\[ \int (f x)^{3/2} \left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 a d \left (f x \right )^{\frac {5}{2}} F_{1}\left (\frac {5}{4}, -\frac {3}{2}, -\frac {3}{2}, \frac {9}{4}, -\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}, -\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 f \sqrt {1+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}}\, \sqrt {1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}}+\frac {2 a e \left (f x \right )^{\frac {9}{2}} F_{1}\left (\frac {9}{4}, -\frac {3}{2}, -\frac {3}{2}, \frac {13}{4}, -\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}, -\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{9 f^{3} \sqrt {1+\frac {2 c \,x^{2}}{b -\sqrt {-4 a c +b^{2}}}}\, \sqrt {1+\frac {2 c \,x^{2}}{b +\sqrt {-4 a c +b^{2}}}}} \]
command
Integrate[(f*x)^(3/2)*(d + e*x^2)*(a + b*x^2 + c*x^4)^(3/2),x]
Mathematica 13.1 output
\[ \frac {2 f \sqrt {f x} \left (5 \left (a+b x^2+c x^4\right ) \left (308 b^4 e-4 b^3 c \left (147 d+55 e x^2\right )+12 b^2 c \left (-167 a e+5 c x^2 \left (7 d+3 e x^2\right )\right )+3 b c^2 \left (16 a \left (77 d+25 e x^2\right )+5 c x^4 \left (399 d+299 e x^2\right )\right )+3 c^2 \left (816 a^2 e+65 c^2 x^6 \left (21 d+17 e x^2\right )+5 a c x^2 \left (637 d+425 e x^2\right )\right )\right )-20 a \left (-147 b^3 c d+924 a b c^2 d+77 b^4 e-501 a b^2 c e+612 a^2 c^2 e\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {1}{4};\frac {1}{2},\frac {1}{2};\frac {5}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )-4 \left (-441 b^4 c d+3297 a b^2 c^2 d-5460 a^2 c^3 d+231 b^5 e-1778 a b^3 c e+3336 a^2 b c^2 e\right ) x^2 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {5}{4};\frac {1}{2},\frac {1}{2};\frac {9}{4};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )\right )}{348075 c^3 \sqrt {a+b x^2+c x^4}} \]
Mathematica 12.3 output
\[ \text {\$Aborted} \]________________________________________________________________________________________