\[ \int \frac {(2-5 x) x^{13/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {2 x^{\frac {11}{2}} \left (74+95 x \right )}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {4 x^{\frac {7}{2}} \left (1484+1685 x \right )}{27 \sqrt {3 x^{2}+5 x +2}}-\frac {1521056 \left (2+3 x \right ) \sqrt {x}}{76545 \sqrt {3 x^{2}+5 x +2}}+\frac {1521056 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticE \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{76545 \sqrt {3 x^{2}+5 x +2}}-\frac {211144 \left (1+x \right )^{\frac {3}{2}} \sqrt {\frac {1}{1+x}}\, \EllipticF \left (\frac {\sqrt {x}}{\sqrt {1+x}}, \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {\frac {2+3 x}{1+x}}}{5103 \sqrt {3 x^{2}+5 x +2}}-\frac {167336 x^{\frac {3}{2}} \sqrt {3 x^{2}+5 x +2}}{2835}+\frac {45820 x^{\frac {5}{2}} \sqrt {3 x^{2}+5 x +2}}{567}+\frac {211144 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{5103} \]
command
integrate((2-5*x)*x^(13/2)/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (5698840 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - 6844752 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) + 27 \, {\left (6075 \, x^{6} - 23652 \, x^{5} + 87570 \, x^{4} - 2983836 \, x^{3} - 8594380 \, x^{2} - 7545152 \, x - 2111440\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}\right )}}{688905 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (5 \, x^{7} - 2 \, x^{6}\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x}}{27 \, x^{6} + 135 \, x^{5} + 279 \, x^{4} + 305 \, x^{3} + 186 \, x^{2} + 60 \, x + 8}, x\right ) \]