3.868 \(\int \frac{x^6}{\left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=271 \[ \frac{\left (-\frac{b \left (b^2-8 a c\right )}{\sqrt{b^2-4 a c}}-6 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\frac{b \left (b^2-8 a c\right )}{\sqrt{b^2-4 a c}}-6 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x^3 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b x}{2 c \left (b^2-4 a c\right )} \]

[Out]

-(b*x)/(2*c*(b^2 - 4*a*c)) + (x^3*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c
*x^4)) + ((b^2 - 6*a*c - (b*(b^2 - 8*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sq
rt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)*Sqrt[b -
Sqrt[b^2 - 4*a*c]]) + ((b^2 - 6*a*c + (b*(b^2 - 8*a*c))/Sqrt[b^2 - 4*a*c])*ArcT
an[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*c^(3/2)*(b^2 - 4
*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

_______________________________________________________________________________________

Rubi [A]  time = 1.18703, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\left (-\frac{b \left (b^2-8 a c\right )}{\sqrt{b^2-4 a c}}-6 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (\frac{b \left (b^2-8 a c\right )}{\sqrt{b^2-4 a c}}-6 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{2 \sqrt{2} c^{3/2} \left (b^2-4 a c\right ) \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{x^3 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b x}{2 c \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + b*x^2 + c*x^4)^2,x]

[Out]

-(b*x)/(2*c*(b^2 - 4*a*c)) + (x^3*(2*a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c
*x^4)) + ((b^2 - 6*a*c - (b*(b^2 - 8*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sq
rt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*c^(3/2)*(b^2 - 4*a*c)*Sqrt[b -
Sqrt[b^2 - 4*a*c]]) + ((b^2 - 6*a*c + (b*(b^2 - 8*a*c))/Sqrt[b^2 - 4*a*c])*ArcT
an[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*c^(3/2)*(b^2 - 4
*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 96.2992, size = 267, normalized size = 0.99 \[ - \frac{b x}{2 c \left (- 4 a c + b^{2}\right )} + \frac{x^{3} \left (2 a + b x^{2}\right )}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} + \frac{\sqrt{2} \left (- 2 a b c + b \left (- 6 a c + b^{2}\right ) + \left (- 6 a c + b^{2}\right ) \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 c^{\frac{3}{2}} \sqrt{b + \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{\sqrt{2} \left (- a b c + \frac{b \left (- 6 a c + b^{2}\right )}{2} - \frac{\left (- 6 a c + b^{2}\right ) \sqrt{- 4 a c + b^{2}}}{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 c^{\frac{3}{2}} \sqrt{b - \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(c*x**4+b*x**2+a)**2,x)

[Out]

-b*x/(2*c*(-4*a*c + b**2)) + x**3*(2*a + b*x**2)/(2*(-4*a*c + b**2)*(a + b*x**2
+ c*x**4)) + sqrt(2)*(-2*a*b*c + b*(-6*a*c + b**2) + (-6*a*c + b**2)*sqrt(-4*a*c
+ b**2))*atan(sqrt(2)*sqrt(c)*x/sqrt(b + sqrt(-4*a*c + b**2)))/(4*c**(3/2)*sqrt
(b + sqrt(-4*a*c + b**2))*(-4*a*c + b**2)**(3/2)) - sqrt(2)*(-a*b*c + b*(-6*a*c
+ b**2)/2 - (-6*a*c + b**2)*sqrt(-4*a*c + b**2)/2)*atan(sqrt(2)*sqrt(c)*x/sqrt(b
- sqrt(-4*a*c + b**2)))/(2*c**(3/2)*sqrt(b - sqrt(-4*a*c + b**2))*(-4*a*c + b**
2)**(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.988531, size = 282, normalized size = 1.04 \[ \frac{-\frac{2 \sqrt{c} x \left (a \left (b-2 c x^2\right )+b^2 x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\sqrt{2} \left (b^2 \sqrt{b^2-4 a c}-6 a c \sqrt{b^2-4 a c}+8 a b c-b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (b^2 \sqrt{b^2-4 a c}-6 a c \sqrt{b^2-4 a c}-8 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{4 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + b*x^2 + c*x^4)^2,x]

