\[ \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {a^{2} x^{2}-b x}\, \left (32 a^{4} x^{2}-88 a^{2} b x +115 b^{2}\right ) \sqrt {x \left (a x +\sqrt {a^{2} x^{2}-b x}\right )}}{40 b^{2} x}+\sqrt {x \left (a x +\sqrt {a^{2} x^{2}-b x}\right )}\, \left (\frac {-32 a^{5} x^{2}+104 a^{3} b x -145 a \,b^{2}}{40 b^{2}}+\frac {9 \sqrt {b}\, \sqrt {-a x +\sqrt {a^{2} x^{2}-b x}}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {a}\, \sqrt {-a x +\sqrt {a^{2} x^{2}-b x}}}{\sqrt {b}}\right ) \sqrt {2}}{16 \sqrt {a}\, x}\right ) \]
command
Integrate[(-(b*x) + a^2*x^2)^(3/2)/(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2),x]
Mathematica 13.1 output
\[ \frac {\sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (-2 \sqrt {a} x \left (115 b^3+8 a^3 b x \left (15 a x-13 \sqrt {x \left (-b+a^2 x\right )}\right )+32 a^5 x^2 \left (-a x+\sqrt {x \left (-b+a^2 x\right )}\right )+29 a b^2 \left (-7 a x+5 \sqrt {x \left (-b+a^2 x\right )}\right )\right )+45 \sqrt {2} b^{5/2} \sqrt {x \left (-b+a^2 x\right )} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}}}{\sqrt {b}}\right )\right )}{80 \sqrt {a} b^2 x \sqrt {x \left (-b+a^2 x\right )}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]________________________________________________________________________________________