\[ \int \left (d+e x+f x^2+g x^3\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {\left (-b g +2 c e \right ) \left (2 c \,x^{2}+b \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{32 c^{2}}+\frac {x \left (7 c f \,x^{2}+3 b f +9 c d \right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{63 c}+\frac {g \left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {5}{2}}}{10 c}+\frac {3 \left (-4 a c +b^{2}\right )^{2} \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 )}{512 c^{\frac {7}{2}}}-\frac {3 \left (-4 a c +b^{2}\right ) \left (-b g +2 c e \right ) \left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{256 c^{3}}+\frac {x \left (9 b^{2} c d +90 a \,c^{2} d -4 b^{3} f +9 a b c f +3 c \left (14 a c f -4 b^{2} f +9 b c d \right ) x^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{315 c^{2}}-\frac {\left (-84 a^{2} c^{2} f +57 a \,b^{2} c f -144 a b \,c^{2} d -8 b^{4} f +18 b^{3} c d \right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{315 c^{\frac {5}{2}} \left (\sqrt {a}+x^{2} \sqrt {c}\right )}+\frac {a^{\frac {1}{4}} \left (-84 a^{2} c^{2} f +57 a \,b^{2} c f -144 a b \,c^{2} d -8 b^{4} f +18 b^{3} 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}}}}{315 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{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 (18 b^{3} c d -144 a b \,c^{2} d -8 b^{4} f +57 a \,b^{2} c f -84 a^{2} c^{2} f +\left (24 a b c f -180 a \,c^{2} d -4 b^{3} f +9 b^{2} c d \right ) \sqrt {a}\, \sqrt {c}\right ) \sqrt {\frac {c \,x^{4}+b \,x^{2}+a}{\left (\sqrt {a}+x^{2} \sqrt {c}\right )^{2}}}}{630 \cos \left (2 \arctan \left (\frac {c^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right )\right ) c^{\frac {11}{4}} \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
\[ \text {Result too large to show} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________