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