3.883 \(\int \frac{x^6}{\left (a+b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=298 \[ \frac{3 x \left (x^2 \left (4 a c+b^2\right )+4 a b\right )}{8 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x^3 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 \left (-\frac{b \left (12 a c+b^2\right )}{\sqrt{b^2-4 a c}}+4 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \left (\frac{b \left (12 a c+b^2\right )}{\sqrt{b^2-4 a c}}+4 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}} \]

[Out]

(x^3*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*x*(4*a*b + (b^2
+ 4*a*c)*x^2))/(8*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*(b^2 + 4*a*c - (b*(
b^2 + 12*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 -
4*a*c]]])/(8*Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*
(b^2 + 4*a*c + (b*(b^2 + 12*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/
Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b
^2 - 4*a*c]])

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Rubi [A]  time = 1.5028, antiderivative size = 298, 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 \[ \frac{3 x \left (x^2 \left (4 a c+b^2\right )+4 a b\right )}{8 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x^3 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{3 \left (-\frac{b \left (12 a c+b^2\right )}{\sqrt{b^2-4 a c}}+4 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{8 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \left (\frac{b \left (12 a c+b^2\right )}{\sqrt{b^2-4 a c}}+4 a c+b^2\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{8 \sqrt{2} \sqrt{c} \left (b^2-4 a c\right )^2 \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + b*x^2 + c*x^4)^3,x]

[Out]

(x^3*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (3*x*(4*a*b + (b^2
+ 4*a*c)*x^2))/(8*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (3*(b^2 + 4*a*c - (b*(
b^2 + 12*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 -
4*a*c]]])/(8*Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^2*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*
(b^2 + 4*a*c + (b*(b^2 + 12*a*c))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/
Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(8*Sqrt[2]*Sqrt[c]*(b^2 - 4*a*c)^2*Sqrt[b + Sqrt[b
^2 - 4*a*c]])

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Rubi in Sympy [A]  time = 76.4623, size = 286, normalized size = 0.96 \[ \frac{x^{3} \left (2 a + b x^{2}\right )}{4 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{2}} + \frac{x \left (12 a b + x^{2} \left (12 a c + 3 b^{2}\right )\right )}{8 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x^{2} + c x^{4}\right )} + \frac{3 \sqrt{2} \left (b \left (12 a c + b^{2}\right ) + \sqrt{- 4 a c + b^{2}} \left (4 a c + b^{2}\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{16 \sqrt{c} \sqrt{b + \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} - \frac{3 \sqrt{2} \left (b \left (12 a c + b^{2}\right ) - \sqrt{- 4 a c + b^{2}} \left (4 a c + b^{2}\right )\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{16 \sqrt{c} \sqrt{b - \sqrt{- 4 a c + b^{2}}} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(c*x**4+b*x**2+a)**3,x)

[Out]

x**3*(2*a + b*x**2)/(4*(-4*a*c + b**2)*(a + b*x**2 + c*x**4)**2) + x*(12*a*b + x
**2*(12*a*c + 3*b**2))/(8*(-4*a*c + b**2)**2*(a + b*x**2 + c*x**4)) + 3*sqrt(2)*
(b*(12*a*c + b**2) + sqrt(-4*a*c + b**2)*(4*a*c + b**2))*atan(sqrt(2)*sqrt(c)*x/
sqrt(b + sqrt(-4*a*c + b**2)))/(16*sqrt(c)*sqrt(b + sqrt(-4*a*c + b**2))*(-4*a*c
+ b**2)**(5/2)) - 3*sqrt(2)*(b*(12*a*c + b**2) - sqrt(-4*a*c + b**2)*(4*a*c + b
**2))*atan(sqrt(2)*sqrt(c)*x/sqrt(b - sqrt(-4*a*c + b**2)))/(16*sqrt(c)*sqrt(b -
sqrt(-4*a*c + b**2))*(-4*a*c + b**2)**(5/2))

