\[ \int \frac {\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {x \left (2 a \,x^{2}+11 b^{2}\right )}{3 \sqrt {a \,x^{2}+b^{2}}\, \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}+\frac {2 x \left (2 a b \,x^{2}+5 b^{3}\right )}{3 \left (a \,x^{2}+b^{2}\right ) \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}-\frac {5 b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}\, \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}\right )}{\sqrt {a}} \]
command
Integrate[((-b^2 + a*x^2)^2*Sqrt[b + Sqrt[b^2 + a*x^2]])/(b^2 + a*x^2)^2,x]
Mathematica 13.1 output
\[ \frac {x \left (11 b^2+2 a x^2\right )}{3 \sqrt {b^2+a x^2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \left (5 b^3+2 a b x^2\right )}{3 \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {5 b^{3/2} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]
Mathematica 12.3 output
\[ \int \frac {\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx \]________________________________________________________________________________________