\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx \]
Optimal antiderivative \[ \frac {\left (-4096 a^{3} c^{6} x^{3}+7680 a^{2} b \,c^{4} x^{2}+3072 a b \,c^{6} x -1890 a^{2} b^{2} x^{2}-504 a \,b^{2} c^{2} x -4224 b^{2} c^{4}+945 b^{3}\right ) \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}+\left (3072 a^{3} c^{5} x^{3}+4096 a^{2} c^{7} x^{2}-2304 a b \,c^{5} x -2048 b \,c^{7}+630 a \,b^{2} c x +432 b^{2} c^{3}\right ) \sqrt {a x +\sqrt {a^{2} x^{2}-b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}+\sqrt {a^{2} x^{2}-b}\, \left (\left (-4096 a^{2} c^{6} x^{2}+7680 a b \,c^{4} x +1024 b \,c^{6}-1890 a \,b^{2} x -504 b^{2} c^{2}\right ) \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}+\left (3072 a^{2} c^{5} x^{2}+4096 a \,c^{7} x -768 b \,c^{5}+630 b^{2} c \right ) \sqrt {a x +\sqrt {a^{2} x^{2}-b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}\right )}{3840 a \,c^{5} \left (a x +\sqrt {a^{2} x^{2}-b}\right )^{\frac {5}{2}}}+\frac {63 b^{2} \arctanh \left (\frac {\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}{\sqrt {c}}\right )}{256 a \,c^{\frac {11}{2}}}-\frac {b \arctanh \left (\frac {\sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}{\sqrt {c}}\right )}{a \,c^{\frac {3}{2}}} \]
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
\[ \frac {\left (945 b^3-4224 b^2 c^4-504 a b^2 c^2 x+3072 a b c^6 x-1890 a^2 b^2 x^2+7680 a^2 b c^4 x^2-4096 a^3 c^6 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (432 b^2 c^3-2048 b c^7+630 a b^2 c x-2304 a b c^5 x+4096 a^2 c^7 x^2+3072 a^3 c^5 x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (-504 b^2 c^2+1024 b c^6-1890 a b^2 x+7680 a b c^4 x-4096 a^2 c^6 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (630 b^2 c-768 b c^5+4096 a c^7 x+3072 a^2 c^5 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{3840 a c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {63 b^2 \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{256 a c^{11/2}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{a c^{3/2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx \]________________________________________________________________________________________