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Mathematica [A]  time = 1.60246, size = 343, normalized size = 1.15 \[ \frac{-\frac{4 \left (a x \left (b-2 c x^2\right )+b^2 x^3\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{8 a b c x+24 a c^2 x^3+4 b^3 x+6 b^2 c x^3}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{3 \sqrt{2} \sqrt{c} \left (b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}-12 a b c-b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{3 \sqrt{2} \sqrt{c} \left (b^2 \sqrt{b^2-4 a c}+4 a c \sqrt{b^2-4 a c}+12 a b c+b^3\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} x}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\left (b^2-4 a c\right )^{5/2} \sqrt{\sqrt{b^2-4 a c}+b}}}{16 c} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + b*x^2 + c*x^4)^3,x]

[Out]

((4*b^3*x + 8*a*b*c*x + 6*b^2*c*x^3 + 24*a*c^2*x^3)/((b^2 - 4*a*c)^2*(a + b*x^2
+ c*x^4)) - (4*(b^2*x^3 + a*x*(b - 2*c*x^2)))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)
^2) + (3*Sqrt[2]*Sqrt[c]*(-b^3 - 12*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 4*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)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (3*Sqrt[2]*Sqrt[c]*(b^3 + 12*a*b*c + b
^2*Sqrt[b^2 - 4*a*c] + 4*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)^(5/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(16*
c)

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Maple [B]  time = 0.128, size = 4017, normalized size = 13.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(c*x^4+b*x^2+a)^3,x)

[Out]

