3.325 \(\int \frac{1}{x^2 \left (a+b x^4+c x^8\right )} \, dx\)

Optimal. Leaf size=363 \[ -\frac{\sqrt [4]{c} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{\sqrt{b^2-4 a c}-b}}-\frac{1}{a x} \]

[Out]

-(1/(a*x)) - (c^(1/4)*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b -
Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - (c^(1
/4)*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c]
)^(1/4)])/(2*2^(3/4)*a*(-b + Sqrt[b^2 - 4*a*c])^(1/4)) + (c^(1/4)*(1 - b/Sqrt[b^
2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3
/4)*a*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) + (c^(1/4)*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTa
nh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b + Sqrt[
b^2 - 4*a*c])^(1/4))

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Rubi [A]  time = 0.896001, antiderivative size = 363, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{\sqrt [4]{c} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{-\sqrt{b^2-4 a c}-b}}-\frac{\sqrt [4]{c} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \left (1-\frac{b}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{-\sqrt{b^2-4 a c}-b}}+\frac{\sqrt [4]{c} \left (\frac{b}{\sqrt{b^2-4 a c}}+1\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2\ 2^{3/4} a \sqrt [4]{\sqrt{b^2-4 a c}-b}}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x^4 + c*x^8)),x]

[Out]

-(1/(a*x)) - (c^(1/4)*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b -
Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) - (c^(1
/4)*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c]
)^(1/4)])/(2*2^(3/4)*a*(-b + Sqrt[b^2 - 4*a*c])^(1/4)) + (c^(1/4)*(1 - b/Sqrt[b^
2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3
/4)*a*(-b - Sqrt[b^2 - 4*a*c])^(1/4)) + (c^(1/4)*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTa
nh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(3/4)*a*(-b + Sqrt[
b^2 - 4*a*c])^(1/4))

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Rubi in Sympy [A]  time = 129.01, size = 357, normalized size = 0.98 \[ - \frac{\sqrt [4]{2} \sqrt [4]{c} \left (b + \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 a \sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt [4]{2} \sqrt [4]{c} \left (b + \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 a \sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt [4]{2} \sqrt [4]{c} \left (b - \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{4 a \sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{\sqrt [4]{2} \sqrt [4]{c} \left (b - \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{4 a \sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(c*x**8+b*x**4+a),x)

[Out]

-2**(1/4)*c**(1/4)*(b + sqrt(-4*a*c + b**2))*atan(2**(1/4)*c**(1/4)*x/(-b + sqrt
(-4*a*c + b**2))**(1/4))/(4*a*(-b + sqrt(-4*a*c + b**2))**(1/4)*sqrt(-4*a*c + b*
*2)) + 2**(1/4)*c**(1/4)*(b + sqrt(-4*a*c + b**2))*atanh(2**(1/4)*c**(1/4)*x/(-b
+ sqrt(-4*a*c + b**2))**(1/4))/(4*a*(-b + sqrt(-4*a*c + b**2))**(1/4)*sqrt(-4*a
*c + b**2)) + 2**(1/4)*c**(1/4)*(b - sqrt(-4*a*c + b**2))*atan(2**(1/4)*c**(1/4)
*x/(-b - sqrt(-4*a*c + b**2))**(1/4))/(4*a*(-b - sqrt(-4*a*c + b**2))**(1/4)*sqr
t(-4*a*c + b**2)) - 2**(1/4)*c**(1/4)*(b - sqrt(-4*a*c + b**2))*atanh(2**(1/4)*c
**(1/4)*x/(-b - sqrt(-4*a*c + b**2))**(1/4))/(4*a*(-b - sqrt(-4*a*c + b**2))**(1
/4)*sqrt(-4*a*c + b**2)) - 1/(a*x)

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Mathematica [C]  time = 0.0527092, size = 71, normalized size = 0.2 \[ -\frac{\text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 c \log (x-\text{$\#$1})+b \log (x-\text{$\#$1})}{2 \text{$\#$1}^5 c+\text{$\#$1} b}\&\right ]}{4 a}-\frac{1}{a x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*x^4 + c*x^8)),x]

[Out]

