\[ \int \frac {x^4 \left (c+d x^2+e x^4+f x^6\right )}{\sqrt {a+b x^2}} \, dx \]
Optimal antiderivative \[ \frac {a^{2} \left (-63 a^{3} f +70 a^{2} b e -80 a \,b^{2} d +96 b^{3} c \right ) \arctanh \! \left (\frac {x \sqrt {b}}{\sqrt {b \,x^{2}+a}}\right )}{256 b^{\frac {11}{2}}}-\frac {a \left (-63 a^{3} f +70 a^{2} b e -80 a \,b^{2} d +96 b^{3} c \right ) x \sqrt {b \,x^{2}+a}}{256 b^{5}}+\frac {\left (-63 a^{3} f +70 a^{2} b e -80 a \,b^{2} d +96 b^{3} c \right ) x^{3} \sqrt {b \,x^{2}+a}}{384 b^{4}}+\frac {\left (63 a^{2} f -70 a b e +80 b^{2} d \right ) x^{5} \sqrt {b \,x^{2}+a}}{480 b^{3}}+\frac {\left (-9 a f +10 b e \right ) x^{7} \sqrt {b \,x^{2}+a}}{80 b^{2}}+\frac {f \,x^{9} \sqrt {b \,x^{2}+a}}{10 b} \]
command
integrate(x**4*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {63 a^{\frac {9}{2}} f x}{256 b^{5} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {35 a^{\frac {7}{2}} e x}{128 b^{4} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {21 a^{\frac {7}{2}} f x^{3}}{256 b^{4} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 a^{\frac {5}{2}} d x}{16 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {35 a^{\frac {5}{2}} e x^{3}}{384 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {21 a^{\frac {5}{2}} f x^{5}}{640 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{\frac {3}{2}} c x}{8 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 a^{\frac {3}{2}} d x^{3}}{48 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {7 a^{\frac {3}{2}} e x^{5}}{192 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{\frac {3}{2}} f x^{7}}{160 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {\sqrt {a} c x^{3}}{8 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {\sqrt {a} d x^{5}}{24 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {\sqrt {a} e x^{7}}{48 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {\sqrt {a} f x^{9}}{80 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {63 a^{5} f \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{256 b^{\frac {11}{2}}} + \frac {35 a^{4} e \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{128 b^{\frac {9}{2}}} - \frac {5 a^{3} d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 b^{\frac {7}{2}}} + \frac {3 a^{2} c \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} + \frac {c x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {d x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {e x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {f x^{11}}{10 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]