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