\[ \int \frac {x^{13/2}}{\left (a+b x^2+c x^4\right )^2} \, dx \]
Optimal antiderivative \[ -\frac {b \,x^{\frac {3}{2}}}{2 c \left (-4 a c +b^{2}\right )}+\frac {x^{\frac {7}{2}} \left (b \,x^{2}+2 a \right )}{2 \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )}-\frac {\arctan \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x}}{\left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (3 b^{3}-20 a b c -\left (-14 a c +3 b^{2}\right ) \sqrt {-4 a c +b^{2}}\right ) 2^{\frac {1}{4}}}{8 c^{\frac {7}{4}} \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}+\frac {\arctanh \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x}}{\left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (3 b^{3}-20 a b c -\left (-14 a c +3 b^{2}\right ) \sqrt {-4 a c +b^{2}}\right ) 2^{\frac {1}{4}}}{8 c^{\frac {7}{4}} \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}+\frac {\arctan \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x}}{\left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (3 b^{3}-20 a b c +\left (-14 a c +3 b^{2}\right ) \sqrt {-4 a c +b^{2}}\right ) 2^{\frac {1}{4}}}{8 c^{\frac {7}{4}} \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}-\frac {\arctanh \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x}}{\left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (3 b^{3}-20 a b c +\left (-14 a c +3 b^{2}\right ) \sqrt {-4 a c +b^{2}}\right ) 2^{\frac {1}{4}}}{8 c^{\frac {7}{4}} \left (-4 a c +b^{2}\right )^{\frac {3}{2}} \left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}} \]
command
integrate(x^(13/2)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]