\[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {4 \sqrt {a^{2} x^{2}-b x}\, \left (-8192 a^{6} x^{3}-5120 a^{4} b \,x^{2}-4032 a^{2} b^{2} x +3003 b^{3}\right ) \sqrt {x \left (a x +\sqrt {a^{2} x^{2}-b x}\right )}}{45045 b^{5} x^{5}}+\frac {4 \left (8192 a^{7} x^{3}+1024 a^{5} b \,x^{2}+448 a^{3} b^{2} x +6699 a \,b^{3}\right ) \sqrt {x \left (a x +\sqrt {a^{2} x^{2}-b x}\right )}}{45045 b^{5} x^{4}} \]
command
Integrate[1/(x^2*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
\[ \frac {4 \sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (3003 b^4+1024 a^5 b x^2 \left (-3 a x+\sqrt {x \left (-b+a^2 x\right )}\right )+8192 a^7 x^3 \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )+64 a^3 b^2 x \left (-17 a x+7 \sqrt {x \left (-b+a^2 x\right )}\right )+21 a b^3 \left (-335 a x+319 \sqrt {x \left (-b+a^2 x\right )}\right )\right )}{45045 b^5 x^4 \sqrt {x \left (-b+a^2 x\right )}} \]
Mathematica 12.3 output
\[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]________________________________________________________________________________________