\[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^2 \, dx \]
Optimal antiderivative \[ \frac {a \left (3 a^{2} d^{2}-20 a b c d +80 b^{2} c^{2}\right ) x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{384 b^{2}}+\frac {\left (3 a^{2} d^{2}-20 a b c d +80 b^{2} c^{2}\right ) x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{480 b^{2}}+\frac {3 d \left (-a d +4 b c \right ) x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{80 b^{2}}+\frac {d x \left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (d \,x^{2}+c \right )}{10 b}+\frac {a^{3} \left (3 a^{2} d^{2}-20 a b c d +80 b^{2} c^{2}\right ) \arctanh \! \left (\frac {x \sqrt {b}}{\sqrt {b \,x^{2}+a}}\right )}{256 b^{\frac {5}{2}}}+\frac {a^{2} \left (3 a^{2} d^{2}-20 a b c d +80 b^{2} c^{2}\right ) x \sqrt {b \,x^{2}+a}}{256 b^{2}} \]
command
integrate((b*x**2+a)**(5/2)*(d*x**2+c)**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ - \frac {3 a^{\frac {9}{2}} d^{2} x}{256 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 a^{\frac {7}{2}} c d x}{64 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {a^{\frac {7}{2}} d^{2} x^{3}}{256 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {a^{\frac {5}{2}} c^{2} x \sqrt {1 + \frac {b x^{2}}{a}}}{2} + \frac {3 a^{\frac {5}{2}} c^{2} x}{16 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {133 a^{\frac {5}{2}} c d x^{3}}{192 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {129 a^{\frac {5}{2}} d^{2} x^{5}}{640 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {35 a^{\frac {3}{2}} b c^{2} x^{3}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {127 a^{\frac {3}{2}} b c d x^{5}}{96 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {73 a^{\frac {3}{2}} b d^{2} x^{7}}{160 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {17 \sqrt {a} b^{2} c^{2} x^{5}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {23 \sqrt {a} b^{2} c d x^{7}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {29 \sqrt {a} b^{2} d^{2} x^{9}}{80 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 a^{5} d^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{256 b^{\frac {5}{2}}} - \frac {5 a^{4} c d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{64 b^{\frac {3}{2}}} + \frac {5 a^{3} c^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 \sqrt {b}} + \frac {b^{3} c^{2} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {b^{3} c d x^{9}}{4 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {b^{3} d^{2} x^{11}}{10 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]