\[ \int \frac {x^{3/2}}{\left (a+b x^2+c x^4\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {3 c^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x}}{\left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (b^{2}+44 a c -\frac {b^{3}}{\sqrt {-4 a c +b^{2}}}+\frac {68 a b c}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {3}{4}}}{64 a \left (-4 a c +b^{2}\right )^{2} \left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}}-\frac {3 c^{\frac {3}{4}} \arctanh \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x}}{\left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (b^{2}+44 a c -\frac {b^{3}}{\sqrt {-4 a c +b^{2}}}+\frac {68 a b c}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {3}{4}}}{64 a \left (-4 a c +b^{2}\right )^{2} \left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}}-\frac {3 c^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x}}{\left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (b^{3}-68 a b c +\left (44 a c +b^{2}\right ) \sqrt {-4 a c +b^{2}}\right ) 2^{\frac {3}{4}}}{64 a \left (-4 a c +b^{2}\right )^{\frac {5}{2}} \left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}}-\frac {3 c^{\frac {3}{4}} \arctanh \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x}}{\left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (b^{3}-68 a b c +\left (44 a c +b^{2}\right ) \sqrt {-4 a c +b^{2}}\right ) 2^{\frac {3}{4}}}{64 a \left (-4 a c +b^{2}\right )^{\frac {5}{2}} \left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}}-\frac {\left (2 c \,x^{2}+b \right ) \sqrt {x}}{4 \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\left (b \left (20 a c +b^{2}\right )+c \left (44 a c +b^{2}\right ) x^{2}\right ) \sqrt {x}}{16 a \left (-4 a c +b^{2}\right )^{2} \left (c \,x^{4}+b \,x^{2}+a \right )} \]
command
integrate(x^(3/2)/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]