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