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