\[ \int \frac {x^4 \left (d+e x^4\right )}{a+b x^4+c x^8} \, dx \]
Optimal antiderivative \[ \frac {e x}{c}-\frac {\arctan \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} x}{\left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (c d -b e +\frac {2 a c e -b^{2} e +b c d}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {3}{4}}}{4 c^{\frac {5}{4}} \left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}}-\frac {\arctanh \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} x}{\left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (c d -b e +\frac {2 a c e -b^{2} e +b c d}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {3}{4}}}{4 c^{\frac {5}{4}} \left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}}-\frac {\arctan \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} x}{\left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (c d -b e +\frac {-2 a c e +b^{2} e -b c d}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {3}{4}}}{4 c^{\frac {5}{4}} \left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}}-\frac {\arctanh \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} x}{\left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (c d -b e +\frac {-2 a c e +b^{2} e -b c d}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {3}{4}}}{4 c^{\frac {5}{4}} \left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}} \]
command
integrate(x^4*(e*x^4+d)/(c*x^8+b*x^4+a),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 \[ \text {Timed out} \]