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