\[ \int \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \]
Optimal antiderivative \[ \frac {\left (60 a^{2} x^{2}-8 a \,c^{2} x -75 b \right ) \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}+\left (6 a c x +16 c^{3}\right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}+\sqrt {a^{2} x^{2}+b}\, \left (\left (60 a x -8 c^{2}\right ) \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}+6 c \sqrt {a x +\sqrt {a^{2} x^{2}+b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}\right )}{105 a \sqrt {a x +\sqrt {a^{2} x^{2}+b}}}-\frac {b \arctanh \left (\frac {\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}}{\sqrt {c}}\right )}{a \sqrt {c}} \]
command
Integrate[Sqrt[a*x + Sqrt[b + a^2*x^2]]*Sqrt[c + Sqrt[a*x + Sqrt[b + a^2*x^2]]],x]
Mathematica 13.1 output
\[ \frac {\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \left (-75 b+60 a^2 x^2+2 a x \left (-4 c^2+30 \sqrt {b+a^2 x^2}+3 c \sqrt {a x+\sqrt {b+a^2 x^2}}\right )+2 c \left (-4 c \sqrt {b+a^2 x^2}+8 c^2 \sqrt {a x+\sqrt {b+a^2 x^2}}+3 \sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}}\right )\right )}{\sqrt {a x+\sqrt {b+a^2 x^2}}}-\frac {105 b \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{\sqrt {c}}}{105 a} \]
Mathematica 12.3 output
\[ \int \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \]________________________________________________________________________________________