\[ \int x^4 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx \]
Optimal antiderivative \[ \frac {\sqrt {a}\, x \sqrt {a^{2} x^{4}+b}\, \left (192 a^{5} x^{10}+264 a^{3} b \,x^{6}+104 a \,b^{2} x^{2}\right ) \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}+\sqrt {a}\, x \left (192 a^{6} x^{12}+360 a^{4} b \,x^{8}+212 a^{2} b^{2} x^{4}+39 b^{3}\right ) \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{1152 a^{\frac {7}{2}} b \,x^{2}+1536 a^{\frac {11}{2}} x^{6}+384 a^{\frac {5}{2}} b \sqrt {a^{2} x^{4}+b}+1536 a^{\frac {9}{2}} x^{4} \sqrt {a^{2} x^{4}+b}}-\frac {13 b^{2} \ln \left (i a \,x^{2}+i \sqrt {a^{2} x^{4}+b}+i \sqrt {2}\, \sqrt {a}\, x \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}\right ) \sqrt {2}}{256 a^{\frac {5}{2}}} \]
command
Integrate[x^4*Sqrt[b + a^2*x^4]*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]],x]
Mathematica 13.1 output
\[ \frac {\frac {2 \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}} \left (39 b^3+192 a^5 x^{10} \left (a x^2+\sqrt {b+a^2 x^4}\right )+24 a^3 b x^6 \left (15 a x^2+11 \sqrt {b+a^2 x^4}\right )+4 a b^2 x^2 \left (53 a x^2+26 \sqrt {b+a^2 x^4}\right )\right )}{3 a b x^2+4 a^3 x^6+b \sqrt {b+a^2 x^4}+4 a^2 x^4 \sqrt {b+a^2 x^4}}-39 \sqrt {2} b^2 \tanh ^{-1}\left (\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {2} \sqrt {a} x}\right )}{768 a^{5/2}} \]
Mathematica 12.3 output
\[ \int x^4 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx \]________________________________________________________________________________________