[Out]

((-2*Sqrt[c]*x*(b^2*x^2 + a*(b - 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4))
+ (Sqrt[2]*(-b^3 + 8*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 6*a*c*Sqrt[b^2 - 4*a*c])*Ar
cTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt
[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(b^3 - 8*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - 6*a
*c*Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(
(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(4*c^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.116, size = 2158, normalized size = 8. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(c*x^4+b*x^2+a)^2,x)

[Out]

(-1/2*(2*a*c-b^2)/c/(4*a*c-b^2)*x^3+1/2*a*b/c/(4*a*c-b^2)*x)/(c*x^4+b*x^2+a)+32*
c^3/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(4*a*b*c^2-b^3*c
+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2)*x*2^(1/2)/c/(
(4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2))*a^3*b-20*c^2/(-
c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2
*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2)*x*2^(1/2)/c/((4*a*c
-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2))*a^2*b^3+4*c/(-c^2*(4*
a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c
-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2)*x*2^(1/2)/c/((4*a*c-b^2)*(
4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2))*a*b^5-1/4/(-c^2*(4*a*c-b^2)^
3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^
(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2)*x*2^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2
-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2))*b^7+6*c/(4*a*c-b^2)*2^(1/2)/((4*a*c-b
^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^
2*c^2)*x*2^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1
/2))*a^2-5/2/(4*a*c-b^2)*2^(1/2)/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)
^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*c^3-2*b^2*c^2)*x*2^(1/2)/c/((4*a*c-b^2)*(4*a*b
*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2))*a*b^2+1/4/c/(4*a*c-b^2)*2^(1/2)/(
(4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2)))^(1/2)*arctan(1/2*(8*a*
c^3-2*b^2*c^2)*x*2^(1/2)/c/((4*a*c-b^2)*(4*a*b*c^2-b^3*c+(-c^2*(4*a*c-b^2)^3)^(1
/2)))^(1/2))*b^4-32*c^3/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*
c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3+2
*b^2*c^2)*x*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2)
)^(1/2))*a^3*b+20*c^2/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c^
2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3+2*b
^2*c^2)*x*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^
(1/2))*a^2*b^3-4*c/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c^2+b
^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3+2*b^2*
c^2)*x*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/
2))*a*b^5+1/4/(-c^2*(4*a*c-b^2)^3)^(1/2)/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c^2+b^3*c+
(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3+2*b^2*c^2)*
x*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2))*b
^7+6*c/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c
-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3+2*b^2*c^2)*x*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c
^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2))*a^2-5/2/(4*a*c-b^2)*2^(1/2)/((-4*a*
b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3
+2*b^2*c^2)*x*2^(1/2)/c/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^
2))^(1/2))*a*b^2+1/4/c/(4*a*c-b^2)*2^(1/2)/((-4*a*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^
3)^(1/2))*(4*a*c-b^2))^(1/2)*arctanh(1/2*(-8*a*c^3+2*b^2*c^2)*x*2^(1/2)/c/((-4*a
*b*c^2+b^3*c+(-c^2*(4*a*c-b^2)^3)^(1/2))*(4*a*c-b^2))^(1/2))*b^4

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{{\left (b^{2} - 2 \, a c\right )} x^{3} + a b x}{2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}\right )}} - \frac{-\int \frac{{\left (b^{2} - 6 \, a c\right )} x^{2} + a b}{c x^{4} + b x^{2} + a}\,{d x}}{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(c*x^4 + b*x^2 + a)^2,x, algorithm="maxima")

[Out]

