\[ \int x^2 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {a}\, x \sqrt {a^{2} x^{4}+b}\, \left (16 a^{3} x^{6}+28 a b \,x^{2}\right ) \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}+\sqrt {a}\, x \left (16 a^{4} x^{8}+36 a^{2} b \,x^{4}+9 b^{2}\right ) \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{48 a^{\frac {3}{2}} b +96 a^{\frac {7}{2}} x^{4}+96 a^{\frac {5}{2}} x^{2} \sqrt {a^{2} x^{4}+b}}-\frac {3 b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {a}\, x \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{\sqrt {b}}\right ) \sqrt {2}}{32 a^{\frac {3}{2}}} \]
command
Integrate[x^2*Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]
Mathematica 13.1 output
\[ \frac {\frac {2 \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}} \left (9 b^2+16 a^3 x^6 \left (a x^2+\sqrt {b+a^2 x^4}\right )+4 a b x^2 \left (9 a x^2+7 \sqrt {b+a^2 x^4}\right )\right )}{b+2 a x^2 \left (a x^2+\sqrt {b+a^2 x^4}\right )}-9 \sqrt {2} b^{3/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )}{96 a^{3/2}} \]
Mathematica 12.3 output
\[ \int x^2 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx \]________________________________________________________________________________________