\[ \int \frac {(f x)^{3/2} \left (d+e x^2\right )}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 d \left (f x \right )^{\frac {5}{2}} F_{1}\left (\frac {5}{4}, \frac {3}{2}, \frac {3}{2}, \frac {9}{4}, -\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}, -\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}\right ) \sqrt {1+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}}\, \sqrt {1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}}{5 a f \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {2 e \left (f x \right )^{\frac {9}{2}} F_{1}\left (\frac {9}{4}, \frac {3}{2}, \frac {3}{2}, \frac {13}{4}, -\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}, -\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}\right ) \sqrt {1+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}}\, \sqrt {1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}}{9 a \,f^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}} \]
command
Integrate[((f*x)^(3/2)*(d + e*x^2))/(a + b*x^2 + c*x^4)^(3/2),x]
Mathematica 13.1 output
\[ -\frac {f \sqrt {f x} \left (5 \left (b d-2 a e+2 c d x^2-b e x^2\right )-5 (b d-2 a e) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {1}{4};\frac {1}{2},\frac {1}{2};\frac {5}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )+(-2 c d+b e) x^2 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {5}{4};\frac {1}{2},\frac {1}{2};\frac {9}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )\right )}{5 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________