\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx \]
Optimal antiderivative \[ -\frac {b \left (-7 b^{2} d +12 a c \right ) \left (-b^{2} d +4 a c \right ) \arctanh \left (\frac {b d +2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}\right ) \sqrt {d}}{128 c^{\frac {9}{2}}}-\frac {2 \left (a +\frac {c}{x}+b \sqrt {\frac {d}{x}}\right )^{\frac {3}{2}}}{5 c x}+\frac {\left (a +\frac {c}{x}+b \sqrt {\frac {d}{x}}\right )^{\frac {3}{2}} \left (32 a c -35 b^{2} d +42 b c \sqrt {\frac {d}{x}}\right )}{120 c^{3}}-\frac {b \left (-7 b^{2} d +12 a c \right ) \left (b d +2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{64 c^{4}} \]
command
Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^3,x]
Mathematica 13.1 output
\[ \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (\frac {2 \sqrt {c} \left (-384 c^4-16 c^3 \left (8 a+3 b \sqrt {\frac {d}{x}}\right ) x+105 b^4 d^2 x^2-10 b^2 c d \left (46 a+7 b \sqrt {\frac {d}{x}}\right ) x^2+8 c^2 x \left (7 b^2 d+32 a^2 x+29 a b \sqrt {\frac {d}{x}} x\right )\right )}{x^2}+\frac {15 b d \left (48 a^2 c^2-40 a b^2 c d+7 b^4 d^2\right ) \log \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )}{\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}}\right )}{1920 c^{9/2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx \]________________________________________________________________________________________