\[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx \]
Optimal antiderivative \[ -\frac {148831 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{141750}-\frac {2252 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{70875}-\frac {31 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{525}+\frac {2 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{35}-\frac {2252 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{4725} \]
command
integrate((3+5*x)^(3/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{4725} \, {\left (6750 \, x^{2} + 6705 \, x - 659\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left ({\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]