\[ \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {4621 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{150}+\frac {139 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{150}+\frac {\left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{\sqrt {1-2 x}}+\frac {9 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{5}+\frac {139 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{10} \]
command
integrate((2+3*x)^(3/2)*(3+5*x)^(3/2)/(1-2*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (30 \, x^{2} + 106 \, x - 253\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{10 \, {\left (2 \, x - 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (15 \, x^{2} + 19 \, x + 6\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{4 \, x^{2} - 4 \, x + 1}, x\right ) \]