\[ \int \left (d+e x+f x^2+g x^3\right ) \sqrt {a+b x^2+c x^4} \, dx \]
Optimal antiderivative \[ \frac {g \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{6 c}-\frac {\left (-4 a c +b^{2}\right ) \left (-b g +2 c e \right ) \arctanh \left (\frac {2 c \,x^{2}+b}{2 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{32 c^{\frac {5}{2}}}+\frac {\left (-b g +2 c e \right ) \left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 c^{2}}+\frac {x \left (3 c f \,x^{2}+b f +5 c d \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 c}+\frac {\left (6 a c f -2 b^{2} f +5 b c d \right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{15 c^{\frac {3}{2}} \left (\sqrt {a}+x^{2} \sqrt {c}\right )}-\frac {a^{\frac {1}{4}} \left (6 a c f -2 b^{2} f +5 b c d \right ) \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2-\frac {b}{\sqrt {a}\, \sqrt {c}}}}{2}\right ) \left (\sqrt {a}+x^{2} \sqrt {c}\right ) \sqrt {\frac {c \,x^{4}+b \,x^{2}+a}{\left (\sqrt {a}+x^{2} \sqrt {c}\right )^{2}}}}{15 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {7}{4}} \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {a^{\frac {1}{4}} \sqrt {\frac {\cos \left (4 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ), \frac {\sqrt {2-\frac {b}{\sqrt {a}\, \sqrt {c}}}}{2}\right ) \left (\sqrt {a}+x^{2} \sqrt {c}\right ) \left (b +2 \sqrt {a}\, \sqrt {c}\right ) \left (5 c d -2 b f +3 f \sqrt {a}\, \sqrt {c}\right ) \sqrt {\frac {c \,x^{4}+b \,x^{2}+a}{\left (\sqrt {a}+x^{2} \sqrt {c}\right )^{2}}}}{30 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {7}{4}} \sqrt {c \,x^{4}+b \,x^{2}+a}} \]
command
Integrate[(d + e*x + f*x^2 + g*x^3)*Sqrt[a + b*x^2 + c*x^4],x]
Mathematica 13.1 output
\[ \frac {2 \sqrt {c} \left (a+b x^2+c x^4\right ) \left (-15 b^2 g+2 b c (15 e+x (8 f+5 g x))+4 c (10 a g+c x (20 d+x (15 e+2 x (6 f+5 g x))))\right )+\frac {-8 i \sqrt {2} \sqrt {c} \left (-b+\sqrt {b^2-4 a c}\right ) \left (-5 b c d+2 b^2 f-6 a c f\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+8 i \sqrt {2} \sqrt {c} \left (-2 b^3 f+b c \left (-5 \sqrt {b^2-4 a c} d+8 a f\right )+b^2 \left (5 c d+2 \sqrt {b^2-4 a c} f\right )-2 a c \left (10 c d+3 \sqrt {b^2-4 a c} f\right )\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-15 \left (b^2-4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} (-2 c e+b g) \sqrt {a+b x^2+c x^4} \log \left (b+2 c x^2-2 \sqrt {c} \sqrt {a+b x^2+c x^4}\right )}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}}{480 c^{5/2} \sqrt {a+b x^2+c x^4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________