\[ \int \frac {(d+e x)^3}{\left (a+c x^4\right )^3} \, dx \]
Optimal antiderivative \[ \frac {x \left (15 d \,e^{2} x^{2}+18 d^{2} e x +7 d^{3}\right )}{32 a^{2} \left (c \,x^{4}+a \right )}+\frac {-a \,e^{3}+c x \left (3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )}{8 a c \left (c \,x^{4}+a \right )^{2}}+\frac {9 d^{2} e \arctan \left (\frac {x^{2} \sqrt {c}}{\sqrt {a}}\right )}{16 a^{\frac {5}{2}} \sqrt {c}}-\frac {3 d \ln \left (-a^{\frac {1}{4}} c^{\frac {1}{4}} x \sqrt {2}+\sqrt {a}+x^{2} \sqrt {c}\right ) \left (-5 e^{2} \sqrt {a}+7 d^{2} \sqrt {c}\right ) \sqrt {2}}{256 a^{\frac {11}{4}} c^{\frac {3}{4}}}+\frac {3 d \ln \left (a^{\frac {1}{4}} c^{\frac {1}{4}} x \sqrt {2}+\sqrt {a}+x^{2} \sqrt {c}\right ) \left (-5 e^{2} \sqrt {a}+7 d^{2} \sqrt {c}\right ) \sqrt {2}}{256 a^{\frac {11}{4}} c^{\frac {3}{4}}}+\frac {3 d \arctan \left (-1+\frac {c^{\frac {1}{4}} x \sqrt {2}}{a^{\frac {1}{4}}}\right ) \left (5 e^{2} \sqrt {a}+7 d^{2} \sqrt {c}\right ) \sqrt {2}}{128 a^{\frac {11}{4}} c^{\frac {3}{4}}}+\frac {3 d \arctan \left (1+\frac {c^{\frac {1}{4}} x \sqrt {2}}{a^{\frac {1}{4}}}\right ) \left (5 e^{2} \sqrt {a}+7 d^{2} \sqrt {c}\right ) \sqrt {2}}{128 a^{\frac {11}{4}} c^{\frac {3}{4}}} \]
command
integrate((e*x+d)^3/(c*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 \[ \text {Timed out} \]