\[ \int \frac {\left (a+b x^2\right ) \left (c+d x^2\right )^2}{\left (e+f x^2\right )^4} \, dx \]
Optimal antiderivative \[ -\frac {\left (-a f +b e \right ) x \left (d \,x^{2}+c \right )^{2}}{6 e f \left (f \,x^{2}+e \right )^{3}}-\frac {\left (d e \left (a f +5 b e \right )-c f \left (5 a f +b e \right )\right ) x \left (d \,x^{2}+c \right )}{24 e^{2} f^{2} \left (f \,x^{2}+e \right )^{2}}-\frac {\left (a f \left (-15 c^{2} f^{2}+4 c d e f +3 d^{2} e^{2}\right )+b e \left (-3 c^{2} f^{2}-4 c d e f +15 d^{2} e^{2}\right )\right ) x}{48 e^{3} f^{3} \left (f \,x^{2}+e \right )}+\frac {\left (b e \left (c^{2} f^{2}+2 c d e f +5 d^{2} e^{2}\right )+a f \left (5 c^{2} f^{2}+2 c d e f +d^{2} e^{2}\right )\right ) \arctan \! \left (\frac {x \sqrt {f}}{\sqrt {e}}\right )}{16 e^{\frac {7}{2}} f^{\frac {7}{2}}} \]
command
integrate((b*x**2+a)*(d*x**2+c)**2/(f*x**2+e)**4,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ - \frac {\sqrt {- \frac {1}{e^{7} f^{7}}} \left (5 a c^{2} f^{3} + 2 a c d e f^{2} + a d^{2} e^{2} f + b c^{2} e f^{2} + 2 b c d e^{2} f + 5 b d^{2} e^{3}\right ) \log {\left (- e^{4} f^{3} \sqrt {- \frac {1}{e^{7} f^{7}}} + x \right )}}{32} + \frac {\sqrt {- \frac {1}{e^{7} f^{7}}} \left (5 a c^{2} f^{3} + 2 a c d e f^{2} + a d^{2} e^{2} f + b c^{2} e f^{2} + 2 b c d e^{2} f + 5 b d^{2} e^{3}\right ) \log {\left (e^{4} f^{3} \sqrt {- \frac {1}{e^{7} f^{7}}} + x \right )}}{32} + \frac {x^{5} \left (15 a c^{2} f^{5} + 6 a c d e f^{4} + 3 a d^{2} e^{2} f^{3} + 3 b c^{2} e f^{4} + 6 b c d e^{2} f^{3} - 33 b d^{2} e^{3} f^{2}\right ) + x^{3} \left (40 a c^{2} e f^{4} + 16 a c d e^{2} f^{3} - 8 a d^{2} e^{3} f^{2} + 8 b c^{2} e^{2} f^{3} - 16 b c d e^{3} f^{2} - 40 b d^{2} e^{4} f\right ) + x \left (33 a c^{2} e^{2} f^{3} - 6 a c d e^{3} f^{2} - 3 a d^{2} e^{4} f - 3 b c^{2} e^{3} f^{2} - 6 b c d e^{4} f - 15 b d^{2} e^{5}\right )}{48 e^{6} f^{3} + 144 e^{5} f^{4} x^{2} + 144 e^{4} f^{5} x^{4} + 48 e^{3} f^{6} x^{6}} \]