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