(3/8*c*(4*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^7+1/8*b*(16*a*c+5*b^2)/(16*a^2*c
^2-8*a*b^2*c+b^4)*x^5-1/8*(4*a*c-19*b^2)*a/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+3/2*a^
2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x)/(c*x^4+b*x^2+a)^2+576*c^5/(-(4*a*c-b^2)^5)^(1/
2)/(16*a^2*c^2-8*a*b^2*c+b^4)*2^(1/2)/((16*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^
2-8*a*b^3*c+b^5+(-(4*a*c-b^2)^5)^(1/2)))^(1/2)*arctan(1/2*(32*a^2*c^3-16*a*b^2*c
^2+2*b^4*c)*x*2^(1/2)/((16*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+
(-(4*a*c-b^2)^5)^(1/2)))^(1/2))*a^5*b-528*c^4/(-(4*a*c-b^2)^5)^(1/2)/(16*a^2*c^2
-8*a*b^2*c+b^4)*2^(1/2)/((16*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^
5+(-(4*a*c-b^2)^5)^(1/2)))^(1/2)*arctan(1/2*(32*a^2*c^3-16*a*b^2*c^2+2*b^4*c)*x*
2^(1/2)/((16*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-b^2)^
5)^(1/2)))^(1/2))*a^4*b^3+168*c^3/(-(4*a*c-b^2)^5)^(1/2)/(16*a^2*c^2-8*a*b^2*c+b
^4)*2^(1/2)/((16*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-b
^2)^5)^(1/2)))^(1/2)*arctan(1/2*(32*a^2*c^3-16*a*b^2*c^2+2*b^4*c)*x*2^(1/2)/((16
*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-b^2)^5)^(1/2)))^(
1/2))*a^3*b^5-18*c^2/(-(4*a*c-b^2)^5)^(1/2)/(16*a^2*c^2-8*a*b^2*c+b^4)*2^(1/2)/(
(16*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-b^2)^5)^(1/2))
)^(1/2)*arctan(1/2*(32*a^2*c^3-16*a*b^2*c^2+2*b^4*c)*x*2^(1/2)/((16*a^2*c^2-8*a*
b^2*c+b^4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-b^2)^5)^(1/2)))^(1/2))*a^2*b^7
+18/(-(4*a*c-b^2)^5)^(1/2)/(16*a^2*c^2-8*a*b^2*c+b^4)*c^2*2^(1/2)/((-16*a^2*b*c^
2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b^2*c+b^4)*c)^(1/2)*arct
anh(1/2*(-32*a^2*c^3+16*a*b^2*c^2-2*b^4*c)*x*2^(1/2)/((-16*a^2*b*c^2+8*a*b^3*c-b
^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b^2*c+b^4)*c)^(1/2))*a^2*b^7+3/16/(-(
4*a*c-b^2)^5)^(1/2)/(16*a^2*c^2-8*a*b^2*c+b^4)*2^(1/2)/((16*a^2*c^2-8*a*b^2*c+b^
4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-b^2)^5)^(1/2)))^(1/2)*arctan(1/2*(32*a
^2*c^3-16*a*b^2*c^2+2*b^4*c)*x*2^(1/2)/((16*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c
^2-8*a*b^3*c+b^5+(-(4*a*c-b^2)^5)^(1/2)))^(1/2))*b^11-3/(16*a^2*c^2-8*a*b^2*c+b^
4)*2^(1/2)/((16*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-b^
2)^5)^(1/2)))^(1/2)*arctan(1/2*(32*a^2*c^3-16*a*b^2*c^2+2*b^4*c)*x*2^(1/2)/((16*
a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-b^2)^5)^(1/2)))^(1
/2))*a^2*b^2*c^2+3/4*c/(-(4*a*c-b^2)^5)^(1/2)/(16*a^2*c^2-8*a*b^2*c+b^4)*2^(1/2)
/((-16*a^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b^2*c+b^4
)*c)^(1/2)*arctanh(1/2*(-32*a^2*c^3+16*a*b^2*c^2-2*b^4*c)*x*2^(1/2)/((-16*a^2*b*
c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b^2*c+b^4)*c)^(1/2))*a
*b^9-3/16/(-(4*a*c-b^2)^5)^(1/2)/(16*a^2*c^2-8*a*b^2*c+b^4)*2^(1/2)/((-16*a^2*b*
c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b^2*c+b^4)*c)^(1/2)*ar
ctanh(1/2*(-32*a^2*c^3+16*a*b^2*c^2-2*b^4*c)*x*2^(1/2)/((-16*a^2*b*c^2+8*a*b^3*c
-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b^2*c+b^4)*c)^(1/2))*b^11-3/(16*a^2
*c^2-8*a*b^2*c+b^4)*2^(1/2)/((-16*a^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2)
