\[ \int \frac {x}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {20 a \sqrt {x \left (a x +\sqrt {a^{2} x^{2}-b x}\right )}}{3 b^{2} x}-\frac {4 \sqrt {a^{2} x^{2}-b x}\, \sqrt {x \left (a x +\sqrt {a^{2} x^{2}-b x}\right )}}{3 b^{2} x^{2}} \]
command
Integrate[x/(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
\[ \frac {4 \sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (b+a \left (-a x+5 \sqrt {x \left (-b+a^2 x\right )}\right )\right )}{3 b^2 x \sqrt {x \left (-b+a^2 x\right )}} \]
Mathematica 12.3 output
\[ \int \frac {x}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]________________________________________________________________________________________