\[ \int \left (a+x^2\right ) \left (x+\sqrt {a+x^2}\right )^n \, dx \]
Optimal antiderivative \[ -\frac {a^{3} \left (x +\sqrt {x^{2}+a}\right )^{-3+n}}{8 \left (3-n \right )}-\frac {3 a^{2} \left (x +\sqrt {x^{2}+a}\right )^{-1+n}}{8 \left (1-n \right )}+\frac {3 a \left (x +\sqrt {x^{2}+a}\right )^{1+n}}{8 \left (1+n \right )}+\frac {\left (x +\sqrt {x^{2}+a}\right )^{3+n}}{24+8 n} \]
command
integrate((x**2+a)*(x+(x**2+a)**(1/2))**n,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {output too large to display} \]
Sympy 1.8 under Python 3.8.8 output \[ \text {Timed out} \]_____________________________________________________