\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^4} \, dx \]
Optimal antiderivative \[ \frac {b^{2} c \arctanh \left (\frac {\sqrt {a +b \sqrt {c \,x^{3}}}}{\sqrt {a}}\right )}{6 a^{\frac {3}{2}}}-\frac {\sqrt {a +b \sqrt {c \,x^{3}}}}{3 x^{3}}-\frac {b c \sqrt {a +b \sqrt {c \,x^{3}}}}{6 a \sqrt {c \,x^{3}}} \]
command
Integrate[Sqrt[a + b*Sqrt[c*x^3]]/x^4,x]
Mathematica 13.1 output
\[ -\frac {\sqrt {a+b \sqrt {c x^3}} \left (2 a+b \sqrt {c x^3}\right )}{6 a x^3}+\frac {b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^3}}}{\sqrt {a}}\right )}{6 a^{3/2}} \]
Mathematica 12.3 output
\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^4} \, dx \]________________________________________________________________________________________