\[ \int \frac {1}{x^{5/2} \left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {7 a d -4 b c}{6 a \,c^{2} \left (-a d +b c \right ) x^{\frac {3}{2}}}-\frac {d}{2 c \left (-a d +b c \right ) x^{\frac {3}{2}} \left (d \,x^{2}+c \right )}+\frac {b^{\frac {11}{4}} \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{2 a^{\frac {7}{4}} \left (-a d +b c \right )^{2}}-\frac {b^{\frac {11}{4}} \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{2 a^{\frac {7}{4}} \left (-a d +b c \right )^{2}}-\frac {d^{\frac {7}{4}} \left (-7 a d +11 b c \right ) \arctan \left (1-\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{8 c^{\frac {11}{4}} \left (-a d +b c \right )^{2}}+\frac {d^{\frac {7}{4}} \left (-7 a d +11 b c \right ) \arctan \left (1+\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{8 c^{\frac {11}{4}} \left (-a d +b c \right )^{2}}+\frac {b^{\frac {11}{4}} \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 a^{\frac {7}{4}} \left (-a d +b c \right )^{2}}-\frac {b^{\frac {11}{4}} \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 a^{\frac {7}{4}} \left (-a d +b c \right )^{2}}-\frac {d^{\frac {7}{4}} \left (-7 a d +11 b c \right ) \ln \left (\sqrt {c}+x \sqrt {d}-c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 c^{\frac {11}{4}} \left (-a d +b c \right )^{2}}+\frac {d^{\frac {7}{4}} \left (-7 a d +11 b c \right ) \ln \left (\sqrt {c}+x \sqrt {d}+c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 c^{\frac {11}{4}} \left (-a d +b c \right )^{2}} \]
command
integrate(1/x^(5/2)/(b*x^2+a)/(d*x^2+c)^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 \[ \text {Timed out} \]