\[ \int \frac {d+e x+f x^2+g x^3}{\left (a+b x^2+c x^4\right )^{5/2}} \, dx \]
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 )}{3 a \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-b e +2 a g -\left (-b g +2 c e \right ) x^{2}}{3 \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {4 \left (-b g +2 c e \right ) \left (2 c \,x^{2}+b \right )}{3 \left (-4 a c +b^{2}\right )^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {x \left (2 b^{4} d -17 a \,b^{2} c d +20 a^{2} c^{2} d +a \,b^{3} f +4 a^{2} b c f +c \left (12 a^{2} c f +a \,b^{2} f -16 a b c d +2 b^{3} d \right ) x^{2}\right )}{3 a^{2} \left (-4 a c +b^{2}\right )^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (12 a^{2} c f +a \,b^{2} f -16 a b c d +2 b^{3} d \right ) x \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 a^{2} \left (-4 a c +b^{2}\right )^{2} \left (\sqrt {a}+x^{2} \sqrt {c}\right )}+\frac {c^{\frac {1}{4}} \left (12 a^{2} c f +a \,b^{2} f -16 a b c d +2 b^{3} 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}}}}{3 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {7}{4}} \left (-4 a c +b^{2}\right )^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {c^{\frac {1}{4}} \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 (\sqrt {a}+x^{2} \sqrt {c}\right ) \left (2 b^{2} d -10 a c d +a b f +6 a^{\frac {3}{2}} f \sqrt {c}-3 b d \sqrt {a}\, \sqrt {c}\right ) \sqrt {\frac {c \,x^{4}+b \,x^{2}+a}{\left (\sqrt {a}+x^{2} \sqrt {c}\right )^{2}}}}{6 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) a^{\frac {7}{4}} \left (-4 a c +b^{2}\right ) \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)^(5/2),x]
Mathematica 13.1 output
\[ \frac {-4 a \left (b^2-4 a c\right ) \left (-2 a^2 g-b d x \left (b+c x^2\right )+2 a c x (d+x (e+f x))+a b (e+x (f-g x))\right )+4 \left (a+b x^2+c x^4\right ) \left (2 b^3 d x \left (b+c x^2\right )+a b x \left (-17 b c d+b^2 f-16 c^2 d x^2+b c f x^2\right )+4 a^2 \left (-b^2 g+c^2 x (5 d+x (4 e+3 f x))+b c (2 e+x (f-2 g x))\right )\right )+\frac {i \sqrt {2} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (a+b x^2+c x^4\right ) \left (-\left (\left (-b+\sqrt {b^2-4 a c}\right ) \left (2 b^3 d-16 a b c d+a b^2 f+12 a^2 c f\right ) E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )+\left (-2 b^4 d+b^3 \left (2 \sqrt {b^2-4 a c} d-a f\right )+4 a b c \left (-4 \sqrt {b^2-4 a c} d+a f\right )+a b^2 \left (18 c d+\sqrt {b^2-4 a c} f\right )+4 a^2 c \left (-10 c d+3 \sqrt {b^2-4 a c} f\right )\right ) F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}}{12 a^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )^{3/2}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________