)*(16*a^2*c^2-8*a*b^2*c+b^4)*c)^(1/2)*arctanh(1/2*(-32*a^2*c^3+16*a*b^2*c^2-2*b^
4*c)*x*2^(1/2)/((-16*a^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2
-8*a*b^2*c+b^4)*c)^(1/2))*a^2*b^2*c^2-576/(-(4*a*c-b^2)^5)^(1/2)/(16*a^2*c^2-8*a
*b^2*c+b^4)*c^5*2^(1/2)/((-16*a^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(1
6*a^2*c^2-8*a*b^2*c+b^4)*c)^(1/2)*arctanh(1/2*(-32*a^2*c^3+16*a*b^2*c^2-2*b^4*c)
*x*2^(1/2)/((-16*a^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a
*b^2*c+b^4)*c)^(1/2))*a^5*b+528/(-(4*a*c-b^2)^5)^(1/2)/(16*a^2*c^2-8*a*b^2*c+b^4
)*c^4*2^(1/2)/((-16*a^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-
8*a*b^2*c+b^4)*c)^(1/2)*arctanh(1/2*(-32*a^2*c^3+16*a*b^2*c^2-2*b^4*c)*x*2^(1/2)
/((-16*a^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b^2*c+b^4
)*c)^(1/2))*a^4*b^3-168/(-(4*a*c-b^2)^5)^(1/2)/(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*2^
(1/2)/((-16*a^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b^2*
c+b^4)*c)^(1/2)*arctanh(1/2*(-32*a^2*c^3+16*a*b^2*c^2-2*b^4*c)*x*2^(1/2)/((-16*a
^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b^2*c+b^4)*c)^(1/
2))*a^3*b^5+12/(16*a^2*c^2-8*a*b^2*c+b^4)*2^(1/2)/((16*a^2*c^2-8*a*b^2*c+b^4)*c*
(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-b^2)^5)^(1/2)))^(1/2)*arctan(1/2*(32*a^2*c^
3-16*a*b^2*c^2+2*b^4*c)*x*2^(1/2)/((16*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2-8*
a*b^3*c+b^5+(-(4*a*c-b^2)^5)^(1/2)))^(1/2))*a^3*c^3-3/4*c/(16*a^2*c^2-8*a*b^2*c+
b^4)*2^(1/2)/((16*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-
b^2)^5)^(1/2)))^(1/2)*arctan(1/2*(32*a^2*c^3-16*a*b^2*c^2+2*b^4*c)*x*2^(1/2)/((1
6*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-b^2)^5)^(1/2)))^
(1/2))*a*b^4+3/16/(16*a^2*c^2-8*a*b^2*c+b^4)*2^(1/2)/((16*a^2*c^2-8*a*b^2*c+b^4)
*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-b^2)^5)^(1/2)))^(1/2)*arctan(1/2*(32*a^2
*c^3-16*a*b^2*c^2+2*b^4*c)*x*2^(1/2)/((16*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2
-8*a*b^3*c+b^5+(-(4*a*c-b^2)^5)^(1/2)))^(1/2))*b^6+12/(16*a^2*c^2-8*a*b^2*c+b^4)
*2^(1/2)/((-16*a^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b
^2*c+b^4)*c)^(1/2)*arctanh(1/2*(-32*a^2*c^3+16*a*b^2*c^2-2*b^4*c)*x*2^(1/2)/((-1
6*a^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b^2*c+b^4)*c)^
(1/2))*a^3*c^3-3/4*c/(16*a^2*c^2-8*a*b^2*c+b^4)*2^(1/2)/((-16*a^2*b*c^2+8*a*b^3*
c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b^2*c+b^4)*c)^(1/2)*arctanh(1/2*(-
32*a^2*c^3+16*a*b^2*c^2-2*b^4*c)*x*2^(1/2)/((-16*a^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*
c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b^2*c+b^4)*c)^(1/2))*a*b^4+3/16/(16*a^2*c^2-8*a
*b^2*c+b^4)*2^(1/2)/((-16*a^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^
2*c^2-8*a*b^2*c+b^4)*c)^(1/2)*arctanh(1/2*(-32*a^2*c^3+16*a*b^2*c^2-2*b^4*c)*x*2
^(1/2)/((-16*a^2*b*c^2+8*a*b^3*c-b^5+(-(4*a*c-b^2)^5)^(1/2))*(16*a^2*c^2-8*a*b^2
*c+b^4)*c)^(1/2))*b^6-3/4*c/(-(4*a*c-b^2)^5)^(1/2)/(16*a^2*c^2-8*a*b^2*c+b^4)*2^
(1/2)/((16*a^2*c^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-b^2)^5)
^(1/2)))^(1/2)*arctan(1/2*(32*a^2*c^3-16*a*b^2*c^2+2*b^4*c)*x*2^(1/2)/((16*a^2*c
^2-8*a*b^2*c+b^4)*c*(16*a^2*b*c^2-8*a*b^3*c+b^5+(-(4*a*c-b^2)^5)^(1/2)))^(1/2))*
a*b^9