-1/2*((b^2 - 2*a*c)*x^3 + a*b*x)/((b^2*c^2 - 4*a*c^3)*x^4 + a*b^2*c - 4*a^2*c^2
+ (b^3*c - 4*a*b*c^2)*x^2) - 1/2*integrate(-((b^2 - 6*a*c)*x^2 + a*b)/(c*x^4 + b
*x^2 + a), x)/(b^2*c - 4*a*c^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.307, size = 3047, normalized size = 11.24 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(c*x^4 + b*x^2 + a)^2,x, algorithm="fricas")

[Out]

-1/4*(2*(b^2 - 2*a*c)*x^3 + 2*a*b*x - sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a*b^2
*c - 4*a^2*c^2 + (b^3*c - 4*a*b*c^2)*x^2)*sqrt(-(b^5 - 15*a*b^3*c + 60*a^2*b*c^2
+ (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((b^4 - 18*a*b^2*c
+ 81*a^2*c^2)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3
- 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log((5*a*b^4 - 81*a^2*b^2*c + 32
4*a^3*c^2)*x + 1/2*sqrt(1/2)*(b^7 - 17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3
- (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7)*sqr
t((b^4 - 18*a*b^2*c + 81*a^2*c^2)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*
a^3*c^9)))*sqrt(-(b^5 - 15*a*b^3*c + 60*a^2*b*c^2 + (b^6*c^3 - 12*a*b^4*c^4 + 48
*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/(b^6*c^6 - 12*a*
b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^
5 - 64*a^3*c^6))) + sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a*b^2*c - 4*a^2*c^2 + (
b^3*c - 4*a*b*c^2)*x^2)*sqrt(-(b^5 - 15*a*b^3*c + 60*a^2*b*c^2 + (b^6*c^3 - 12*a
*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/(b^
6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 +
48*a^2*b^2*c^5 - 64*a^3*c^6))*log((5*a*b^4 - 81*a^2*b^2*c + 324*a^3*c^2)*x - 1/2
*sqrt(1/2)*(b^7 - 17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3 - (b^8*c^3 - 24*a*
b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7)*sqrt((b^4 - 18*a*b^2*
c + 81*a^2*c^2)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt(-(
b^5 - 15*a*b^3*c + 60*a^2*b*c^2 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*
a^3*c^6)*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b
^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)))
- sqrt(1/2)*((b^2*c^2 - 4*a*c^3)*x^4 + a*b^2*c - 4*a^2*c^2 + (b^3*c - 4*a*b*c^2)
*x^2)*sqrt(-(b^5 - 15*a*b^3*c + 60*a^2*b*c^2 - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*
b^2*c^5 - 64*a^3*c^6)*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/(b^6*c^6 - 12*a*b^4*c
^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 6
4*a^3*c^6))*log((5*a*b^4 - 81*a^2*b^2*c + 324*a^3*c^2)*x + 1/2*sqrt(1/2)*(b^7 -
17*a*b^5*c + 88*a^2*b^3*c^2 - 144*a^3*b*c^3 + (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*
b^4*c^5 - 640*a^3*b^2*c^6 + 768*a^4*c^7)*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/(b
^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt(-(b^5 - 15*a*b^3*c +
60*a^2*b*c^2 - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((b^4
- 18*a*b^2*c + 81*a^2*c^2)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^
9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))) + sqrt(1/2)*((b^2*
c^2 - 4*a*c^3)*x^4 + a*b^2*c - 4*a^2*c^2 + (b^3*c - 4*a*b*c^2)*x^2)*sqrt(-(b^5 -
15*a*b^3*c + 60*a^2*b*c^2 - (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c
^6)*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^
8 - 64*a^3*c^9)))/(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))*log((5
*a*b^4 - 81*a^2*b^2*c + 324*a^3*c^2)*x - 1/2*sqrt(1/2)*(b^7 - 17*a*b^5*c + 88*a^
2*b^3*c^2 - 144*a^3*b*c^3 + (b^8*c^3 - 24*a*b^6*c^4 + 192*a^2*b^4*c^5 - 640*a^3*
b^2*c^6 + 768*a^4*c^7)*sqrt((b^4 - 18*a*b^2*c + 81*a^2*c^2)/(b^6*c^6 - 12*a*b^4*
c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))*sqrt(-(b^5 - 15*a*b^3*c + 60*a^2*b*c^2 - (b
^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*sqrt((b^4 - 18*a*b^2*c + 81
*a^2*c^2)/(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)))/(b^6*c^3 - 12
*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6))))/((b^2*c^2 - 4*a*c^3)*x^4 + a*b^2*c
- 4*a^2*c^2 + (b^3*c - 4*a*b*c^2)*x^2)

