\[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx \]
Optimal antiderivative \[ \frac {b \left (63 b^{4} d^{2}-280 a \,b^{2} c d +240 a^{2} c^{2}\right ) \arctanh \left (\frac {b d +2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}\right ) \sqrt {d}}{128 c^{\frac {11}{2}}}+\frac {9 b \left (\frac {d}{x}\right )^{\frac {3}{2}} \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{20 c^{2} d}-\frac {2 \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{5 c \,x^{2}}+\frac {\left (-63 b^{2} d +64 a c \right ) \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{120 c^{3} x}-\frac {\left (1024 a^{2} c^{2}-2940 a \,b^{2} c d +945 b^{4} d^{2}+14 b c \left (-45 b^{2} d +92 a c \right ) \sqrt {\frac {d}{x}}\right ) \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{960 c^{5}} \]
command
Integrate[1/(Sqrt[a + b*Sqrt[d/x] + c/x]*x^4),x]
Mathematica 13.1 output
\[ \frac {-2 \sqrt {c} \left (384 c^5-16 c^4 \left (8 a+3 b \sqrt {\frac {d}{x}}\right ) x+945 b^4 d^2 \left (a+b \sqrt {\frac {d}{x}}\right ) x^3-105 b^2 c d x^2 \left (-3 b^2 d+28 a^2 x+34 a b \sqrt {\frac {d}{x}} x\right )+8 c^3 x \left (9 b^2 d+64 a^2 x+43 a b \sqrt {\frac {d}{x}} x\right )+2 c^2 x^2 \left (-574 a b^2 d-63 b^3 d \sqrt {\frac {d}{x}}+512 a^3 x+1156 a^2 b \sqrt {\frac {d}{x}} x\right )\right )-15 b \left (240 a^2 c^2-280 a b^2 c d+63 b^4 d^2\right ) x^3 \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \log \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )}{1920 c^{11/2} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^3} \]
Mathematica 12.3 output
\[ \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x^4} \, dx \]________________________________________________________________________________________