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