\[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx \]
Optimal antiderivative \[ \frac {\left (35 b^{4} d^{2}-120 a \,b^{2} c d +48 a^{2} c^{2}\right ) \arctanh \left (\frac {2 a +b \sqrt {\frac {d}{x}}}{2 \sqrt {a}\, \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}\right )}{64 a^{\frac {9}{2}}}-\frac {7 b \,d^{2} \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{12 a^{2} \left (\frac {d}{x}\right )^{\frac {3}{2}}}-\frac {\left (-35 b^{2} d +36 a c \right ) x \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{48 a^{3}}+\frac {x^{2} \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{2 a}+\frac {5 b d \left (-21 b^{2} d +44 a c \right ) \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{96 a^{4} \sqrt {\frac {d}{x}}} \]
command
Integrate[x/Sqrt[a + b*Sqrt[d/x] + c/x],x]
Mathematica 13.1 output
\[ \frac {\sqrt {a} d \left (-105 b^3 d \left (b d+c \sqrt {\frac {d}{x}}\right )+48 a^4 x^2-8 a^3 x \left (3 c+b \sqrt {\frac {d}{x}} x\right )+a^2 \left (-72 c^2+14 b^2 d x+92 b c \sqrt {\frac {d}{x}} x\right )-5 a b \left (-58 b c d-44 c^2 \sqrt {\frac {d}{x}}+7 b^2 d \sqrt {\frac {d}{x}} x\right )\right )-3 \sqrt {d} \left (48 a^2 c^2-120 a b^2 c d+35 b^4 d^2\right ) \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {d}{x}}-\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{96 a^{9/2} d \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \]
Mathematica 12.3 output
\[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \, dx \]________________________________________________________________________________________