\[ \int \frac {\sqrt {b^2+a x^2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \]
Optimal antiderivative \[ -\frac {2 b x}{3 \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}+\frac {2 x \sqrt {a \,x^{2}+b^{2}}}{3 \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}+\frac {2 \sqrt {2}\, b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {a}\, x \sqrt {2}}{2 \sqrt {b}\, \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}-\frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}\, \sqrt {2}}{2 \sqrt {b}}\right )}{\sqrt {a}} \]
command
integrate((a*x^2+b^2)^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {3 \, \sqrt {2} a b x \sqrt {-\frac {b}{a}} \log \left (-\frac {a x^{3} + 4 \, b^{2} x - 4 \, \sqrt {a x^{2} + b^{2}} b x - 2 \, {\left (2 \, \sqrt {2} \sqrt {a x^{2} + b^{2}} b \sqrt {-\frac {b}{a}} - \sqrt {2} {\left (a x^{2} + 2 \, b^{2}\right )} \sqrt {-\frac {b}{a}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{3}}\right ) + 4 \, {\left (a x^{2} + 2 \, b^{2} - 2 \, \sqrt {a x^{2} + b^{2}} b\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{6 \, a x}, -\frac {3 \, \sqrt {2} a b x \sqrt {\frac {b}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {b + \sqrt {a x^{2} + b^{2}}} \sqrt {\frac {b}{a}}}{x}\right ) - 2 \, {\left (a x^{2} + 2 \, b^{2} - 2 \, \sqrt {a x^{2} + b^{2}} b\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{3 \, a x}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]