\[ \int \frac {x^{11} \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx \]
Optimal antiderivative \[ -\frac {1024 c^{3} \left (d \,x^{3}+c \right )^{\frac {3}{2}}}{9 d^{4}}-\frac {38 c^{2} \left (d \,x^{3}+c \right )^{\frac {5}{2}}}{5 d^{4}}-\frac {4 c \left (d \,x^{3}+c \right )^{\frac {7}{2}}}{7 d^{4}}-\frac {2 \left (d \,x^{3}+c \right )^{\frac {9}{2}}}{27 d^{4}}+\frac {9216 c^{\frac {9}{2}} \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{d^{4}}-\frac {3072 c^{4} \sqrt {d \,x^{3}+c}}{d^{4}} \]
command
integrate(x**11*(d*x**3+c)**(3/2)/(-d*x**3+8*c),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ - \frac {9216 c^{5} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{3 \sqrt {- c}} \right )}}{d^{4} \sqrt {- c}} - \frac {3072 c^{4} \sqrt {c + d x^{3}}}{d^{4}} - \frac {1024 c^{3} \left (c + d x^{3}\right )^{\frac {3}{2}}}{9 d^{4}} - \frac {38 c^{2} \left (c + d x^{3}\right )^{\frac {5}{2}}}{5 d^{4}} - \frac {4 c \left (c + d x^{3}\right )^{\frac {7}{2}}}{7 d^{4}} - \frac {2 \left (c + d x^{3}\right )^{\frac {9}{2}}}{27 d^{4}} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________