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