\[ \int \frac {x \left (a+b x^2+c x^4\right )^{3/2}}{d+e x^2} \, dx \]
Optimal antiderivative \[ \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{6 e}-\frac {\left (-b e +2 c d \right ) \left (8 c^{2} d^{2}-b^{2} e^{2}-4 c e \left (-3 a e +2 b d \right )\right ) \arctanh \left (\frac {2 c \,x^{2}+b}{2 \sqrt {c}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{32 c^{\frac {3}{2}} e^{4}}+\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right )^{\frac {3}{2}} \arctanh \left (\frac {b d -2 a e +\left (-b e +2 c d \right ) x^{2}}{2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{2 e^{4}}+\frac {\left (8 c^{2} d^{2}+b^{2} e^{2}-2 c e \left (-4 a e +5 b d \right )-2 c e \left (-b e +2 c d \right ) x^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{16 c \,e^{3}} \]
command
integrate(x*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [-\frac {{\left (3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d e^{2} + {\left (b^{3} - 12 \, a b c\right )} e^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 48 \, {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )} \sqrt {c d^{2} - b d e + a e^{2}} \log \left (-\frac {8 \, c^{2} d^{2} x^{4} + 8 \, b c d^{2} x^{2} + {\left (b^{2} + 4 \, a c\right )} d^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c d x^{2} + b d - {\left (b x^{2} + 2 \, a\right )} e\right )} \sqrt {c d^{2} - b d e + a e^{2}} + {\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} e^{2} - 2 \, {\left (4 \, b c d x^{4} + {\left (3 \, b^{2} + 4 \, a c\right )} d x^{2} + 4 \, a b d\right )} e}{x^{4} e^{2} + 2 \, d x^{2} e + d^{2}}\right ) - 4 \, {\left (24 \, c^{3} d^{2} e + {\left (8 \, c^{3} x^{4} + 14 \, b c^{2} x^{2} + 3 \, b^{2} c + 32 \, a c^{2}\right )} e^{3} - 6 \, {\left (2 \, c^{3} d x^{2} + 5 \, b c^{2} d\right )} e^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}\right )} e^{\left (-4\right )}}{192 \, c^{2}}, \frac {{\left (3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d e^{2} + {\left (b^{3} - 12 \, a b c\right )} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 24 \, {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )} \sqrt {c d^{2} - b d e + a e^{2}} \log \left (-\frac {8 \, c^{2} d^{2} x^{4} + 8 \, b c d^{2} x^{2} + {\left (b^{2} + 4 \, a c\right )} d^{2} + 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c d x^{2} + b d - {\left (b x^{2} + 2 \, a\right )} e\right )} \sqrt {c d^{2} - b d e + a e^{2}} + {\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} e^{2} - 2 \, {\left (4 \, b c d x^{4} + {\left (3 \, b^{2} + 4 \, a c\right )} d x^{2} + 4 \, a b d\right )} e}{x^{4} e^{2} + 2 \, d x^{2} e + d^{2}}\right ) + 2 \, {\left (24 \, c^{3} d^{2} e + {\left (8 \, c^{3} x^{4} + 14 \, b c^{2} x^{2} + 3 \, b^{2} c + 32 \, a c^{2}\right )} e^{3} - 6 \, {\left (2 \, c^{3} d x^{2} + 5 \, b c^{2} d\right )} e^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}\right )} e^{\left (-4\right )}}{96 \, c^{2}}, \frac {{\left (96 \, {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}} \arctan \left (-\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c d x^{2} + b d - {\left (b x^{2} + 2 \, a\right )} e\right )} \sqrt {-c d^{2} + b d e - a e^{2}}}{2 \, {\left (c^{2} d^{2} x^{4} + b c d^{2} x^{2} + a c d^{2} + {\left (a c x^{4} + a b x^{2} + a^{2}\right )} e^{2} - {\left (b c d x^{4} + b^{2} d x^{2} + a b d\right )} e\right )}}\right ) - 3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d e^{2} + {\left (b^{3} - 12 \, a b c\right )} e^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (24 \, c^{3} d^{2} e + {\left (8 \, c^{3} x^{4} + 14 \, b c^{2} x^{2} + 3 \, b^{2} c + 32 \, a c^{2}\right )} e^{3} - 6 \, {\left (2 \, c^{3} d x^{2} + 5 \, b c^{2} d\right )} e^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}\right )} e^{\left (-4\right )}}{192 \, c^{2}}, \frac {{\left (48 \, {\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}} \arctan \left (-\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c d x^{2} + b d - {\left (b x^{2} + 2 \, a\right )} e\right )} \sqrt {-c d^{2} + b d e - a e^{2}}}{2 \, {\left (c^{2} d^{2} x^{4} + b c d^{2} x^{2} + a c d^{2} + {\left (a c x^{4} + a b x^{2} + a^{2}\right )} e^{2} - {\left (b c d x^{4} + b^{2} d x^{2} + a b d\right )} e\right )}}\right ) + 3 \, {\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d e^{2} + {\left (b^{3} - 12 \, a b c\right )} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, {\left (24 \, c^{3} d^{2} e + {\left (8 \, c^{3} x^{4} + 14 \, b c^{2} x^{2} + 3 \, b^{2} c + 32 \, a c^{2}\right )} e^{3} - 6 \, {\left (2 \, c^{3} d x^{2} + 5 \, b c^{2} d\right )} e^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}\right )} e^{\left (-4\right )}}{96 \, c^{2}}\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]