-(1/(a*x)) - RootSum[a + b*#1^4 + c*#1^8 & , (b*Log[x - #1] + c*Log[x - #1]*#1^4
)/(b*#1 + 2*c*#1^5) & ]/(4*a)

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Maple [C]  time = 0.002, size = 63, normalized size = 0.2 \[ -{\frac{1}{4\,a}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+{{\it \_Z}}^{4}b+a \right ) }{\frac{ \left ({{\it \_R}}^{6}c+{{\it \_R}}^{2}b \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b}}}-{\frac{1}{ax}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(c*x^8+b*x^4+a),x)

[Out]

-1/4/a*sum((_R^6*c+_R^2*b)/(2*_R^7*c+_R^3*b)*ln(x-_R),_R=RootOf(_Z^8*c+_Z^4*b+a)
)-1/a/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{\int \frac{c x^{6} + b x^{2}}{c x^{8} + b x^{4} + a}\,{d x}}{a} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^8 + b*x^4 + a)*x^2),x, algorithm="maxima")

[Out]

-integrate((c*x^6 + b*x^2)/(c*x^8 + b*x^4 + a), x)/a - 1/(a*x)

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Fricas [A]  time = 0.754121, size = 6971, normalized size = 19.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^8 + b*x^4 + a)*x^2),x, algorithm="fricas")

[Out]