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 \,{\left (b^{2} c + 4 \, a c^{2}\right )} x^{7} +{\left (5 \, b^{3} + 16 \, a b c\right )} x^{5} + 12 \, a^{2} b x +{\left (19 \, a b^{2} - 4 \, a^{2} c\right )} x^{3}}{8 \,{\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{8} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{6} + a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{4} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x^{2}\right )}} + \frac{3 \, \int \frac{{\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, a b}{c x^{4} + b x^{2} + a}\,{d x}}{8 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(c*x^4 + b*x^2 + a)^3,x, algorithm="maxima")

[Out]

1/8*(3*(b^2*c + 4*a*c^2)*x^7 + (5*b^3 + 16*a*b*c)*x^5 + 12*a^2*b*x + (19*a*b^2 -
4*a^2*c)*x^3)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^
2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c +
32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2) + 3/8*integrate(((
b^2 + 4*a*c)*x^2 - 4*a*b)/(c*x^4 + b*x^2 + a), x)/(b^4 - 8*a*b^2*c + 16*a^2*c^2)

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Fricas [A]  time = 0.324563, size = 4223, normalized size = 14.17 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(c*x^4 + b*x^2 + a)^3,x, algorithm="fricas")

[Out]

1/16*(6*(b^2*c + 4*a*c^2)*x^7 + 2*(5*b^3 + 16*a*b*c)*x^5 + 24*a^2*b*x + 2*(19*a*
b^2 - 4*a^2*c)*x^3 + 3*sqrt(1/2)*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(
b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (
b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2)*
sqrt(-(b^5 + 40*a*b^3*c + 80*a^2*b*c^2 + (b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^
3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)/sqrt(b^10*c^2 - 20*a*b^8*
c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7))/(b^1
0*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024
*a^5*c^6))*log(3*(5*b^4 + 40*a*b^2*c + 16*a^2*c^2)*x + 3*sqrt(1/2)*(2*b^7 - 24*a
*b^5*c + 96*a^2*b^3*c^2 - 128*a^3*b*c^3 + (3*b^12*c - 56*a*b^10*c^2 + 400*a^2*b^
8*c^3 - 1280*a^3*b^6*c^4 + 1280*a^4*b^4*c^5 + 2048*a^5*b^2*c^6 - 4096*a^6*c^7)/s
qrt(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c
^6 - 1024*a^5*c^7))*sqrt(-(b^5 + 40*a*b^3*c + 80*a^2*b*c^2 + (b^10*c - 20*a*b^8*
c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)/sqrt(
b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 -
1024*a^5*c^7))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 128
0*a^4*b^2*c^5 - 1024*a^5*c^6))) - 3*sqrt(1/2)*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c
^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 1
6*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3
*b*c^2)*x^2)*sqrt(-(b^5 + 40*a*b^3*c + 80*a^2*b*c^2 + (b^10*c - 20*a*b^8*c^2 + 1
60*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)/sqrt(b^10*c^
2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a
^5*c^7))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b
^2*c^5 - 1024*a^5*c^6))*log(3*(5*b^4 + 40*a*b^2*c + 16*a^2*c^2)*x - 3*sqrt(1/2)*
(2*b^7 - 24*a*b^5*c + 96*a^2*b^3*c^2 - 128*a^3*b*c^3 + (3*b^12*c - 56*a*b^10*c^2
+ 400*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 1280*a^4*b^4*c^5 + 2048*a^5*b^2*c^6 - 40
96*a^6*c^7)/sqrt(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1
280*a^4*b^2*c^6 - 1024*a^5*c^7))*sqrt(-(b^5 + 40*a*b^3*c + 80*a^2*b*c^2 + (b^10*
c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a
^5*c^6)/sqrt(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*
a^4*b^2*c^6 - 1024*a^5*c^7))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*
b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6))) + 3*sqrt(1/2)*((b^4*c^2 - 8*a*b^2*c
^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*
a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b
^3*c + 16*a^3*b*c^2)*x^2)*sqrt(-(b^5 + 40*a*b^3*c + 80*a^2*b*c^2 - (b^10*c - 20*
a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6)
/sqrt(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2
*c^6 - 1024*a^5*c^7))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4
+ 1280*a^4*b^2*c^5 - 1024*a^5*c^6))*log(3*(5*b^4 + 40*a*b^2*c + 16*a^2*c^2)*x +
3*sqrt(1/2)*(2*b^7 - 24*a*b^5*c + 96*a^2*b^3*c^2 - 128*a^3*b*c^3 - (3*b^12*c -
56*a*b^10*c^2 + 400*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 1280*a^4*b^4*c^5 + 2048*a^5
*b^2*c^6 - 4096*a^6*c^7)/sqrt(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^
3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7))*sqrt(-(b^5 + 40*a*b^3*c + 80*a^2*b
*c^2 - (b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2
*c^5 - 1024*a^5*c^6)/sqrt(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^
4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c
^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6))) - 3*sqrt(1/2)*((b^4*c^
2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 +
a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*
b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2)*sqrt(-(b^5 + 40*a*b^3*c + 80*a^2*b*c^2 -
(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 -
1024*a^5*c^6)/sqrt(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4 - 640*a^3*b^4*c^5 +
1280*a^4*b^2*c^6 - 1024*a^5*c^7))/(b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 64
0*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6))*log(3*(5*b^4 + 40*a*b^2*c + 16
*a^2*c^2)*x - 3*sqrt(1/2)*(2*b^7 - 24*a*b^5*c + 96*a^2*b^3*c^2 - 128*a^3*b*c^3 -
(3*b^12*c - 56*a*b^10*c^2 + 400*a^2*b^8*c^3 - 1280*a^3*b^6*c^4 + 1280*a^4*b^4*c
^5 + 2048*a^5*b^2*c^6 - 4096*a^6*c^7)/sqrt(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6
*c^4 - 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7))*sqrt(-(b^5 + 40*a*b^3
*c + 80*a^2*b*c^2 - (b^10*c - 20*a*b^8*c^2 + 160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 +
1280*a^4*b^2*c^5 - 1024*a^5*c^6)/sqrt(b^10*c^2 - 20*a*b^8*c^3 + 160*a^2*b^6*c^4
- 640*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 - 1024*a^5*c^7))/(b^10*c - 20*a*b^8*c^2 +
160*a^2*b^6*c^3 - 640*a^3*b^4*c^4 + 1280*a^4*b^2*c^5 - 1024*a^5*c^6))))/((b^4*c^
2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 +
a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*
b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2)

