\[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3} \, dx \]
Optimal antiderivative \[ -\frac {b \left (-5 b^{2} d +12 a c \right ) \arctanh \left (\frac {b d +2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}\right ) \sqrt {d}}{8 c^{\frac {7}{2}}}-\frac {2 \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{3 c x}+\frac {\left (16 a c -15 b^{2} d +10 b c \sqrt {\frac {d}{x}}\right ) \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{12 c^{3}} \]
command
Integrate[1/(Sqrt[a + b*Sqrt[d/x] + c/x]*x^3),x]
Mathematica 13.1 output
\[ \frac {2 \sqrt {c} \left (-8 c^3+2 c^2 \left (4 a+b \sqrt {\frac {d}{x}}\right ) x-15 b^2 d \left (a+b \sqrt {\frac {d}{x}}\right ) x^2+c x \left (-5 b^2 d+16 a^2 x+26 a b \sqrt {\frac {d}{x}} x\right )\right )+3 b \left (12 a c-5 b^2 d\right ) x^2 \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \log \left (c^3 \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )\right )}{24 c^{7/2} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^2} \]
Mathematica 12.3 output
\[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3} \, dx \]________________________________________________________________________________________