\[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{5/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {a^{2} x^{2}-b x}\, \left (121339 a^{10} x^{5}-148243 a^{8} b \,x^{4}+12416 a^{6} b^{2} x^{3}+5248 a^{4} b^{3} x^{2}+2688 a^{2} b^{4} x -4368 b^{5}\right ) \sqrt {x \left (a x +\sqrt {a^{2} x^{2}-b x}\right )}}{16380 b^{7} x^{5} \left (-a^{2} x +b \right )^{2}}+\sqrt {x \left (a x +\sqrt {a^{2} x^{2}-b x}\right )}\, \left (-\frac {283847 a^{9} x^{4}-229768 a^{7} b \,x^{3}-24840 a^{5} b^{2} x^{2}-9352 a^{3} b^{3} x -4872 a \,b^{4}}{8190 b^{7} x^{4} \left (-a^{2} x +b \right )}+\frac {109 a^{\frac {15}{2}} \sqrt {-a x +\sqrt {a^{2} x^{2}-b x}}\, \arctan \left (\frac {\sqrt {a}\, \sqrt {-a x +\sqrt {a^{2} x^{2}-b x}}}{\sqrt {b}}\right )}{4 b^{\frac {15}{2}} x}\right ) \]
command
Integrate[1/((-(b*x) + a^2*x^2)^(5/2)*(a*x^2 + x*Sqrt[-(b*x) + a^2*x^2])^(3/2)),x]
Mathematica 13.1 output
\[ \frac {\sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (\sqrt {b} \left (-4368 b^5+16 a^5 b^2 x^2 \left (776 a x-3105 \sqrt {x \left (-b+a^2 x\right )}\right )+16 a^3 b^3 x \left (328 a x-1169 \sqrt {x \left (-b+a^2 x\right )}\right )+336 a b^4 \left (8 a x-29 \sqrt {x \left (-b+a^2 x\right )}\right )-a^7 b x^3 \left (148243 a x+459536 \sqrt {x \left (-b+a^2 x\right )}\right )+a^9 x^4 \left (121339 a x+567694 \sqrt {x \left (-b+a^2 x\right )}\right )\right )+446355 a^{15/2} x^2 \left (x \left (-b+a^2 x\right )\right )^{3/2} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}}}{\sqrt {b}}\right )\right )}{16380 b^{15/2} x^3 \left (x \left (-b+a^2 x\right )\right )^{3/2}} \]
Mathematica 12.3 output
\[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{5/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]________________________________________________________________________________________