\[ \int \frac {\left (a+b x+c x^2\right )^4}{d+e x^3} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-6 a^{2} b c e -2 a \,b^{3} e +2 a \,c^{3} d +3 b^{2} c^{2} d \right ) x}{e^{2}}-\frac {\left (-6 a^{2} c^{2} e -12 a \,b^{2} c e -b^{4} e +4 b \,c^{3} d \right ) x^{2}}{2 e^{2}}-\frac {c \left (-12 a b c e -4 b^{3} e +c^{3} d \right ) x^{3}}{3 e^{2}}+\frac {c^{2} \left (2 a c +3 b^{2}\right ) x^{4}}{2 e}+\frac {4 b \,c^{3} x^{5}}{5 e}+\frac {c^{4} x^{6}}{6 e}+\frac {\left (e^{\frac {1}{3}} \left (a^{4} e^{2}-12 a^{2} b c d e -4 a \,b^{3} d e +4 a \,c^{3} d^{2}+6 b^{2} c^{2} d^{2}\right )+d^{\frac {1}{3}} \left (b^{4} d e +12 a \,b^{2} c d e +6 a^{2} c^{2} d e -4 b \left (a^{3} e^{2}+c^{3} d^{2}\right )\right )\right ) \ln \left (d^{\frac {1}{3}}+e^{\frac {1}{3}} x \right )}{3 d^{\frac {2}{3}} e^{\frac {8}{3}}}-\frac {\left (e^{\frac {1}{3}} \left (a^{4} e^{2}-12 a^{2} b c d e -4 a \,b^{3} d e +4 a \,c^{3} d^{2}+6 b^{2} c^{2} d^{2}\right )+d^{\frac {1}{3}} \left (b^{4} d e +12 a \,b^{2} c d e +6 a^{2} c^{2} d e -4 b \left (a^{3} e^{2}+c^{3} d^{2}\right )\right )\right ) \ln \left (d^{\frac {2}{3}}-d^{\frac {1}{3}} e^{\frac {1}{3}} x +e^{\frac {2}{3}} x^{2}\right )}{6 d^{\frac {2}{3}} e^{\frac {8}{3}}}+\frac {\left (c^{4} d^{2}-12 a b \,c^{2} d e +6 a^{2} b^{2} e^{2}-4 c e \left (-a^{3} e +b^{3} d \right )\right ) \ln \left (e \,x^{3}+d \right )}{3 e^{3}}-\frac {\left (b \,d^{\frac {1}{3}}+a \,e^{\frac {1}{3}}\right ) \left (4 c^{3} d^{2}+6 c^{2} \left (b \,d^{\frac {5}{3}} e^{\frac {1}{3}}-a \,d^{\frac {4}{3}} e^{\frac {2}{3}}\right )-12 a b c d e -e \left (b^{3} d +3 a \,b^{2} d^{\frac {2}{3}} e^{\frac {1}{3}}-3 a^{2} b \,d^{\frac {1}{3}} e^{\frac {2}{3}}-a^{3} e \right )\right ) \arctan \left (\frac {\left (d^{\frac {1}{3}}-2 e^{\frac {1}{3}} x \right ) \sqrt {3}}{3 d^{\frac {1}{3}}}\right ) \sqrt {3}}{3 d^{\frac {2}{3}} e^{\frac {8}{3}}} \]
command
integrate((c*x^2+b*x+a)^4/(e*x^3+d),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 \[ \text {Timed out} \]