\[ \int \frac {c+d x+e x^2+f x^3}{\left (a-b x^4\right )^3} \, dx \]
Optimal antiderivative \[ \frac {x \left (5 e \,x^{2}+6 d x +7 c \right )}{32 a^{2} \left (-b \,x^{4}+a \right )}+\frac {a f +b x \left (e \,x^{2}+d x +c \right )}{8 a b \left (-b \,x^{4}+a \right )^{2}}+\frac {3 d \arctanh \! \left (\frac {x^{2} \sqrt {b}}{\sqrt {a}}\right )}{16 a^{\frac {5}{2}} \sqrt {b}}+\frac {\arctan \! \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right ) \left (-5 e \sqrt {a}+21 c \sqrt {b}\right )}{64 a^{\frac {11}{4}} b^{\frac {3}{4}}}+\frac {\arctanh \! \left (\frac {b^{\frac {1}{4}} x}{a^{\frac {1}{4}}}\right ) \left (5 e \sqrt {a}+21 c \sqrt {b}\right )}{64 a^{\frac {11}{4}} b^{\frac {3}{4}}} \]
command
integrate((f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ - \operatorname {RootSum} {\left (268435456 t^{4} a^{11} b^{3} + t^{2} \left (- 6881280 a^{6} b^{2} c e - 4718592 a^{6} b^{2} d^{2}\right ) + t \left (- 153600 a^{4} b d e^{2} - 2709504 a^{3} b^{2} c^{2} d\right ) - 625 a^{2} e^{4} + 22050 a b c^{2} e^{2} - 60480 a b c d^{2} e + 20736 a b d^{4} - 194481 b^{2} c^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 262144000 t^{3} a^{10} b^{2} e^{3} - 4624220160 t^{3} a^{9} b^{3} c^{2} e + 12683575296 t^{3} a^{9} b^{3} c d^{2} + 309657600 t^{2} a^{7} b^{2} c d e^{2} - 283115520 t^{2} a^{7} b^{2} d^{3} e - 1820786688 t^{2} a^{6} b^{3} c^{3} d + 5040000 t a^{5} b c e^{4} + 6912000 t a^{5} b d^{2} e^{3} + 118540800 t a^{4} b^{2} c^{3} e^{2} - 365783040 t a^{4} b^{2} c^{2} d^{2} e - 111476736 t a^{4} b^{2} c d^{4} + 522764928 t a^{3} b^{3} c^{5} + 112500 a^{3} d e^{5} - 4536000 a^{2} b c d^{3} e^{2} + 2488320 a^{2} b d^{5} e + 58344300 a b^{2} c^{4} d e - 80015040 a b^{2} c^{3} d^{3}}{15625 a^{3} e^{6} + 275625 a^{2} b c^{2} e^{4} - 3024000 a^{2} b c d^{2} e^{3} + 2073600 a^{2} b d^{4} e^{2} - 4862025 a b^{2} c^{4} e^{2} + 53343360 a b^{2} c^{3} d^{2} e - 36578304 a b^{2} c^{2} d^{4} - 85766121 b^{3} c^{6}} \right )} \right )\right )} - \frac {- 4 a^{2} f - 11 a b c x - 10 a b d x^{2} - 9 a b e x^{3} + 7 b^{2} c x^{5} + 6 b^{2} d x^{6} + 5 b^{2} e x^{7}}{32 a^{4} b - 64 a^{3} b^{2} x^{4} + 32 a^{2} b^{3} x^{8}} \]