\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^{17/2}} \, dx \]
Optimal antiderivative \[ -\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{15 c d \left (2 c d x +b d \right )^{\frac {15}{2}}}-\frac {\sqrt {c \,x^{2}+b x +a}}{110 c^{2} d^{3} \left (2 c d x +b d \right )^{\frac {11}{2}}}+\frac {\sqrt {c \,x^{2}+b x +a}}{385 c^{2} \left (-4 a c +b^{2}\right ) d^{5} \left (2 c d x +b d \right )^{\frac {7}{2}}}+\frac {\sqrt {c \,x^{2}+b x +a}}{231 c^{2} \left (-4 a c +b^{2}\right )^{2} d^{7} \left (2 c d x +b d \right )^{\frac {3}{2}}}+\frac {\EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{462 c^{3} \left (-4 a c +b^{2}\right )^{\frac {7}{4}} d^{\frac {17}{2}} \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^(17/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {5 \, \sqrt {2} {\left (256 \, c^{8} x^{8} + 1024 \, b c^{7} x^{7} + 1792 \, b^{2} c^{6} x^{6} + 1792 \, b^{3} c^{5} x^{5} + 1120 \, b^{4} c^{4} x^{4} + 448 \, b^{5} c^{3} x^{3} + 112 \, b^{6} c^{2} x^{2} + 16 \, b^{7} c x + b^{8}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + 2 \, {\left (640 \, c^{8} x^{6} + 1920 \, b c^{7} x^{5} - 5 \, b^{6} c^{2} - 10 \, a b^{4} c^{3} + 896 \, a^{2} b^{2} c^{4} - 2464 \, a^{3} c^{5} + 192 \, {\left (13 \, b^{2} c^{6} - 2 \, a c^{7}\right )} x^{4} + 256 \, {\left (7 \, b^{3} c^{5} - 3 \, a b c^{6}\right )} x^{3} + 2 \, {\left (253 \, b^{4} c^{4} + 664 \, a b^{2} c^{5} - 1904 \, a^{2} c^{6}\right )} x^{2} - 2 \, {\left (35 \, b^{5} c^{3} - 856 \, a b^{3} c^{4} + 1904 \, a^{2} b c^{5}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{4620 \, {\left (256 \, {\left (b^{4} c^{12} - 8 \, a b^{2} c^{13} + 16 \, a^{2} c^{14}\right )} d^{9} x^{8} + 1024 \, {\left (b^{5} c^{11} - 8 \, a b^{3} c^{12} + 16 \, a^{2} b c^{13}\right )} d^{9} x^{7} + 1792 \, {\left (b^{6} c^{10} - 8 \, a b^{4} c^{11} + 16 \, a^{2} b^{2} c^{12}\right )} d^{9} x^{6} + 1792 \, {\left (b^{7} c^{9} - 8 \, a b^{5} c^{10} + 16 \, a^{2} b^{3} c^{11}\right )} d^{9} x^{5} + 1120 \, {\left (b^{8} c^{8} - 8 \, a b^{6} c^{9} + 16 \, a^{2} b^{4} c^{10}\right )} d^{9} x^{4} + 448 \, {\left (b^{9} c^{7} - 8 \, a b^{7} c^{8} + 16 \, a^{2} b^{5} c^{9}\right )} d^{9} x^{3} + 112 \, {\left (b^{10} c^{6} - 8 \, a b^{8} c^{7} + 16 \, a^{2} b^{6} c^{8}\right )} d^{9} x^{2} + 16 \, {\left (b^{11} c^{5} - 8 \, a b^{9} c^{6} + 16 \, a^{2} b^{7} c^{7}\right )} d^{9} x + {\left (b^{12} c^{4} - 8 \, a b^{10} c^{5} + 16 \, a^{2} b^{8} c^{6}\right )} d^{9}\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {2 \, c d x + b d} {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}}{512 \, c^{9} d^{9} x^{9} + 2304 \, b c^{8} d^{9} x^{8} + 4608 \, b^{2} c^{7} d^{9} x^{7} + 5376 \, b^{3} c^{6} d^{9} x^{6} + 4032 \, b^{4} c^{5} d^{9} x^{5} + 2016 \, b^{5} c^{4} d^{9} x^{4} + 672 \, b^{6} c^{3} d^{9} x^{3} + 144 \, b^{7} c^{2} d^{9} x^{2} + 18 \, b^{8} c d^{9} x + b^{9} d^{9}}, x\right ) \]