\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx \]
Optimal antiderivative \[ -\frac {6515539 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{17718750}-\frac {104663 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{8859375}+\frac {2 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {5}{2}} \sqrt {3+5 x}}{45}+\frac {403 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{118125}+\frac {178 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{4725}-\frac {87476 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{590625} \]
command
integrate((1-2*x)^(3/2)*(2+3*x)^(5/2)/(3+5*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {1}{590625} \, {\left (472500 \, x^{3} + 193500 \, x^{2} - 378045 \, x - 110554\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{\sqrt {5 \, x + 3}}, x\right ) \]