1/4*(4*a*x*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + (a^5*b^4 - 8*a^
6*b^2*c + 16*a^7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a
^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^4 -
8*a^6*b^2*c + 16*a^7*c^2)))*arctan(1/2*sqrt(1/2)*(b^11 - 13*a*b^9*c + 63*a^2*b^7
*c^2 - 138*a^3*b^5*c^3 + 128*a^4*b^3*c^4 - 32*a^5*b*c^5 - (a^5*b^10 - 16*a^6*b^8
*c + 98*a^7*b^6*c^2 - 280*a^8*b^4*c^3 + 352*a^9*b^2*c^4 - 128*a^10*c^5)*sqrt((b^
8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^
4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5
*a^2*b*c^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2
*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2
- 64*a^13*c^3)))/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)))*sqrt(-(b^5 - 5*a*b^3*c +
5*a^2*b*c^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a
^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^
2 - 64*a^13*c^3)))/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2))/((b^4*c^4 - 3*a*b^2*c^5
+ a^2*c^6)*x + sqrt(1/2)*(b^4*c^4 - 3*a*b^2*c^5 + a^2*c^6)*sqrt((2*(b^4*c^3 - 3
*a*b^2*c^4 + a^2*c^5)*x^2 - sqrt(1/2)*(b^9 - 10*a*b^7*c + 34*a^2*b^5*c^2 - 43*a^
3*b^3*c^3 + 12*a^4*b*c^4 - (a^5*b^8 - 13*a^6*b^6*c + 60*a^7*b^4*c^2 - 112*a^8*b^
2*c^3 + 64*a^9*c^4)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4
*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))*sqrt(-(b^5 -
5*a*b^3*c + 5*a^2*b*c^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*sqrt((b^8 - 6*a*b
^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*
a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)))/(b^4*c^3 -
3*a*b^2*c^4 + a^2*c^5)))) - 4*a*x*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*
b*c^2 - (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*
c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*
a^13*c^3)))/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)))*arctan(-1/2*sqrt(1/2)*(b^11 -
13*a*b^9*c + 63*a^2*b^7*c^2 - 138*a^3*b^5*c^3 + 128*a^4*b^3*c^4 - 32*a^5*b*c^5
+ (a^5*b^10 - 16*a^6*b^8*c + 98*a^7*b^6*c^2 - 280*a^8*b^4*c^3 + 352*a^9*b^2*c^4
- 128*a^10*c^5)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4
)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))*sqrt(sqrt(1/2)*sq
rt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 - (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*sqrt((
b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*
b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)))*
sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 - (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*sqrt
((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^1
1*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2))
/((b^4*c^4 - 3*a*b^2*c^5 + a^2*c^6)*x + sqrt(1/2)*(b^4*c^4 - 3*a*b^2*c^5 + a^2*c
^6)*sqrt((2*(b^4*c^3 - 3*a*b^2*c^4 + a^2*c^5)*x^2 - sqrt(1/2)*(b^9 - 10*a*b^7*c
+ 34*a^2*b^5*c^2 - 43*a^3*b^3*c^3 + 12*a^4*b*c^4 + (a^5*b^8 - 13*a^6*b^6*c + 60*
a^7*b^4*c^2 - 112*a^8*b^2*c^3 + 64*a^9*c^4)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c
^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a
^13*c^3)))*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 - (a^5*b^4 - 8*a^6*b^2*c + 16*a^
7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b
^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^4 - 8*a^6*b^2*c + 1
6*a^7*c^2)))/(b^4*c^3 - 3*a*b^2*c^4 + a^2*c^5)))) + a*x*sqrt(sqrt(1/2)*sqrt(-(b^
5 - 5*a*b^3*c + 5*a^2*b*c^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*sqrt((b^8 - 6
*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c +
48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)))*log(1/2
*sqrt(1/2)*(b^11 - 13*a*b^9*c + 63*a^2*b^7*c^2 - 138*a^3*b^5*c^3 + 128*a^4*b^3*c
^4 - 32*a^5*b*c^5 - (a^5*b^10 - 16*a^6*b^8*c + 98*a^7*b^6*c^2 - 280*a^8*b^4*c^3
+ 352*a^9*b^2*c^4 - 128*a^10*c^5)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3
*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))
*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + (a^5*b^4 - 8*a^6*b^2*c +
16*a^7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a
^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^4 - 8*a^6*b^2*
c + 16*a^7*c^2)))*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + (a^5*b^4 - 8*a^6*b^2*c
+ 16*a^7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/
(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^4 - 8*a^6*b^
2*c + 16*a^7*c^2)) + (b^4*c^4 - 3*a*b^2*c^5 + a^2*c^6)*x) - a*x*sqrt(sqrt(1/2)*s
qrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*sqrt(
(b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11
*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)))
*log(-1/2*sqrt(1/2)*(b^11 - 13*a*b^9*c + 63*a^2*b^7*c^2 - 138*a^3*b^5*c^3 + 128*
a^4*b^3*c^4 - 32*a^5*b*c^5 - (a^5*b^10 - 16*a^6*b^8*c + 98*a^7*b^6*c^2 - 280*a^8
*b^4*c^3 + 352*a^9*b^2*c^4 - 128*a^10*c^5)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^
2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^
13*c^3)))*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + (a^5*b^4 - 8*a^6
*b^2*c + 16*a^7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^
4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^4 - 8
*a^6*b^2*c + 16*a^7*c^2)))*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + (a^5*b^4 - 8*a
^6*b^2*c + 16*a^7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 +
a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^4 -
8*a^6*b^2*c + 16*a^7*c^2)) + (b^4*c^4 - 3*a*b^2*c^5 + a^2*c^6)*x) + a*x*sqrt(sq
rt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 - (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c
^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6
- 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^4 - 8*a^6*b^2*c + 16*a
^7*c^2)))*log(1/2*sqrt(1/2)*(b^11 - 13*a*b^9*c + 63*a^2*b^7*c^2 - 138*a^3*b^5*c^
3 + 128*a^4*b^3*c^4 - 32*a^5*b*c^5 + (a^5*b^10 - 16*a^6*b^8*c + 98*a^7*b^6*c^2 -
280*a^8*b^4*c^3 + 352*a^9*b^2*c^4 - 128*a^10*c^5)*sqrt((b^8 - 6*a*b^6*c + 11*a^
2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2
- 64*a^13*c^3)))*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 - (a^5*b^4
- 8*a^6*b^2*c + 16*a^7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*
c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5
*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)))*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 - (a^5*b
^4 - 8*a^6*b^2*c + 16*a^7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^
2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a
^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)) + (b^4*c^4 - 3*a*b^2*c^5 + a^2*c^6)*x) - a*x
*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 - (a^5*b^4 - 8*a^6*b^2*c +
16*a^7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a
^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^3)))/(a^5*b^4 - 8*a^6*b^2*
c + 16*a^7*c^2)))*log(-1/2*sqrt(1/2)*(b^11 - 13*a*b^9*c + 63*a^2*b^7*c^2 - 138*a
^3*b^5*c^3 + 128*a^4*b^3*c^4 - 32*a^5*b*c^5 + (a^5*b^10 - 16*a^6*b^8*c + 98*a^7*
b^6*c^2 - 280*a^8*b^4*c^3 + 352*a^9*b^2*c^4 - 128*a^10*c^5)*sqrt((b^8 - 6*a*b^6*
c + 11*a^2*b^4*c^2 - 6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^1
2*b^2*c^2 - 64*a^13*c^3)))*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 -
(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 - 6
*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*c^
3)))/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)))*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2
- (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*sqrt((b^8 - 6*a*b^6*c + 11*a^2*b^4*c^2 -
6*a^3*b^2*c^3 + a^4*c^4)/(a^10*b^6 - 12*a^11*b^4*c + 48*a^12*b^2*c^2 - 64*a^13*
c^3)))/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)) + (b^4*c^4 - 3*a*b^2*c^5 + a^2*c^6)
*x) - 4)/(a*x)

