\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {98 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{81}+\frac {16 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{81}-\frac {2 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}{9 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {82 \sqrt {1-2 x}\, \sqrt {3+5 x}}{27 \sqrt {2+3 x}} \]
command
integrate((1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (129 \, x + 79\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{27 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8}, x\right ) \]