\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {49 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{81}+\frac {8 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{81}-\frac {2 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{3 \sqrt {2+3 x}}+\frac {40 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{27} \]
command
integrate((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (15 \, x + 13\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{27 \, \sqrt {3 \, x + 2}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{9 \, x^{2} + 12 \, x + 4}, x\right ) \]