\[ \int \sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \]
Optimal antiderivative \[ \frac {\left (40320 a^{4} c^{2} x^{4}-2560 a^{3} c^{4} x^{3}-2048 a^{2} c^{6} x^{2}+114240 a^{2} b \,c^{2} x^{2}-1920 a b \,c^{4} x -1024 b \,c^{6}+3465 a \,b^{2} x +32760 b^{2} c^{2}\right ) \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}+\left (2240 a^{3} c^{3} x^{3}+1536 a^{2} c^{5} x^{2}+2048 a \,c^{7} x +38640 a b \,c^{3} x +768 b \,c^{5}-2310 b^{2} c \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}+\sqrt {a^{2} x^{2}+b}\, \left (\left (40320 a^{3} c^{2} x^{3}-2560 a^{2} c^{4} x^{2}-2048 a \,c^{6} x +94080 a b \,c^{2} x -640 b \,c^{4}+3465 b^{2}\right ) \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}+\left (2240 a^{2} c^{3} x^{2}+1536 a \,c^{5} x +2048 c^{7}+37520 b \,c^{3}\right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}\right )}{55440 a \,c^{2} \left (a x +\sqrt {a^{2} x^{2}+b}\right )^{\frac {3}{2}}}-\frac {b^{2} \arctanh \left (\frac {\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}+b}}}}{\sqrt {c}}\right )}{16 a \,c^{\frac {5}{2}}} \]
command
Integrate[Sqrt[b + a^2*x^2]*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
\[ \frac {\frac {\sqrt {c} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \left (105 b^2 \left (312 c^2-22 c \sqrt {a x+\sqrt {b+a^2 x^2}}+33 \left (a x+\sqrt {b+a^2 x^2}\right )\right )+16 b c^2 \left (-64 c^4+48 c^3 \sqrt {a x+\sqrt {b+a^2 x^2}}-40 c^2 \left (3 a x+\sqrt {b+a^2 x^2}\right )+420 a x \left (17 a x+14 \sqrt {b+a^2 x^2}\right )+35 c \sqrt {a x+\sqrt {b+a^2 x^2}} \left (69 a x+67 \sqrt {b+a^2 x^2}\right )\right )+64 c^2 \left (a x+\sqrt {b+a^2 x^2}\right ) \left (630 a^3 x^3+32 c^5 \sqrt {a x+\sqrt {b+a^2 x^2}}+8 a c^3 x \left (-4 c+3 \sqrt {a x+\sqrt {b+a^2 x^2}}\right )+5 a^2 c x^2 \left (-8 c+7 \sqrt {a x+\sqrt {b+a^2 x^2}}\right )\right )\right )}{\left (a x+\sqrt {b+a^2 x^2}\right )^{3/2}}-3465 b^2 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}}}{\sqrt {c}}\right )}{55440 a c^{5/2}} \]
Mathematica 12.3 output
\[ \int \sqrt {b+a^2 x^2} \sqrt {a x+\sqrt {b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {b+a^2 x^2}}} \, dx \]________________________________________________________________________________________