\[ \int \frac {\sqrt {x}}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {d \left (a d +b c \right ) x^{\frac {3}{2}}}{2 a c \left (-a d +b c \right )^{2} \left (d \,x^{2}+c \right )}+\frac {b \,x^{\frac {3}{2}}}{2 a \left (-a d +b c \right ) \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}-\frac {b^{\frac {5}{4}} \left (-9 a d +b c \right ) \arctan \left (1-\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{8 a^{\frac {5}{4}} \left (-a d +b c \right )^{3}}+\frac {b^{\frac {5}{4}} \left (-9 a d +b c \right ) \arctan \left (1+\frac {b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{a^{\frac {1}{4}}}\right ) \sqrt {2}}{8 a^{\frac {5}{4}} \left (-a d +b c \right )^{3}}-\frac {d^{\frac {5}{4}} \left (-a d +9 b c \right ) \arctan \left (1-\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{8 c^{\frac {5}{4}} \left (-a d +b c \right )^{3}}+\frac {d^{\frac {5}{4}} \left (-a d +9 b c \right ) \arctan \left (1+\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{8 c^{\frac {5}{4}} \left (-a d +b c \right )^{3}}+\frac {b^{\frac {5}{4}} \left (-9 a d +b c \right ) \ln \left (\sqrt {a}+x \sqrt {b}-a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 a^{\frac {5}{4}} \left (-a d +b c \right )^{3}}-\frac {b^{\frac {5}{4}} \left (-9 a d +b c \right ) \ln \left (\sqrt {a}+x \sqrt {b}+a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 a^{\frac {5}{4}} \left (-a d +b c \right )^{3}}+\frac {d^{\frac {5}{4}} \left (-a d +9 b c \right ) \ln \left (\sqrt {c}+x \sqrt {d}-c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 c^{\frac {5}{4}} \left (-a d +b c \right )^{3}}-\frac {d^{\frac {5}{4}} \left (-a d +9 b c \right ) \ln \left (\sqrt {c}+x \sqrt {d}+c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{16 c^{\frac {5}{4}} \left (-a d +b c \right )^{3}} \]
command
integrate(x^(1/2)/(b*x^2+a)^2/(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 \[ \text {Timed out} \]