\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {1473539 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{5906250}-\frac {31288 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{2953125}-\frac {2 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {7}{2}}}{5 \sqrt {3+5 x}}+\frac {5153 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{39375}+\frac {958 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{1575}-\frac {8 \left (2+3 x \right )^{\frac {7}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{45}-\frac {12601 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{196875} \]
command
integrate((1-2*x)^(3/2)*(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (472500 \, x^{4} + 517500 \, x^{3} - 252225 \, x^{2} - 377530 \, x - 83787\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{196875 \, \sqrt {5 \, x + 3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{25 \, x^{2} + 30 \, x + 9}, x\right ) \]