\[ \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a+b x^4\right )^4} \, dx \]
Optimal antiderivative \[ \frac {x \left (b f \,x^{3}+b e \,x^{2}+b d x -a g +b c \right )}{12 a b \left (b \,x^{4}+a \right )^{3}}+\frac {x \left (45 b e \,x^{2}+60 b d x +7 a g +77 b c \right )}{384 a^{3} b \left (b \,x^{4}+a \right )}+\frac {-8 a f +x \left (9 b e \,x^{2}+10 b d x +a g +11 b c \right )}{96 a^{2} b \left (b \,x^{4}+a \right )^{2}}+\frac {5 d \arctan \left (\frac {x^{2} \sqrt {b}}{\sqrt {a}}\right )}{32 a^{\frac {7}{2}} \sqrt {b}}-\frac {\ln \left (-a^{\frac {1}{4}} b^{\frac {1}{4}} x \sqrt {2}+\sqrt {a}+x^{2} \sqrt {b}\right ) \left (77 b c +7 a g -15 e \sqrt {a}\, \sqrt {b}\right ) \sqrt {2}}{1024 a^{\frac {15}{4}} b^{\frac {5}{4}}}+\frac {\ln \left (a^{\frac {1}{4}} b^{\frac {1}{4}} x \sqrt {2}+\sqrt {a}+x^{2} \sqrt {b}\right ) \left (77 b c +7 a g -15 e \sqrt {a}\, \sqrt {b}\right ) \sqrt {2}}{1024 a^{\frac {15}{4}} b^{\frac {5}{4}}}+\frac {\arctan \left (-1+\frac {b^{\frac {1}{4}} x \sqrt {2}}{a^{\frac {1}{4}}}\right ) \left (77 b c +7 a g +15 e \sqrt {a}\, \sqrt {b}\right ) \sqrt {2}}{512 a^{\frac {15}{4}} b^{\frac {5}{4}}}+\frac {\arctan \left (1+\frac {b^{\frac {1}{4}} x \sqrt {2}}{a^{\frac {1}{4}}}\right ) \left (77 b c +7 a g +15 e \sqrt {a}\, \sqrt {b}\right ) \sqrt {2}}{512 a^{\frac {15}{4}} b^{\frac {5}{4}}} \]
command
integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^4+a)^4,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 \[ \text {Timed out} \]