\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^4} \, dx \]
Optimal antiderivative \[ \frac {b \left (-b^{2} d +4 a c \right ) \left (33 b^{4} d^{2}-120 a \,b^{2} c d +80 a^{2} c^{2}\right ) \arctanh \left (\frac {b d +2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c}\, \sqrt {d}\, \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}\right ) \sqrt {d}}{1024 c^{\frac {13}{2}}}+\frac {11 b \left (\frac {d}{x}\right )^{\frac {3}{2}} \left (a +\frac {c}{x}+b \sqrt {\frac {d}{x}}\right )^{\frac {3}{2}}}{42 c^{2} d}-\frac {2 \left (a +\frac {c}{x}+b \sqrt {\frac {d}{x}}\right )^{\frac {3}{2}}}{7 c \,x^{2}}+\frac {\left (-33 b^{2} d +32 a c \right ) \left (a +\frac {c}{x}+b \sqrt {\frac {d}{x}}\right )^{\frac {3}{2}}}{140 c^{3} x}-\frac {\left (a +\frac {c}{x}+b \sqrt {\frac {d}{x}}\right )^{\frac {3}{2}} \left (1024 a^{2} c^{2}-3276 a \,b^{2} c d +1155 b^{4} d^{2}+18 b c \left (-77 b^{2} d +148 a c \right ) \sqrt {\frac {d}{x}}\right )}{6720 c^{5}}+\frac {b \left (33 b^{4} d^{2}-120 a \,b^{2} c d +80 a^{2} c^{2}\right ) \left (b d +2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a +\frac {c}{x}+b \sqrt {\frac {d}{x}}}}{512 c^{6}} \]
command
Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^4,x]
Mathematica 13.1 output
\[ \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} \left (-\frac {2 \sqrt {c} \left (15360 c^6+256 c^5 \left (12 a+5 b \sqrt {\frac {d}{x}}\right ) x-3465 b^6 d^3 x^3+210 b^4 c d^2 \left (104 a+11 b \sqrt {\frac {d}{x}}\right ) x^3-168 b^2 c^2 d x^2 \left (11 b^2 d+206 a^2 x+72 a b \sqrt {\frac {d}{x}} x\right )-64 c^4 x \left (22 b^2 d+64 a^2 x+79 a b \sqrt {\frac {d}{x}} x\right )+16 c^3 x^2 \left (486 a b^2 d+99 b^3 d \sqrt {\frac {d}{x}}+512 a^3 x+794 a^2 b \sqrt {\frac {d}{x}} x\right )\right )}{x^3}+\frac {105 b d \left (-320 a^3 c^3+560 a^2 b^2 c^2 d-252 a b^4 c d^2+33 b^6 d^3\right ) \log \left (b d+2 c \sqrt {\frac {d}{x}}-2 \sqrt {c} \sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}\right )}{\sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}}}\right )}{107520 c^{13/2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^4} \, dx \]________________________________________________________________________________________