\[ \int \frac {1}{\sqrt {x} \left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx \]
Optimal antiderivative \[ -\frac {b^{\frac {11}{4}} \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{2 a^{\frac {3}{4}} \left (-a d +b c \right )^{3}}+\frac {b^{\frac {11}{4}} \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{2 a^{\frac {3}{4}} \left (-a d +b c \right )^{3}}+\frac {d^{\frac {3}{4}} \left (21 a^{2} d^{2}-66 a b c d +77 b^{2} c^{2}\right ) \arctan \left (1-\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{64 c^{\frac {11}{4}} \left (-a d +b c \right )^{3}}-\frac {d^{\frac {3}{4}} \left (21 a^{2} d^{2}-66 a b c d +77 b^{2} c^{2}\right ) \arctan \left (1+\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{64 c^{\frac {11}{4}} \left (-a d +b c \right )^{3}}-\frac {b^{\frac {11}{4}} \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 a^{\frac {3}{4}} \left (-a d +b c \right )^{3}}+\frac {b^{\frac {11}{4}} \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{4 a^{\frac {3}{4}} \left (-a d +b c \right )^{3}}+\frac {d^{\frac {3}{4}} \left (21 a^{2} d^{2}-66 a b c d +77 b^{2} c^{2}\right ) \ln \left (\sqrt {c}+x \sqrt {d}-c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{128 c^{\frac {11}{4}} \left (-a d +b c \right )^{3}}-\frac {d^{\frac {3}{4}} \left (21 a^{2} d^{2}-66 a b c d +77 b^{2} c^{2}\right ) \ln \left (\sqrt {c}+x \sqrt {d}+c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{128 c^{\frac {11}{4}} \left (-a d +b c \right )^{3}}-\frac {d \sqrt {x}}{4 c \left (-a d +b c \right ) \left (d \,x^{2}+c \right )^{2}}-\frac {d \left (-7 a d +15 b c \right ) \sqrt {x}}{16 c^{2} \left (-a d +b c \right )^{2} \left (d \,x^{2}+c \right )} \]
command
integrate(1/(b*x^2+a)/(d*x^2+c)^3/x^(1/2),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 \[ \text {Timed out} \]