\[ \int \frac {c+d x+e x^2}{\left (a-b x^4\right )^4} \, dx \]
Optimal antiderivative \[ \frac {x \left (e \,x^{2}+d x +c \right )}{12 a \left (-b \,x^{4}+a \right )^{3}}+\frac {x \left (9 e \,x^{2}+10 d x +11 c \right )}{96 a^{2} \left (-b \,x^{4}+a \right )^{2}}+\frac {x \left (45 e \,x^{2}+60 d x +77 c \right )}{384 a^{3} \left (-b \,x^{4}+a \right )}+\frac {5 d \arctanh \! \left (\frac {x^{2} \sqrt {b}}{\sqrt {a}}\right )}{32 a^{\frac {7}{2}} \sqrt {b}}+\frac {\arctan \! \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right ) \left (-15 e \sqrt {a}+77 c \sqrt {b}\right )}{256 a^{\frac {15}{4}} b^{\frac {3}{4}}}+\frac {\arctanh \! \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right ) \left (15 e \sqrt {a}+77 c \sqrt {b}\right )}{256 a^{\frac {15}{4}} b^{\frac {3}{4}}} \]
command
integrate((e*x**2+d*x+c)/(-b*x**4+a)**4,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \operatorname {RootSum} {\left (68719476736 t^{4} a^{15} b^{3} + t^{2} \left (- 1211105280 a^{8} b^{2} c e - 838860800 a^{8} b^{2} d^{2}\right ) + t \left (18432000 a^{5} b d e^{2} + 485703680 a^{4} b^{2} c^{2} d\right ) - 50625 a^{2} e^{4} + 2668050 a b c^{2} e^{2} - 7392000 a b c d^{2} e + 2560000 a b d^{4} - 35153041 b^{2} c^{4}, \left ( t \mapsto t \log {\left (x + \frac {452984832000 t^{3} a^{13} b^{2} e^{3} + 11936653639680 t^{3} a^{12} b^{3} c^{2} e - 33071248179200 t^{3} a^{12} b^{3} c d^{2} + 544997376000 t^{2} a^{9} b^{2} c d e^{2} - 503316480000 t^{2} a^{9} b^{2} d^{3} e - 4787095470080 t^{2} a^{8} b^{3} c^{3} d - 5987520000 t a^{6} b c e^{4} - 8294400000 t a^{6} b d^{2} e^{3} - 210370406400 t a^{5} b^{2} c^{3} e^{2} + 655699968000 t a^{5} b^{2} c^{2} d^{2} e + 201850880000 t a^{5} b^{2} c d^{4} - 1385873488384 t a^{4} b^{3} c^{5} + 91125000 a^{3} d e^{5} - 5544000000 a^{2} b c d^{3} e^{2} + 3072000000 a^{2} b d^{5} e + 105459123000 a b^{2} c^{4} d e - 146090560000 a b^{2} c^{3} d^{3}}{11390625 a^{3} e^{6} + 300155625 a^{2} b c^{2} e^{4} - 3326400000 a^{2} b c d^{2} e^{3} + 2304000000 a^{2} b d^{4} e^{2} - 7909434225 a b^{2} c^{4} e^{2} + 87654336000 a b^{2} c^{3} d^{2} e - 60712960000 a b^{2} c^{2} d^{4} - 208422380089 b^{3} c^{6}} \right )} \right )\right )} + \frac {- 153 a^{2} c x - 132 a^{2} d x^{2} - 113 a^{2} e x^{3} + 198 a b c x^{5} + 160 a b d x^{6} + 126 a b e x^{7} - 77 b^{2} c x^{9} - 60 b^{2} d x^{10} - 45 b^{2} e x^{11}}{- 384 a^{6} + 1152 a^{5} b x^{4} - 1152 a^{4} b^{2} x^{8} + 384 a^{3} b^{3} x^{12}} \]