\[ \int \frac {(2+3 x)^{5/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {4621 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{330}-\frac {139 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{330}+\frac {\left (2+3 x \right )^{\frac {5}{2}} \sqrt {3+5 x}}{3 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {100 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}-\frac {133 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{22} \]
command
integrate((2+3*x)^(5/2)*(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
\[ -\frac {{\left (198 \, x^{2} - 2060 \, x + 711\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{66 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}, x\right ) \]