\[ \int \frac {\sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{7/2}} \, dx \]
Optimal antiderivative \[ -\frac {4636 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{6615}-\frac {124 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{6615}-\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}}{15 \left (2+3 x \right )^{\frac {5}{2}}}+\frac {74 \sqrt {1-2 x}\, \sqrt {3+5 x}}{315 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {4636 \sqrt {1-2 x}\, \sqrt {3+5 x}}{2205 \sqrt {2+3 x}} \]
command
integrate((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (20862 \, x^{2} + 28593 \, x + 9643\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2205 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16}, x\right ) \]