3.869 \(\int \frac{x^4}{\left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=237 \[ \frac{x \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (b-\frac{4 a c+b^2}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (b \sqrt{b^2-4 a c}+4 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]

[Out]

(x*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((b - (b^2 + 4*a*c)/Sq
rt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqr
t[2]*Sqrt[c]*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b^2 + 4*a*c + b*Sqrt
[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[
2]*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

_______________________________________________________________________________________

Rubi [A]  time = 0.757968, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{x \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\left (b-\frac{4 a c+b^2}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (b \sqrt{b^2-4 a c}+4 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^2 + c*x^4)^2,x]

[Out]

(x*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((b - (b^2 + 4*a*c)/Sq
rt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqr
t[2]*Sqrt[c]*(b^2 - 4*a*c)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b^2 + 4*a*c + b*Sqrt
[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[
2]*Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 55.512, size = 218, normalized size = 0.92 \[ \frac{x \left (2 a + b x^{2}\right )}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} + \frac{\sqrt{2} \left (4 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 \sqrt{c} \sqrt{b + \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{\sqrt{2} \left (4 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{4 \sqrt{c} \sqrt{b - \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(c*x**4+b*x**2+a)**2,x)

[Out]

x*(2*a + b*x**2)/(2*(-4*a*c + b**2)*(a + b*x**2 + c*x**4)) + sqrt(2)*(4*a*c + b*
*2 + b*sqrt(-4*a*c + b**2))*atan(sqrt(2)*sqrt(c)*x/sqrt(b + sqrt(-4*a*c + b**2))
)/(4*sqrt(c)*sqrt(b + sqrt(-4*a*c + b**2))*(-4*a*c + b**2)**(3/2)) - sqrt(2)*(4*
a*c + b**2 - b*sqrt(-4*a*c + b**2))*atan(sqrt(2)*sqrt(c)*x/sqrt(b - sqrt(-4*a*c
+ b**2)))/(4*sqrt(c)*sqrt(b - sqrt(-4*a*c + b**2))*(-4*a*c + b**2)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.767157, size = 235, normalized size = 0.99 \[ \frac{1}{4} \left (\frac{2 \left (2 a x+b x^3\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \left (b \sqrt{b^2-4 a c}-4 a c-b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (b \sqrt{b^2-4 a c}+4 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{c} \left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^2 + c*x^4)^2,x]

[Out]

((2*(2*a*x + b*x^3))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-b^2 - 4*a*
c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]]
)/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(b^2 + 4*
a*c + b*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]
]])/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/4

_______________________________________________________________________________________

Maple [B]  time = 0.055, size = 1641, normalized size = 6.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(c*x^4+b*x^2+a)^2,x)

[Out]

(-1/2*b/(4*a*c-b^2)*x^3-a/(4*a*c-b^2)*x)/(c*x^4+b*x^2+a)+16/(-(4*a*c-b^2)^3)^(1/
2)/(4*a*c-b^2)*c^3*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)
^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^
(1/2))*(4*a*c-b^2)*c)^(1/2))*a^3-4/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*c^2*2^(1/2
)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*
c^2+2*b^2*c)*x*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/
2))*a^2*b^2-c/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c
-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)/((
-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*b^4*a+1/4/(-(4*a*c-b^
2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^
2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2
)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*b^6-c/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c+b^3+(-(4*
a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)
/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2))*b*a+1/4/(4*a*c-b^2
)*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)*c)^(1/2)*arctanh(1/
2*(-8*a*c^2+2*b^2*c)*x*2^(1/2)/((-4*a*b*c+b^3+(-(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2
)*c)^(1/2))*b^3-16/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*c^3*2^(1/2)/((4*a*c-b^2)*c
*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x*2^(1
/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*a^3+4/(-(4*a*c-b
^2)^3)^(1/2)/(4*a*c-b^2)*c^2*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3
)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b
^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*a^2*b^2+c/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*
2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8
*a*c^2-2*b^2*c)*x*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(
1/2))*b^4*a-1/4/(-(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*c*(4*a*b
*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x*2^(1/2)/((4
*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))*b^6-c/(4*a*c-b^2)*2^(1/
2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^
2-2*b^2*c)*x*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2)))^(1/2))
*b*a+1/4/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*a*c-b^2)^3)^(1/2))
)^(1/2)*arctan(1/2*(8*a*c^2-2*b^2*c)*x*2^(1/2)/((4*a*c-b^2)*c*(4*a*b*c-b^3+(-(4*
a*c-b^2)^3)^(1/2)))^(1/2))*b^3

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b x^{3} + 2 \, a x}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )}} + \frac{\int \frac{b x^{2} - 2 \, a}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (b^{2} - 4 \, a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(c*x^4 + b*x^2 + a)^2,x, algorithm="maxima")

[Out]

1/2*(b*x^3 + 2*a*x)/((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x
^2) + 1/2*integrate((b*x^2 - 2*a)/(c*x^4 + b*x^2 + a), x)/(b^2 - 4*a*c)

_______________________________________________________________________________________

Fricas [A]  time = 0.280536, size = 2252, normalized size = 9.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(c*x^4 + b*x^2 + a)^2,x, algorithm="fricas")

