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