\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {53279 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{984375}-\frac {110014 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{984375}-\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (2+3 x \right )^{\frac {3}{2}}}{5 \sqrt {3+5 x}}-\frac {32 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}{175}-\frac {1972 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{4375}+\frac {106772 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{65625} \]
command
integrate((1-2*x)^(5/2)*(2+3*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (22500 \, x^{3} - 31350 \, x^{2} + 9545 \, x + 9168\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{65625 \, \sqrt {5 \, x + 3}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{25 \, x^{2} + 30 \, x + 9}, x\right ) \]