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