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Sympy [A]  time = 47.6673, size = 627, normalized size = 2.1 \[ \frac{12 a^{2} b x + x^{7} \left (12 a c^{2} + 3 b^{2} c\right ) + x^{5} \left (16 a b c + 5 b^{3}\right ) + x^{3} \left (- 4 a^{2} c + 19 a b^{2}\right )}{128 a^{4} c^{2} - 64 a^{3} b^{2} c + 8 a^{2} b^{4} + x^{8} \left (128 a^{2} c^{4} - 64 a b^{2} c^{3} + 8 b^{4} c^{2}\right ) + x^{6} \left (256 a^{2} b c^{3} - 128 a b^{3} c^{2} + 16 b^{5} c\right ) + x^{4} \left (256 a^{3} c^{3} - 48 a b^{4} c + 8 b^{6}\right ) + x^{2} \left (256 a^{3} b c^{2} - 128 a^{2} b^{3} c + 16 a b^{5}\right )} + \operatorname{RootSum}{\left (t^{4} \left (68719476736 a^{10} c^{11} - 171798691840 a^{9} b^{2} c^{10} + 193273528320 a^{8} b^{4} c^{9} - 128849018880 a^{7} b^{6} c^{8} + 56371445760 a^{6} b^{8} c^{7} - 16911433728 a^{5} b^{10} c^{6} + 3523215360 a^{4} b^{12} c^{5} - 503316480 a^{3} b^{14} c^{4} + 47185920 a^{2} b^{16} c^{3} - 2621440 a b^{18} c^{2} + 65536 b^{20} c\right ) + t^{2} \left (- 188743680 a^{7} b c^{7} + 141557760 a^{6} b^{3} c^{6} - 2359296 a^{5} b^{5} c^{5} - 26542080 a^{4} b^{7} c^{4} + 9584640 a^{3} b^{9} c^{3} - 1290240 a^{2} b^{11} c^{2} + 46080 a b^{13} c + 2304 b^{15}\right ) + 20736 a^{5} c^{4} + 103680 a^{4} b^{2} c^{3} + 142560 a^{3} b^{4} c^{2} + 32400 a^{2} b^{6} c + 2025 a b^{8}, \left ( t \mapsto t \log{\left (x + \frac{33554432 t^{3} a^{6} c^{7} - 16777216 t^{3} a^{5} b^{2} c^{6} - 10485760 t^{3} a^{4} b^{4} c^{5} + 10485760 t^{3} a^{3} b^{6} c^{4} - 3276800 t^{3} a^{2} b^{8} c^{3} + 458752 t^{3} a b^{10} c^{2} - 24576 t^{3} b^{12} c - 64512 t a^{3} b c^{3} - 43776 t a^{2} b^{3} c^{2} - 21312 t a b^{5} c - 144 t b^{7}}{432 a^{2} c^{2} + 1080 a b^{2} c + 135 b^{4}} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(c*x**4+b*x**2+a)**3,x)

