\[ \int \frac {\sqrt {x} \left (a+b x^2\right )^2}{\left (c+d x^2\right )^3} \, dx \]
Optimal antiderivative \[ \frac {\left (-a d +b c \right )^{2} x^{\frac {3}{2}}}{4 c \,d^{2} \left (d \,x^{2}+c \right )^{2}}-\frac {\left (-a d +b c \right ) \left (5 a d +11 b c \right ) x^{\frac {3}{2}}}{16 c^{2} d^{2} \left (d \,x^{2}+c \right )}-\frac {\left (5 a^{2} d^{2}+6 a b c d +21 b^{2} c^{2}\right ) \arctan \left (1-\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{64 c^{\frac {9}{4}} d^{\frac {11}{4}}}+\frac {\left (5 a^{2} d^{2}+6 a b c d +21 b^{2} c^{2}\right ) \arctan \left (1+\frac {d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}}{c^{\frac {1}{4}}}\right ) \sqrt {2}}{64 c^{\frac {9}{4}} d^{\frac {11}{4}}}+\frac {\left (5 a^{2} d^{2}+6 a b c d +21 b^{2} c^{2}\right ) \ln \left (\sqrt {c}+x \sqrt {d}-c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{128 c^{\frac {9}{4}} d^{\frac {11}{4}}}-\frac {\left (5 a^{2} d^{2}+6 a b c d +21 b^{2} c^{2}\right ) \ln \left (\sqrt {c}+x \sqrt {d}+c^{\frac {1}{4}} d^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}\right ) \sqrt {2}}{128 c^{\frac {9}{4}} d^{\frac {11}{4}}} \]
command
integrate((b*x**2+a)**2*x**(1/2)/(d*x**2+c)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {4 a b x^{\frac {3}{2}}}{4 c^{2} d + 4 c d^{2} x^{2}} + \frac {4 a b \operatorname {RootSum} {\left (65536 t^{4} c^{5} d^{3} + 1, \left ( t \mapsto t \log {\left (4096 t^{3} c^{4} d^{2} + \sqrt {x} \right )} \right )\right )}}{d} - \frac {4 b^{2} c x^{\frac {3}{2}}}{4 c^{2} d^{2} + 4 c d^{3} x^{2}} - \frac {4 b^{2} c \operatorname {RootSum} {\left (65536 t^{4} c^{5} d^{3} + 1, \left ( t \mapsto t \log {\left (4096 t^{3} c^{4} d^{2} + \sqrt {x} \right )} \right )\right )}}{d^{2}} + \frac {2 b^{2} \operatorname {RootSum} {\left (256 t^{4} c d^{3} + 1, \left ( t \mapsto t \log {\left (64 t^{3} c d^{2} + \sqrt {x} \right )} \right )\right )}}{d^{2}} + \frac {18 c x^{\frac {3}{2}} \left (a d - b c\right )^{2}}{32 c^{4} d^{2} + 64 c^{3} d^{3} x^{2} + 32 c^{2} d^{4} x^{4}} + \frac {10 x^{\frac {7}{2}} \left (a d - b c\right )^{2}}{32 c^{4} d + 64 c^{3} d^{2} x^{2} + 32 c^{2} d^{3} x^{4}} + \frac {2 \left (a d - b c\right )^{2} \operatorname {RootSum} {\left (268435456 t^{4} c^{9} d^{3} + 625, \left ( t \mapsto t \log {\left (\frac {2097152 t^{3} c^{7} d^{2}}{125} + \sqrt {x} \right )} \right )\right )}}{d^{2}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________