\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{\sqrt {b d+2 c d x}} \, dx \]
Optimal antiderivative \[ -\frac {5 \left (-4 a c +b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \sqrt {2 c d x +b d}}{154 c^{2} d}+\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}} \sqrt {2 c d x +b d}}{11 c d}+\frac {5 \left (-4 a c +b^{2}\right )^{2} \sqrt {2 c d x +b d}\, \sqrt {c \,x^{2}+b x +a}}{308 c^{3} d}-\frac {5 \left (-4 a c +b^{2}\right )^{\frac {13}{4}} \EllipticF \left (\frac {\sqrt {2 c d x +b d}}{\left (-4 a c +b^{2}\right )^{\frac {1}{4}} \sqrt {d}}, i\right ) \sqrt {-\frac {c \left (c \,x^{2}+b x +a \right )}{-4 a c +b^{2}}}}{308 c^{4} \sqrt {d}\, \sqrt {c \,x^{2}+b x +a}} \]
command
integrate((c*x^2+b*x+a)^(5/2)/(2*c*d*x+b*d)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {5 \, \sqrt {2} {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) - 2 \, {\left (28 \, c^{6} x^{4} + 56 \, b c^{5} x^{3} + 5 \, b^{4} c^{2} - 50 \, a b^{2} c^{3} + 148 \, a^{2} c^{4} + 6 \, {\left (3 \, b^{2} c^{4} + 16 \, a c^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{3} c^{3} - 48 \, a b c^{4}\right )} x\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}}{616 \, c^{5} d} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt {c x^{2} + b x + a}}{\sqrt {2 \, c d x + b d}}, x\right ) \]