\[ \int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \, dx \]
Optimal antiderivative \[ \frac {a \left (-5 a^{3} d^{3}+36 a^{2} b c \,d^{2}-120 a \,b^{2} c^{2} d +320 b^{3} c^{3}\right ) x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{1536 b^{3}}+\frac {\left (-5 a^{3} d^{3}+36 a^{2} b c \,d^{2}-120 a \,b^{2} c^{2} d +320 b^{3} c^{3}\right ) x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{1920 b^{3}}+\frac {d \left (15 a^{2} d^{2}-68 a b c d +152 b^{2} c^{2}\right ) x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{960 b^{3}}+\frac {d \left (-5 a d +16 b c \right ) x \left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (d \,x^{2}+c \right )}{120 b^{2}}+\frac {d x \left (b \,x^{2}+a \right )^{\frac {7}{2}} \left (d \,x^{2}+c \right )^{2}}{12 b}+\frac {a^{3} \left (-5 a^{3} d^{3}+36 a^{2} b c \,d^{2}-120 a \,b^{2} c^{2} d +320 b^{3} c^{3}\right ) \arctanh \! \left (\frac {x \sqrt {b}}{\sqrt {b \,x^{2}+a}}\right )}{1024 b^{\frac {7}{2}}}+\frac {a^{2} \left (-5 a^{3} d^{3}+36 a^{2} b c \,d^{2}-120 a \,b^{2} c^{2} d +320 b^{3} c^{3}\right ) x \sqrt {b \,x^{2}+a}}{1024 b^{3}} \]
command
integrate((b*x**2+a)**(5/2)*(d*x**2+c)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {5 a^{\frac {11}{2}} d^{3} x}{1024 b^{3} \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {9 a^{\frac {9}{2}} c d^{2} x}{256 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {5 a^{\frac {9}{2}} d^{3} x^{3}}{3072 b^{2} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {15 a^{\frac {7}{2}} c^{2} d x}{128 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {3 a^{\frac {7}{2}} c d^{2} x^{3}}{256 b \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {a^{\frac {7}{2}} d^{3} x^{5}}{1536 b \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {a^{\frac {5}{2}} c^{3} x \sqrt {1 + \frac {b x^{2}}{a}}}{2} + \frac {3 a^{\frac {5}{2}} c^{3} x}{16 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {133 a^{\frac {5}{2}} c^{2} d x^{3}}{128 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {387 a^{\frac {5}{2}} c d^{2} x^{5}}{640 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {55 a^{\frac {5}{2}} d^{3} x^{7}}{384 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {35 a^{\frac {3}{2}} b c^{3} x^{3}}{48 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {127 a^{\frac {3}{2}} b c^{2} d x^{5}}{64 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {219 a^{\frac {3}{2}} b c d^{2} x^{7}}{160 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {67 a^{\frac {3}{2}} b d^{3} x^{9}}{192 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {17 \sqrt {a} b^{2} c^{3} x^{5}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {23 \sqrt {a} b^{2} c^{2} d x^{7}}{16 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {87 \sqrt {a} b^{2} c d^{2} x^{9}}{80 \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {7 \sqrt {a} b^{2} d^{3} x^{11}}{24 \sqrt {1 + \frac {b x^{2}}{a}}} - \frac {5 a^{6} d^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{1024 b^{\frac {7}{2}}} + \frac {9 a^{5} c d^{2} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{256 b^{\frac {5}{2}}} - \frac {15 a^{4} c^{2} d \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{128 b^{\frac {3}{2}}} + \frac {5 a^{3} c^{3} \operatorname {asinh}{\left (\frac {\sqrt {b} x}{\sqrt {a}} \right )}}{16 \sqrt {b}} + \frac {b^{3} c^{3} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 b^{3} c^{2} d x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {3 b^{3} c d^{2} x^{11}}{10 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} + \frac {b^{3} d^{3} x^{13}}{12 \sqrt {a} \sqrt {1 + \frac {b x^{2}}{a}}} \]