\[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx \]
Optimal antiderivative \[ \frac {-f \,x^{3}-e \,x^{2}-d x -c}{8 b \left (b \,x^{4}+a \right )^{2}}+\frac {x \left (3 f \,x^{2}+2 e x +d \right )}{32 a b \left (b \,x^{4}+a \right )}+\frac {e \arctan \left (\frac {x^{2} \sqrt {b}}{\sqrt {a}}\right )}{16 a^{\frac {3}{2}} b^{\frac {3}{2}}}-\frac {3 \ln \left (-a^{\frac {1}{4}} b^{\frac {1}{4}} x \sqrt {2}+\sqrt {a}+x^{2} \sqrt {b}\right ) \left (-f \sqrt {a}+d \sqrt {b}\right ) \sqrt {2}}{256 a^{\frac {7}{4}} b^{\frac {7}{4}}}+\frac {3 \ln \left (a^{\frac {1}{4}} b^{\frac {1}{4}} x \sqrt {2}+\sqrt {a}+x^{2} \sqrt {b}\right ) \left (-f \sqrt {a}+d \sqrt {b}\right ) \sqrt {2}}{256 a^{\frac {7}{4}} b^{\frac {7}{4}}}+\frac {3 \arctan \left (-1+\frac {b^{\frac {1}{4}} x \sqrt {2}}{a^{\frac {1}{4}}}\right ) \left (f \sqrt {a}+d \sqrt {b}\right ) \sqrt {2}}{128 a^{\frac {7}{4}} b^{\frac {7}{4}}}+\frac {3 \arctan \left (1+\frac {b^{\frac {1}{4}} x \sqrt {2}}{a^{\frac {1}{4}}}\right ) \left (f \sqrt {a}+d \sqrt {b}\right ) \sqrt {2}}{128 a^{\frac {7}{4}} b^{\frac {7}{4}}} \]
command
integrate(x^3*(f*x^3+e*x^2+d*x+c)/(b*x^4+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 \[ \text {Timed out} \]