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