{2} x +b \right ) \sqrt {a^{2} x^{2}-b x}\, \sqrt {x \left (a x +\sqrt {a^{2} x^{2}-b x}\right )}}{3 b^{2} x^{2}} \]
command
Integrate[Sqrt[-(b*x) + a^2*x^2]/(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2),x]
Mathematica 13.1 output
\[ -\frac {4 \left (b+a \left (7 a x+4 \sqrt {x \left (-b+a^2 x\right )}\right )\right )}{3 b \sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {-b x+a^2 x^2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]________________________________________________________________________________________