\[ \int \frac {\left (x+\sqrt {b+x^2}\right )^a}{\sqrt {b+x^2}} \, dx \]
Optimal antiderivative \[ \frac {\left (x +\sqrt {x^{2}+b}\right )^{a}}{a} \]
command
integrate((x+(x^2+b)^(1/2))^a/(x^2+b)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (x + \sqrt {x^{2} + b}\right )}^{a}}{a} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (x + \sqrt {x^{2} + b}\right )}^{a}}{\sqrt {x^{2} + b}}\,{d x} \]________________________________________________________________________________________