\[ \int \frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {112543103 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{288750}-\frac {6770629 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{577500}+\frac {\left (2+3 x \right )^{\frac {9}{2}} \sqrt {3+5 x}}{3 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {166 \left (2+3 x \right )^{\frac {7}{2}} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}-\frac {139163 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{3850}-\frac {1327 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{154}-\frac {6478333 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{38500} \]
command
integrate((2+3*x)^(9/2)*(3+5*x)^(1/2)/(1-2*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (1336500 \, x^{4} + 6664680 \, x^{3} + 19375686 \, x^{2} - 94671446 \, x + 35797779\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{115500 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}, x\right ) \]