\[ \int x^4 \left (a+b x^2\right )^2 \left (c+d x^2\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {\left (24 a^{2} d^{2}+b c \left (-24 a d +7 b c \right )\right ) x^{5} \left (d \,x^{2}+c \right )^{\frac {3}{2}}}{192 d^{2}}-\frac {b \left (-24 a d +7 b c \right ) x^{5} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{120 d^{2}}+\frac {b^{2} x^{7} \left (d \,x^{2}+c \right )^{\frac {5}{2}}}{12 d}+\frac {c^{4} \left (24 a^{2} d^{2}+b c \left (-24 a d +7 b c \right )\right ) \arctanh \! \left (\frac {x \sqrt {d}}{\sqrt {d \,x^{2}+c}}\right )}{1024 d^{\frac {9}{2}}}-\frac {c^{3} \left (24 a^{2} d^{2}+b c \left (-24 a d +7 b c \right )\right ) x \sqrt {d \,x^{2}+c}}{1024 d^{4}}+\frac {c^{2} \left (24 a^{2} d^{2}+b c \left (-24 a d +7 b c \right )\right ) x^{3} \sqrt {d \,x^{2}+c}}{1536 d^{3}}+\frac {c \left (24 a^{2} d^{2}+b c \left (-24 a d +7 b c \right )\right ) x^{5} \sqrt {d \,x^{2}+c}}{384 d^{2}} \]
command
integrate(x**4*(b*x**2+a)**2*(d*x**2+c)**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ - \frac {3 a^{2} c^{\frac {7}{2}} x}{128 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a^{2} c^{\frac {5}{2}} x^{3}}{128 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {13 a^{2} c^{\frac {3}{2}} x^{5}}{64 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {5 a^{2} \sqrt {c} d x^{7}}{16 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a^{2} c^{4} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{128 d^{\frac {5}{2}}} + \frac {a^{2} d^{2} x^{9}}{8 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {3 a b c^{\frac {9}{2}} x}{128 d^{3} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {a b c^{\frac {7}{2}} x^{3}}{128 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {a b c^{\frac {5}{2}} x^{5}}{320 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {23 a b c^{\frac {3}{2}} x^{7}}{80 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {19 a b \sqrt {c} d x^{9}}{40 \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {3 a b c^{5} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{128 d^{\frac {7}{2}}} + \frac {a b d^{2} x^{11}}{5 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {7 b^{2} c^{\frac {11}{2}} x}{1024 d^{4} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {7 b^{2} c^{\frac {9}{2}} x^{3}}{3072 d^{3} \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {7 b^{2} c^{\frac {7}{2}} x^{5}}{7680 d^{2} \sqrt {1 + \frac {d x^{2}}{c}}} - \frac {b^{2} c^{\frac {5}{2}} x^{7}}{1920 d \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {107 b^{2} c^{\frac {3}{2}} x^{9}}{960 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {23 b^{2} \sqrt {c} d x^{11}}{120 \sqrt {1 + \frac {d x^{2}}{c}}} + \frac {7 b^{2} c^{6} \operatorname {asinh}{\left (\frac {\sqrt {d} x}{\sqrt {c}} \right )}}{1024 d^{\frac {9}{2}}} + \frac {b^{2} d^{2} x^{13}}{12 \sqrt {c} \sqrt {1 + \frac {d x^{2}}{c}}} \]