\[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {x \left (b^{2} d -2 a c d -a b f +c \left (-2 a f +b d \right ) x^{2}\right )}{a \left (-4 a c +b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {-b e +2 a g -\left (-b g +2 c e \right ) x^{2}}{\left (-4 a c +b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (-2 a f +b d \right ) x \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{a \left (-4 a c +b^{2}\right ) \left (\sqrt {a}+x^{2} \sqrt {c}\right )}+\frac {c^{\frac {1}{4}} \left (-2 a f +b d \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2-\frac {b}{\sqrt {a}\, \sqrt {c}}}}{2}\right ) \left (\sqrt {a}+x^{2} \sqrt {c}\right ) \sqrt {\frac {c \,x^{4}+b \,x^{2}+a}{\left (\sqrt {a}+x^{2} \sqrt {c}\right )^{2}}}}{\cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {3}{4}} \left (-4 a c +b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2-\frac {b}{\sqrt {a}\, \sqrt {c}}}}{2}\right ) \left (-f \sqrt {a}+d \sqrt {c}\right ) \left (\sqrt {a}+x^{2} \sqrt {c}\right ) \sqrt {\frac {c \,x^{4}+b \,x^{2}+a}{\left (\sqrt {a}+x^{2} \sqrt {c}\right )^{2}}}}{2 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {3}{4}} c^{\frac {1}{4}} \left (b -2 \sqrt {a}\, \sqrt {c}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}} \]
command
Integrate[(d + e*x + f*x^2 + g*x^3)/(a + b*x^2 + c*x^4)^(3/2),x]
Mathematica 13.1 output
\[ -\frac {4 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (-2 a^2 g-b d x \left (b+c x^2\right )+2 a c x (d+x (e+f x))+a b (e+x (f-g x))\right )+i \left (-b+\sqrt {b^2-4 a c}\right ) (b d-2 a f) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (-b^2 d+4 a c d+b \sqrt {b^2-4 a c} d-2 a \sqrt {b^2-4 a c} f\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{4 a \left (b^2-4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________