\[ \int x^3 (2+3 x)^{3/2} \sqrt {1+4 x} \, dx \]
Optimal antiderivative \[ \frac {\left (4103-7968 x \right ) \left (2+3 x \right )^{\frac {5}{2}} \left (1+4 x \right )^{\frac {3}{2}}}{829440}+\frac {x^{2} \left (2+3 x \right )^{\frac {5}{2}} \left (1+4 x \right )^{\frac {3}{2}}}{72}+\frac {1067875 \arcsinh \! \left (\frac {\sqrt {15}\, \sqrt {1+4 x}}{5}\right ) \sqrt {3}}{254803968}+\frac {42715 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1+4 x}}{15925248}-\frac {8543 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {1+4 x}}{995328}+\frac {213575 \sqrt {2+3 x}\, \sqrt {1+4 x}}{42467328} \]
command
integrate(x**3*(2+3*x)**(3/2)*(1+4*x)**(1/2),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \text {Timed out} \]
Sympy 1.8 under Python 3.8.8 output
\[ \frac {3 \left (4 x + 1\right )^{\frac {13}{2}}}{4096 \sqrt {12 x + 8}} + \frac {7 \left (4 x + 1\right )^{\frac {11}{2}}}{40960 \sqrt {12 x + 8}} - \frac {869 \left (4 x + 1\right )^{\frac {9}{2}}}{196608 \sqrt {12 x + 8}} + \frac {2027 \left (4 x + 1\right )^{\frac {7}{2}}}{1179648 \sqrt {12 x + 8}} + \frac {119135 \left (4 x + 1\right )^{\frac {5}{2}}}{14155776 \sqrt {12 x + 8}} - \frac {904775 \left (4 x + 1\right )^{\frac {3}{2}}}{84934656 \sqrt {12 x + 8}} - \frac {1067875 \sqrt {4 x + 1}}{84934656 \sqrt {12 x + 8}} + \frac {1067875 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {15} \sqrt {4 x + 1}}{5} \right )}}{254803968} \]