\[ \int \frac {a g+f x^3-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {g x}{\sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {f \left (b \,x^{2}+2 a \right )}{\left (-4 a c +b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}} \]
command
Integrate[(a*g + f*x^3 - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]
Mathematica 13.1 output
\[ \frac {b x (b g+f x)+2 a (f-2 c g x)}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________