\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {264260033 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{974531250}-\frac {7261561 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{487265625}-\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (2+3 x \right )^{\frac {7}{2}}}{5 \sqrt {3+5 x}}-\frac {48 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {7}{2}} \sqrt {3+5 x}}{275}+\frac {2020841 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{6496875}+\frac {346636 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{259875}-\frac {2972 \left (2+3 x \right )^{\frac {7}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{7425}-\frac {703672 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{32484375} \]
command
integrate((1-2*x)^(5/2)*(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
\[ \frac {{\left (127575000 \, x^{5} + 56227500 \, x^{4} - 141221250 \, x^{3} - 32807925 \, x^{2} + 71568535 \, x + 26378214\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{32484375 \, \sqrt {5 \, x + 3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{25 \, x^{2} + 30 \, x + 9}, x\right ) \]