\[ \int \frac {1}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {4 \left (32 a^{3} x +39 a b \right ) \sqrt {x \left (a x +\sqrt {a^{2} x^{2}-b x}\right )}}{105 b^{3} x^{2}}-\frac {4 \left (-32 a^{2} x +15 b \right ) \sqrt {a^{2} x^{2}-b x}\, \sqrt {x \left (a x +\sqrt {a^{2} x^{2}-b x}\right )}}{105 b^{3} x^{3}} \]
command
Integrate[1/(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
\[ \frac {4 \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )^2 \left (15 b^2+32 a^3 x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )+a b \left (-47 a x+39 \sqrt {x \left (-b+a^2 x\right )}\right )\right )}{105 b^3 \sqrt {x \left (-b+a^2 x\right )} \left (x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )\right )^{3/2}} \]
Mathematica 12.3 output
\[ \int \frac {1}{\sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]________________________________________________________________________________________