\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^2} \, dx \]
Optimal antiderivative \[ \frac {b \left (-b^{2} d +4 a c \right ) \arctanh \left (\frac {b d +2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}\right ) \sqrt {d}}{8 c^{\frac {5}{2}}}-\frac {2 \left (a +\frac {c}{x}+b \sqrt {\frac {d}{x}}\right )^{\frac {3}{2}}}{3 c}+\frac {b \left (b d +2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{4 c^{2}} \]
command
Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^2,x]
Mathematica 13.1 output
\[ \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (-\frac {2 \sqrt {c} \left (8 c^2-3 b^2 d x+2 c \left (4 a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}+\frac {3 b d \left (-4 a c+b^2 d\right ) \log \left (c^2 \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )\right )}{\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}}\right )}{24 c^{5/2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^2} \, dx \]________________________________________________________________________________________