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