[Out]

1/4*(2*b*x^3 + sqrt(1/2)*((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b
*c)*x^2)*sqrt(-(b^3 + 12*a*b*c + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3
*c^4)/sqrt(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))/(b^6*c - 12*a*
b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log((3*b^2 + 4*a*c)*x + sqrt(1/2)*(b^4 -
8*a*b^2*c + 16*a^2*c^2 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^
4)/sqrt(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*sqrt(-(b^3 + 12*a
*b*c + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)/sqrt(b^6*c^2 - 12*a*
b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 -
64*a^3*c^4))) - sqrt(1/2)*((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a
*b*c)*x^2)*sqrt(-(b^3 + 12*a*b*c + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a
^3*c^4)/sqrt(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))/(b^6*c - 12*
a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log((3*b^2 + 4*a*c)*x - sqrt(1/2)*(b^4
- 8*a*b^2*c + 16*a^2*c^2 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*
c^4)/sqrt(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*sqrt(-(b^3 + 12
*a*b*c + (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)/sqrt(b^6*c^2 - 12*
a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3
- 64*a^3*c^4))) + sqrt(1/2)*((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4
*a*b*c)*x^2)*sqrt(-(b^3 + 12*a*b*c - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64
*a^3*c^4)/sqrt(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))/(b^6*c - 1
2*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log((3*b^2 + 4*a*c)*x + sqrt(1/2)*(b
^4 - 8*a*b^2*c + 16*a^2*c^2 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*
b*c^4)/sqrt(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*sqrt(-(b^3 +
12*a*b*c - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)/sqrt(b^6*c^2 - 1
2*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c
^3 - 64*a^3*c^4))) - sqrt(1/2)*((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 -
4*a*b*c)*x^2)*sqrt(-(b^3 + 12*a*b*c - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 -
64*a^3*c^4)/sqrt(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))/(b^6*c -
12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4))*log((3*b^2 + 4*a*c)*x - sqrt(1/2)*
(b^4 - 8*a*b^2*c + 16*a^2*c^2 - 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^
3*b*c^4)/sqrt(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))*sqrt(-(b^3
+ 12*a*b*c - (b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2*c^3 - 64*a^3*c^4)/sqrt(b^6*c^2 -
12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5))/(b^6*c - 12*a*b^4*c^2 + 48*a^2*b^2
*c^3 - 64*a^3*c^4))) + 4*a*x)/((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 -
4*a*b*c)*x^2)

_______________________________________________________________________________________

Sympy [A]  time = 11.2805, size = 294, normalized size = 1.24 \[ - \frac{2 a x + b x^{3}}{8 a^{2} c - 2 a b^{2} + x^{4} \left (8 a c^{2} - 2 b^{2} c\right ) + x^{2} \left (8 a b c - 2 b^{3}\right )} + \operatorname{RootSum}{\left (t^{4} \left (1048576 a^{6} c^{7} - 1572864 a^{5} b^{2} c^{6} + 983040 a^{4} b^{4} c^{5} - 327680 a^{3} b^{6} c^{4} + 61440 a^{2} b^{8} c^{3} - 6144 a b^{10} c^{2} + 256 b^{12} c\right ) + t^{2} \left (- 12288 a^{4} b c^{4} + 8192 a^{3} b^{3} c^{3} - 1536 a^{2} b^{5} c^{2} + 16 b^{9}\right ) + 16 a^{3} c^{2} + 24 a^{2} b^{2} c + 9 a b^{4}, \left ( t \mapsto t \log{\left (x + \frac{16384 t^{3} a^{3} b c^{4} - 12288 t^{3} a^{2} b^{3} c^{3} + 3072 t^{3} a b^{5} c^{2} - 256 t^{3} b^{7} c + 64 t a^{2} c^{2} - 128 t a b^{2} c - 4 t b^{4}}{4 a c + 3 b^{2}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(c*x**4+b*x**2+a)**2,x)

[Out]

-(2*a*x + b*x**3)/(8*a**2*c - 2*a*b**2 + x**4*(8*a*c**2 - 2*b**2*c) + x**2*(8*a*
b*c - 2*b**3)) + RootSum(_t**4*(1048576*a**6*c**7 - 1572864*a**5*b**2*c**6 + 983
040*a**4*b**4*c**5 - 327680*a**3*b**6*c**4 + 61440*a**2*b**8*c**3 - 6144*a*b**10
*c**2 + 256*b**12*c) + _t**2*(-12288*a**4*b*c**4 + 8192*a**3*b**3*c**3 - 1536*a*
*2*b**5*c**2 + 16*b**9) + 16*a**3*c**2 + 24*a**2*b**2*c + 9*a*b**4, Lambda(_t, _
t*log(x + (16384*_t**3*a**3*b*c**4 - 12288*_t**3*a**2*b**3*c**3 + 3072*_t**3*a*b
**5*c**2 - 256*_t**3*b**7*c + 64*_t*a**2*c**2 - 128*_t*a*b**2*c - 4*_t*b**4)/(4*
a*c + 3*b**2))))

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(c*x^4 + b*x^2 + a)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError