\[ \int \frac {d+e x^2}{\sqrt {f x} \left (a+b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 e \left (f x \right )^{\frac {5}{2}} F_{1}\left (\frac {5}{4}, \frac {3}{2}, \frac {3}{2}, \frac {9}{4}, -\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}, -\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}\right ) \sqrt {1+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}}\, \sqrt {1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}}{5 a \,f^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {2 d F_{1}\left (\frac {1}{4}, \frac {3}{2}, \frac {3}{2}, \frac {5}{4}, -\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}, -\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}\right ) \sqrt {f x}\, \sqrt {1+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}}\, \sqrt {1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}}{a f \sqrt {c \,x^{4}+b \,x^{2}+a}} \]
command
Integrate[(d + e*x^2)/(Sqrt[f*x]*(a + b*x^2 + c*x^4)^(3/2)),x]
Mathematica 13.1 output
\[ \frac {x \left (-5 b^2 d+5 b \left (a e-c d x^2\right )+10 a c \left (d+e x^2\right )-5 \left (b^2 d-6 a c d+a b e\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {1}{4};\frac {1}{2},\frac {1}{2};\frac {5}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )+c (b d-2 a e) x^2 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {5}{4};\frac {1}{2},\frac {1}{2};\frac {9}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )\right )}{5 a \left (-b^2+4 a c\right ) \sqrt {f x} \sqrt {a+b x^2+c x^4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________