\[ \int \frac {1}{\left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {\left (\frac {c}{d}+x \right ) \left (c^{3}-4 a \,d^{2}-c \,d^{2} \left (\frac {c}{d}+x \right )^{2}\right )}{16 a c \left (4 a \,d^{2}+c^{3}\right ) \left (d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c \right )}-\frac {d \arctanh \! \left (\frac {c \sqrt {2}+d x \sqrt {2}+c^{\frac {1}{4}} \sqrt {c^{\frac {3}{2}}+\sqrt {4 a \,d^{2}+c^{3}}}}{c^{\frac {1}{4}} \sqrt {c^{\frac {3}{2}}-\sqrt {4 a \,d^{2}+c^{3}}}}\right ) \left (c^{3}+12 a \,d^{2}+c^{\frac {3}{2}} \sqrt {4 a \,d^{2}+c^{3}}\right ) \sqrt {2}}{64 a \,c^{\frac {7}{4}} \left (4 a \,d^{2}+c^{3}\right )^{\frac {3}{2}} \sqrt {c^{\frac {3}{2}}-\sqrt {4 a \,d^{2}+c^{3}}}}+\frac {d \arctanh \! \left (\frac {-\left (d x +c \right ) \sqrt {2}+c^{\frac {1}{4}} \sqrt {c^{\frac {3}{2}}+\sqrt {4 a \,d^{2}+c^{3}}}}{c^{\frac {1}{4}} \sqrt {c^{\frac {3}{2}}-\sqrt {4 a \,d^{2}+c^{3}}}}\right ) \left (c^{3}+12 a \,d^{2}+c^{\frac {3}{2}} \sqrt {4 a \,d^{2}+c^{3}}\right ) \sqrt {2}}{64 a \,c^{\frac {7}{4}} \left (4 a \,d^{2}+c^{3}\right )^{\frac {3}{2}} \sqrt {c^{\frac {3}{2}}-\sqrt {4 a \,d^{2}+c^{3}}}}-\frac {d \ln \! \left (d^{2} \left (\frac {c}{d}+x \right )^{2}+\sqrt {c}\, \sqrt {4 a \,d^{2}+c^{3}}-c^{\frac {1}{4}} d \left (\frac {c}{d}+x \right ) \sqrt {2}\, \sqrt {c^{\frac {3}{2}}+\sqrt {4 a \,d^{2}+c^{3}}}\right ) \left (c^{3}+12 a \,d^{2}-c^{\frac {3}{2}} \sqrt {4 a \,d^{2}+c^{3}}\right ) \sqrt {2}}{128 a \,c^{\frac {7}{4}} \left (4 a \,d^{2}+c^{3}\right )^{\frac {3}{2}} \sqrt {c^{\frac {3}{2}}+\sqrt {4 a \,d^{2}+c^{3}}}}+\frac {d \ln \! \left (d^{2} \left (\frac {c}{d}+x \right )^{2}+\sqrt {c}\, \sqrt {4 a \,d^{2}+c^{3}}+c^{\frac {1}{4}} d \left (\frac {c}{d}+x \right ) \sqrt {2}\, \sqrt {c^{\frac {3}{2}}+\sqrt {4 a \,d^{2}+c^{3}}}\right ) \left (c^{3}+12 a \,d^{2}-c^{\frac {3}{2}} \sqrt {4 a \,d^{2}+c^{3}}\right ) \sqrt {2}}{128 a \,c^{\frac {7}{4}} \left (4 a \,d^{2}+c^{3}\right )^{\frac {3}{2}} \sqrt {c^{\frac {3}{2}}+\sqrt {4 a \,d^{2}+c^{3}}}} \]
command
integrate(1/(d**2*x**4+4*c*d*x**3+4*c**2*x**2+4*a*c)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {4 a c d + 3 c^{2} d x^{2} + c d^{2} x^{3} + x \left (4 a d^{2} + 2 c^{3}\right )}{256 a^{3} c^{2} d^{2} + 64 a^{2} c^{5} + x^{4} \left (64 a^{2} c d^{4} + 16 a c^{4} d^{2}\right ) + x^{3} \left (256 a^{2} c^{2} d^{3} + 64 a c^{5} d\right ) + x^{2} \left (256 a^{2} c^{3} d^{2} + 64 a c^{6}\right )} + \operatorname {RootSum} {\left (t^{4} \left (1073741824 a^{9} c^{7} d^{6} + 805306368 a^{8} c^{10} d^{4} + 201326592 a^{7} c^{13} d^{2} + 16777216 a^{6} c^{16}\right ) + t^{2} \left (491520 a^{5} c^{5} d^{4} + 122880 a^{4} c^{8} d^{2} + 8192 a^{3} c^{11}\right ) + 81 a^{2} d^{4} + 18 a c^{3} d^{2} + c^{6}, \left ( t \mapsto t \log {\left (x + \frac {- 67108864 t^{3} a^{7} c^{7} d^{8} - 58720256 t^{3} a^{6} c^{10} d^{6} - 18874368 t^{3} a^{5} c^{13} d^{4} - 2621440 t^{3} a^{4} c^{16} d^{2} - 131072 t^{3} a^{3} c^{19} + 27648 t a^{4} c^{2} d^{8} - 9216 t a^{3} c^{5} d^{6} - 5440 t a^{2} c^{8} d^{4} - 736 t a c^{11} d^{2} - 32 t c^{14} + 324 a^{2} c d^{7} + 81 a c^{4} d^{5} + 5 c^{7} d^{3}}{324 a^{2} d^{8} + 81 a c^{3} d^{6} + 5 c^{6} d^{4}} \right )} \right )\right )} \]