\[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx \]
Optimal antiderivative \[ -\frac {\left (-3 b^{2} d +4 a c \right ) \arctanh \left (\frac {2 a +b \sqrt {\frac {d}{x}}}{2 \sqrt {a}\, \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}\right )}{4 a^{\frac {5}{2}}}+\frac {x \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{a}-\frac {3 b d \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{2 a^{2} \sqrt {\frac {d}{x}}} \]
command
Integrate[1/Sqrt[a + b*Sqrt[d/x] + c/x],x]
Mathematica 13.1 output
\[ \frac {\sqrt {a} d \left (2 a-3 b \sqrt {\frac {d}{x}}\right ) \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )+\sqrt {d} \left (4 a c-3 b^2 d\right ) \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {d}{x}}-\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{2 a^{5/2} d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \]
Mathematica 12.3 output
\[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx \]________________________________________________________________________________________