\[ \int \frac {x^{15/2}}{\left (a+b x^2+c x^4\right )^3} \, dx \]
Optimal antiderivative \[ \frac {x^{\frac {9}{2}} \left (b \,x^{2}+2 a \right )}{4 \left (-4 a c +b^{2}\right ) \left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {3 x^{\frac {5}{2}} \left (8 a b +\left (12 a c +b^{2}\right ) x^{2}\right )}{16 \left (-4 a c +b^{2}\right )^{2} \left (c \,x^{4}+b \,x^{2}+a \right )}-\frac {3 \arctan \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x}}{\left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (b^{3}-28 a b c +\frac {-24 a^{2} c^{2}-30 a \,b^{2} c +b^{4}}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {3}{4}}}{64 c^{\frac {5}{4}} \left (-4 a c +b^{2}\right )^{2} \left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}}-\frac {3 \arctanh \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x}}{\left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (b^{3}-28 a b c +\frac {-24 a^{2} c^{2}-30 a \,b^{2} c +b^{4}}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {3}{4}}}{64 c^{\frac {5}{4}} \left (-4 a c +b^{2}\right )^{2} \left (-b -\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}}-\frac {3 \arctan \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x}}{\left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (b^{3}-28 a b c +\frac {24 a^{2} c^{2}+30 a \,b^{2} c -b^{4}}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {3}{4}}}{64 c^{\frac {5}{4}} \left (-4 a c +b^{2}\right )^{2} \left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}}-\frac {3 \arctanh \left (\frac {2^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x}}{\left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {1}{4}}}\right ) \left (b^{3}-28 a b c +\frac {24 a^{2} c^{2}+30 a \,b^{2} c -b^{4}}{\sqrt {-4 a c +b^{2}}}\right ) 2^{\frac {3}{4}}}{64 c^{\frac {5}{4}} \left (-4 a c +b^{2}\right )^{2} \left (-b +\sqrt {-4 a c +b^{2}}\right )^{\frac {3}{4}}}-\frac {3 \left (12 a c +b^{2}\right ) \sqrt {x}}{16 c \left (-4 a c +b^{2}\right )^{2}} \]
command
integrate(x^(15/2)/(c*x^4+b*x^2+a)^3,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 \[ \text {Timed out} \]