\[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {4071079 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{52500}+\frac {673523 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{288750}+\frac {\left (2+3 x \right )^{\frac {7}{2}} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {2517 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{350}+\frac {12 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{7}+\frac {29293 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{875} \]
command
integrate((2+3*x)^(7/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (6750 \, x^{3} + 26010 \, x^{2} + 54757 \, x - 109756\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1750 \, {\left (2 \, x - 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{4 \, x^{2} - 4 \, x + 1}, x\right ) \]