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Sympy [A]  time = 56.7829, size = 304, normalized size = 0.84 \[ \operatorname{RootSum}{\left (t^{8} \left (16777216 a^{9} c^{4} - 16777216 a^{8} b^{2} c^{3} + 6291456 a^{7} b^{4} c^{2} - 1048576 a^{6} b^{6} c + 65536 a^{5} b^{8}\right ) + t^{4} \left (20480 a^{4} b c^{4} - 30720 a^{3} b^{3} c^{3} + 15616 a^{2} b^{5} c^{2} - 3328 a b^{7} c + 256 b^{9}\right ) + c^{5}, \left ( t \mapsto t \log{\left (x + \frac{- 2097152 t^{7} a^{10} c^{5} + 5767168 t^{7} a^{9} b^{2} c^{4} - 4587520 t^{7} a^{8} b^{4} c^{3} + 1605632 t^{7} a^{7} b^{6} c^{2} - 262144 t^{7} a^{6} b^{8} c + 16384 t^{7} a^{5} b^{10} - 2304 t^{3} a^{5} b c^{5} + 8256 t^{3} a^{4} b^{3} c^{4} - 8832 t^{3} a^{3} b^{5} c^{3} + 4032 t^{3} a^{2} b^{7} c^{2} - 832 t^{3} a b^{9} c + 64 t^{3} b^{11}}{a^{2} c^{6} - 3 a b^{2} c^{5} + b^{4} c^{4}} \right )} \right )\right )} - \frac{1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(c*x**8+b*x**4+a),x)

[Out]

RootSum(_t**8*(16777216*a**9*c**4 - 16777216*a**8*b**2*c**3 + 6291456*a**7*b**4*
c**2 - 1048576*a**6*b**6*c + 65536*a**5*b**8) + _t**4*(20480*a**4*b*c**4 - 30720
*a**3*b**3*c**3 + 15616*a**2*b**5*c**2 - 3328*a*b**7*c + 256*b**9) + c**5, Lambd
a(_t, _t*log(x + (-2097152*_t**7*a**10*c**5 + 5767168*_t**7*a**9*b**2*c**4 - 458
7520*_t**7*a**8*b**4*c**3 + 1605632*_t**7*a**7*b**6*c**2 - 262144*_t**7*a**6*b**
8*c + 16384*_t**7*a**5*b**10 - 2304*_t**3*a**5*b*c**5 + 8256*_t**3*a**4*b**3*c**
4 - 8832*_t**3*a**3*b**5*c**3 + 4032*_t**3*a**2*b**7*c**2 - 832*_t**3*a*b**9*c +
64*_t**3*b**11)/(a**2*c**6 - 3*a*b**2*c**5 + b**4*c**4)))) - 1/(a*x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{8} + b x^{4} + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^8 + b*x^4 + a)*x^2),x, algorithm="giac")

[Out]

integrate(1/((c*x^8 + b*x^4 + a)*x^2), x)