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