\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {5 x}{12 b^{2} \left (a \,x^{2}+b^{2}\right ) \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}+\frac {x \left (15 a \,x^{2}+23 b^{2}\right )}{24 b^{3} \left (a \,x^{2}+b^{2}\right )^{\frac {3}{2}} \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}+\frac {5 \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}\, \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}\right )}{8 \sqrt {a}\, b^{\frac {7}{2}}} \]
command
Integrate[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^(5/2),x]
Mathematica 13.1 output
\[ \frac {x \left (23 b^2+15 a x^2+10 b \sqrt {b^2+a x^2}\right )}{24 b^3 \left (b^2+a x^2\right )^{3/2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {5 \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{8 \sqrt {a} b^{7/2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^{5/2}} \, dx \]________________________________________________________________________________________