\[ \int \frac {b^2+a x^2}{\left (-b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx \]
Optimal antiderivative \[ \frac {2 x}{\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}+\frac {2 \sqrt {2}\, \sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x \sqrt {2}}{2 \sqrt {b}\, \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}-\frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}\, \sqrt {2}}{2 \sqrt {b}}\right )}{\sqrt {a}}-\frac {2 \sqrt {1+\sqrt {2}}\, \sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {-2+2 \sqrt {2}}\, \sqrt {b}\, \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}-\frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{\sqrt {-2+2 \sqrt {2}}\, \sqrt {b}}\right )}{\sqrt {a}}-\frac {2 \sqrt {1+\sqrt {2}}\, \sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {2+2 \sqrt {2}}\, \sqrt {b}\, \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}-\frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{\sqrt {2+2 \sqrt {2}}\, \sqrt {b}}\right )}{\sqrt {a}}+\frac {2 \sqrt {\sqrt {2}-1}\, \sqrt {b}\, \arctanh \left (\frac {\sqrt {a}\, x}{\sqrt {-2+2 \sqrt {2}}\, \sqrt {b}\, \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}-\frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{\sqrt {-2+2 \sqrt {2}}\, \sqrt {b}}\right )}{\sqrt {a}}-\frac {2 \sqrt {\sqrt {2}-1}\, \sqrt {b}\, \arctanh \left (\frac {\sqrt {a}\, x}{\sqrt {2+2 \sqrt {2}}\, \sqrt {b}\, \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}-\frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{\sqrt {2+2 \sqrt {2}}\, \sqrt {b}}\right )}{\sqrt {a}} \]
command
Integrate[(b^2 + a*x^2)/((-b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]),x]
Mathematica 13.1 output
\[ \frac {2 x}{\sqrt {b+\sqrt {b^2+a x^2}}}+\frac {\sqrt {2} \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}-\frac {2 \sqrt {1+\sqrt {2}} \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {-1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {1+\sqrt {2}} \sqrt {a}} \]
Mathematica 12.3 output
\[ \int \frac {b^2+a x^2}{\left (-b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx \]________________________________________________________________________________________