\[ \int \frac {x^{-1+\frac {n}{2}} \left (-a h+c f x^{n/2}+c g x^{3 n/2}+c h x^{2 n}\right )}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (c \left (-2 a g +b f \right )+\left (-4 a c +b^{2}\right ) h \,x^{\frac {n}{2}}+c \left (-b g +2 c f \right ) x^{n}\right )}{\left (-4 a c +b^{2}\right ) n \sqrt {a +b \,x^{n}+c \,x^{2 n}}} \]
command
Integrate[(x^(-1 + n/2)*(-(a*h) + c*f*x^(n/2) + c*g*x^((3*n)/2) + c*h*x^(2*n)))/(a + b*x^n + c*x^(2*n))^(3/2),x]
Mathematica 13.1 output
\[ -\frac {2 \left (b c f-2 a c g+b^2 h x^{n/2}-4 a c h x^{n/2}+2 c^2 f x^n-b c g x^n\right )}{\left (b^2-4 a c\right ) n \sqrt {a+b x^n+c x^{2 n}}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________