_______________________________________________________________________________________

Sympy [A]  time = 15.2759, size = 379, normalized size = 1.4 \[ - \frac{- a b x + x^{3} \left (2 a c - b^{2}\right )}{8 a^{2} c^{2} - 2 a b^{2} c + x^{4} \left (8 a c^{3} - 2 b^{2} c^{2}\right ) + x^{2} \left (8 a b c^{2} - 2 b^{3} c\right )} + \operatorname{RootSum}{\left (t^{4} \left (1048576 a^{6} c^{9} - 1572864 a^{5} b^{2} c^{8} + 983040 a^{4} b^{4} c^{7} - 327680 a^{3} b^{6} c^{6} + 61440 a^{2} b^{8} c^{5} - 6144 a b^{10} c^{4} + 256 b^{12} c^{3}\right ) + t^{2} \left (- 61440 a^{5} b c^{5} + 61440 a^{4} b^{3} c^{4} - 24064 a^{3} b^{5} c^{3} + 4608 a^{2} b^{7} c^{2} - 432 a b^{9} c + 16 b^{11}\right ) + 1296 a^{5} c^{2} - 360 a^{4} b^{2} c + 25 a^{3} b^{4}, \left ( t \mapsto t \log{\left (x + \frac{49152 t^{3} a^{4} c^{7} - 40960 t^{3} a^{3} b^{2} c^{6} + 12288 t^{3} a^{2} b^{4} c^{5} - 1536 t^{3} a b^{6} c^{4} + 64 t^{3} b^{8} c^{3} - 1728 t a^{3} b c^{3} + 656 t a^{2} b^{3} c^{2} - 88 t a b^{5} c + 4 t b^{7}}{324 a^{3} c^{2} - 81 a^{2} b^{2} c + 5 a b^{4}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(c*x**4+b*x**2+a)**2,x)

[Out]

-(-a*b*x + x**3*(2*a*c - b**2))/(8*a**2*c**2 - 2*a*b**2*c + x**4*(8*a*c**3 - 2*b
**2*c**2) + x**2*(8*a*b*c**2 - 2*b**3*c)) + RootSum(_t**4*(1048576*a**6*c**9 - 1
572864*a**5*b**2*c**8 + 983040*a**4*b**4*c**7 - 327680*a**3*b**6*c**6 + 61440*a*
*2*b**8*c**5 - 6144*a*b**10*c**4 + 256*b**12*c**3) + _t**2*(-61440*a**5*b*c**5 +
61440*a**4*b**3*c**4 - 24064*a**3*b**5*c**3 + 4608*a**2*b**7*c**2 - 432*a*b**9*
c + 16*b**11) + 1296*a**5*c**2 - 360*a**4*b**2*c + 25*a**3*b**4, Lambda(_t, _t*l
og(x + (49152*_t**3*a**4*c**7 - 40960*_t**3*a**3*b**2*c**6 + 12288*_t**3*a**2*b*
*4*c**5 - 1536*_t**3*a*b**6*c**4 + 64*_t**3*b**8*c**3 - 1728*_t*a**3*b*c**3 + 65
6*_t*a**2*b**3*c**2 - 88*_t*a*b**5*c + 4*_t*b**7)/(324*a**3*c**2 - 81*a**2*b**2*
c + 5*a*b**4))))

_______________________________________________________________________________________

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^6/(c*x^4 + b*x^2 + a)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError