\[ \int \frac {\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{\sqrt {f x}} \, dx \]
Optimal antiderivative \[ \frac {2 a e \left (f x \right )^{\frac {5}{2}} F_{1}\left (\frac {5}{4}, -\frac {3}{2}, -\frac {3}{2}, \frac {9}{4}, -\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}, -\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 f^{3} \sqrt {1+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}}\, \sqrt {1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}}+\frac {2 a d F_{1}\left (\frac {1}{4}, -\frac {3}{2}, -\frac {3}{2}, \frac {5}{4}, -\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}, -\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}\right ) \sqrt {f x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{f \sqrt {1+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}}\, \sqrt {1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}} \]
command
Integrate[((d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2))/Sqrt[f*x],x]
Mathematica 13.1 output
\[ \frac {2 x \left (5 \left (a+b x^2+c x^4\right ) \left (-28 b^3 e+4 b^2 c \left (17 d+5 e x^2\right )+c^2 \left (867 a d+455 a e x^2+255 c d x^4+195 c e x^6\right )+b c \left (176 a e+5 c x^2 \left (85 d+57 e x^2\right )\right )\right )+20 a \left (-17 b^2 c d+612 a c^2 d+7 b^3 e-44 a b c e\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {1}{4};\frac {1}{2},\frac {1}{2};\frac {5}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )+4 \left (-51 b^3 c d+476 a b c^2 d+21 b^4 e-157 a b^2 c e+260 a^2 c^2 e\right ) x^2 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {5}{4};\frac {1}{2},\frac {1}{2};\frac {9}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )\right )}{16575 c^2 \sqrt {f x} \sqrt {a+b x^2+c x^4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________