\[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x \, dx \]
Optimal antiderivative \[ -\frac {\left (-5 b^{2} d +4 a c \right ) \left (-b^{2} d +4 a c \right ) \arctanh \left (\frac {2 a +b \sqrt {\frac {d}{x}}}{2 \sqrt {a}\, \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}\right )}{64 a^{\frac {7}{2}}}-\frac {5 b \,d^{2} \left (a +\frac {c}{x}+b \sqrt {\frac {d}{x}}\right )^{\frac {3}{2}}}{12 a^{2} \left (\frac {d}{x}\right )^{\frac {3}{2}}}+\frac {x^{2} \left (a +\frac {c}{x}+b \sqrt {\frac {d}{x}}\right )^{\frac {3}{2}}}{2 a}-\frac {\left (-5 b^{2} d +4 a c \right ) x \left (2 a +b \sqrt {\frac {d}{x}}\right ) \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{32 a^{3}} \]
command
Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]*x,x]
Mathematica 13.1 output
\[ \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\sqrt {a} x \left (-2 a b \left (5 b d+26 c \sqrt {\frac {d}{x}}\right )+15 b^3 d \sqrt {\frac {d}{x}}+48 a^3 x+8 a^2 \left (3 c+b \sqrt {\frac {d}{x}} x\right )\right )+\frac {3 \sqrt {d} \left (16 a^2 c^2-24 a b^2 c d+5 b^4 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {d}{x}}-\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}}\right )}{96 a^{7/2}} \]
Mathematica 12.3 output
\[ \int \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x \, dx \]________________________________________________________________________________________