\[ \int \frac {x^7}{\left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ \frac {2 x^{4} \left (b x +2 a \right )}{\left (-4 a c +b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}}+\frac {3 \left (-4 a c +5 b^{2}\right ) x \arctanh \left (\frac {2 c x +b}{2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}}\right ) \sqrt {c \,x^{2}+b x +a}}{8 c^{\frac {7}{2}} \sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}}+\frac {\left (-12 a c +5 b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}}{2 c^{2} \left (-4 a c +b^{2}\right )}-\frac {b \left (-52 a c +15 b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}}{4 c^{3} \left (-4 a c +b^{2}\right ) x}-\frac {2 b x \sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}}{c \left (-4 a c +b^{2}\right )} \]
command
integrate(x^7/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {{\left (15 \, b^{4} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 72 \, a b^{2} c \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 48 \, a^{2} c^{2} \log \left ({\left | -b + 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 30 \, \sqrt {a} b^{3} \sqrt {c} - 104 \, a^{\frac {3}{2}} b c^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{8 \, {\left (b^{2} c^{\frac {7}{2}} - 4 \, a c^{\frac {9}{2}}\right )}} + \frac {{\left ({\left (\frac {2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x}{b^{2} c^{3} \mathrm {sgn}\left (x\right ) - 4 \, a c^{4} \mathrm {sgn}\left (x\right )} - \frac {5 \, {\left (b^{3} c - 4 \, a b c^{2}\right )}}{b^{2} c^{3} \mathrm {sgn}\left (x\right ) - 4 \, a c^{4} \mathrm {sgn}\left (x\right )}\right )} x - \frac {15 \, b^{4} - 62 \, a b^{2} c + 24 \, a^{2} c^{2}}{b^{2} c^{3} \mathrm {sgn}\left (x\right ) - 4 \, a c^{4} \mathrm {sgn}\left (x\right )}\right )} x - \frac {15 \, a b^{3} - 52 \, a^{2} b c}{b^{2} c^{3} \mathrm {sgn}\left (x\right ) - 4 \, a c^{4} \mathrm {sgn}\left (x\right )}}{4 \, \sqrt {c x^{2} + b x + a}} - \frac {3 \, {\left (5 \, b^{2} - 4 \, a c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} - b \right |}\right )}{8 \, c^{\frac {7}{2}} \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {x^{7}}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}\,{d x} \]________________________________________________________________________________________