[Out]

(12*a**2*b*x + x**7*(12*a*c**2 + 3*b**2*c) + x**5*(16*a*b*c + 5*b**3) + x**3*(-4
*a**2*c + 19*a*b**2))/(128*a**4*c**2 - 64*a**3*b**2*c + 8*a**2*b**4 + x**8*(128*
a**2*c**4 - 64*a*b**2*c**3 + 8*b**4*c**2) + x**6*(256*a**2*b*c**3 - 128*a*b**3*c
**2 + 16*b**5*c) + x**4*(256*a**3*c**3 - 48*a*b**4*c + 8*b**6) + x**2*(256*a**3*
b*c**2 - 128*a**2*b**3*c + 16*a*b**5)) + RootSum(_t**4*(68719476736*a**10*c**11
- 171798691840*a**9*b**2*c**10 + 193273528320*a**8*b**4*c**9 - 128849018880*a**7
*b**6*c**8 + 56371445760*a**6*b**8*c**7 - 16911433728*a**5*b**10*c**6 + 35232153
60*a**4*b**12*c**5 - 503316480*a**3*b**14*c**4 + 47185920*a**2*b**16*c**3 - 2621
440*a*b**18*c**2 + 65536*b**20*c) + _t**2*(-188743680*a**7*b*c**7 + 141557760*a*
*6*b**3*c**6 - 2359296*a**5*b**5*c**5 - 26542080*a**4*b**7*c**4 + 9584640*a**3*b
**9*c**3 - 1290240*a**2*b**11*c**2 + 46080*a*b**13*c + 2304*b**15) + 20736*a**5*
c**4 + 103680*a**4*b**2*c**3 + 142560*a**3*b**4*c**2 + 32400*a**2*b**6*c + 2025*
a*b**8, Lambda(_t, _t*log(x + (33554432*_t**3*a**6*c**7 - 16777216*_t**3*a**5*b*
*2*c**6 - 10485760*_t**3*a**4*b**4*c**5 + 10485760*_t**3*a**3*b**6*c**4 - 327680
0*_t**3*a**2*b**8*c**3 + 458752*_t**3*a*b**10*c**2 - 24576*_t**3*b**12*c - 64512
*_t*a**3*b*c**3 - 43776*_t*a**2*b**3*c**2 - 21312*_t*a*b**5*c - 144*_t*b**7)/(43
2*a**2*c**2 + 1080*a*b**2*c + 135*b**4))))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 31.299, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(c*x^4 + b*x^2 + a)^3,x, algorithm="giac")

[Out]

Done