\[ \int x^7 \left (c \sqrt {a+b x^2}\right )^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {2 a^{3} \left (b \,x^{2}+a \right ) \left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{7 b^{4}}+\frac {6 a^{2} \left (b \,x^{2}+a \right )^{2} \left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{11 b^{4}}-\frac {2 a \left (b \,x^{2}+a \right )^{3} \left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{5 b^{4}}+\frac {2 \left (b \,x^{2}+a \right )^{4} \left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{19 b^{4}} \]
command
integrate(x**7*(c*(b*x**2+a)**(1/2))**(3/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {256 a^{4} \left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{7315 b^{4}} + \frac {192 a^{3} x^{2} \left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{7315 b^{3}} - \frac {24 a^{2} x^{4} \left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{1045 b^{2}} + \frac {2 a x^{6} \left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{95 b} + \frac {2 x^{8} \left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{19} & \text {for}\: b \neq 0 \\\frac {x^{8} \left (\sqrt {a} c\right )^{\frac {3}{2}}}{8} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________