3.458 \(\int \frac{1}{c+\frac{a}{x^8}+\frac{b}{x^4}} \, dx\)

Optimal. Leaf size=376 \[ \frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{x}{c} \]

[Out]

x/c + ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sq
rt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*c^(5/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) + ((
b - (b^2 - 2*a*c)/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^(1/4)*c^(5/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) + ((b + (b^2
- 2*a*c)/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^(1/4)*c^(5/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) + ((b - (b^2 - 2*a*
c)/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^(1/4)*c^(5/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))

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Rubi [A]  time = 1.36441, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{\left (\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}+b\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{\left (b-\frac{b^2-2 a c}{\sqrt{b^2-4 a c}}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{2 \sqrt [4]{2} c^{5/4} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{x}{c} \]

Antiderivative was successfully verified.

[In]  Int[(c + a/x^8 + b/x^4)^(-1),x]

[Out]

x/c + ((b + (b^2 - 2*a*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b - Sq
rt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)*c^(5/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) + ((
b - (b^2 - 2*a*c)/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^(1/4)*c^(5/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) + ((b + (b^2
- 2*a*c)/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^(1/4)*c^(5/4)*(-b - Sqrt[b^2 - 4*a*c])^(3/4)) + ((b - (b^2 - 2*a*
c)/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^(1/4)*c^(5/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4))

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Rubi in Sympy [A]  time = 137.529, size = 382, normalized size = 1.02 \[ \frac{x}{c} - \frac{2^{\frac{3}{4}} \left (- 2 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 c^{\frac{5}{4}} \left (- b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} - \frac{2^{\frac{3}{4}} \left (- 2 a c + b^{2} - b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b + \sqrt{- 4 a c + b^{2}}}} \right )}}{4 c^{\frac{5}{4}} \left (- b + \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{3}{4}} \left (- 2 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{4 c^{\frac{5}{4}} \left (- b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} + \frac{2^{\frac{3}{4}} \left (- 2 a c + b^{2} + b \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{- b - \sqrt{- 4 a c + b^{2}}}} \right )}}{4 c^{\frac{5}{4}} \left (- b - \sqrt{- 4 a c + b^{2}}\right )^{\frac{3}{4}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x/c - 2**(3/4)*(-2*a*c + b**2 - 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*c**(5/4)*(-b + sqrt(-4*a*c + b**2))**(3/4)
*sqrt(-4*a*c + b**2)) - 2**(3/4)*(-2*a*c + b**2 - 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*c**(5/4)*(-b + sqrt(-4*
a*c + b**2))**(3/4)*sqrt(-4*a*c + b**2)) + 2**(3/4)*(-2*a*c + b**2 + 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*c**(5
/4)*(-b - sqrt(-4*a*c + b**2))**(3/4)*sqrt(-4*a*c + b**2)) + 2**(3/4)*(-2*a*c +
b**2 + 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*c**(5/4)*(-b - sqrt(-4*a*c + b**2))**(3/4)*sqrt(-4*a*c + b**2))

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Mathematica [C]  time = 0.0617522, size = 70, normalized size = 0.19 \[ \frac{x}{c}-\frac{\text{RootSum}\left [\text{$\#$1}^8 c+\text{$\#$1}^4 b+a\&,\frac{\text{$\#$1}^4 b \log (x-\text{$\#$1})+a \log (x-\text{$\#$1})}{2 \text{$\#$1}^7 c+\text{$\#$1}^3 b}\&\right ]}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

x/c - RootSum[a + b*#1^4 + c*#1^8 & , (a*Log[x - #1] + b*Log[x - #1]*#1^4)/(b*#1
^3 + 2*c*#1^7) & ]/(4*c)

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Maple [C]  time = 0.037, size = 59, normalized size = 0.2 \[{\frac{x}{c}}+{\frac{1}{4\,c}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{8}+{{\it \_Z}}^{4}b+a \right ) }{\frac{ \left ( -{{\it \_R}}^{4}b-a \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{7}c+{{\it \_R}}^{3}b}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+a/x^8+b/x^4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.440067, size = 5154, normalized size = 13.71 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(4*c*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + (b^4*c^5 - 8*a*b^
2*c^6 + 16*a^2*c^7)*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)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*
a*b^2*c^6 + 16*a^2*c^7)))*arctan(-1/2*(b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*
c^3 - (b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*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)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*
a^3*c^13)))*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + (b^4*c^5 - 8*a
*b^2*c^6 + 16*a^2*c^7)*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)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 -
8*a*b^2*c^6 + 16*a^2*c^7)))/((a*b^4 - 3*a^2*b^2*c + a^3*c^2)*x + sqrt(1/2)*(a*b
^4 - 3*a^2*b^2*c + a^3*c^2)*sqrt((2*(a^2*b^4 - 3*a^3*b^2*c + a^4*c^2)*x^2 + sqrt
(1/2)*(b^8 - 9*a*b^6*c + 27*a^2*b^4*c^2 - 30*a^3*b^2*c^3 + 8*a^4*c^4 - (b^7*c^5
- 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*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)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 -
64*a^3*c^13)))*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + (b^4*c^5 - 8*a*b^2*c^6 + 1
6*a^2*c^7)*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)/(b^
6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6
+ 16*a^2*c^7)))/(a^2*b^4 - 3*a^3*b^2*c + a^4*c^2)))) - 4*c*sqrt(sqrt(1/2)*sqrt(
-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 - (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*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)/(b^6*c^10 - 12*a*b^4*c^
11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))*arc
tan(1/2*(b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3 + (b^5*c^5 - 8*a*b^3*c^6 +
16*a^2*b*c^7)*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)
/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt(sqrt(1/2)*sqr
t(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 - (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*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)/(b^6*c^10 - 12*a*b^4*
c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))/(
(a*b^4 - 3*a^2*b^2*c + a^3*c^2)*x + sqrt(1/2)*(a*b^4 - 3*a^2*b^2*c + a^3*c^2)*sq
rt((2*(a^2*b^4 - 3*a^3*b^2*c + a^4*c^2)*x^2 + sqrt(1/2)*(b^8 - 9*a*b^6*c + 27*a^
2*b^4*c^2 - 30*a^3*b^2*c^3 + 8*a^4*c^4 + (b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^
7 - 64*a^3*b*c^8)*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)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt(-(b^5 - 5*
a*b^3*c + 5*a^2*b*c^2 - (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*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)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^
2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))/(a^2*b^4 - 3*
a^3*b^2*c + a^4*c^2)))) + c*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2
+ (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*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)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^
13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))*log((a*b^4 - 3*a^2*b^2*c + a^3*c^2)
*x + 1/2*(b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3 - (b^5*c^5 - 8*a*b^3*c^6
+ 16*a^2*b*c^7)*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
)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt(sqrt(1/2)*sq
rt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*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)/(b^6*c^10 - 12*a*b^4
*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7))))
- c*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + (b^4*c^5 - 8*a*b^2*c^
6 + 16*a^2*c^7)*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
)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^
2*c^6 + 16*a^2*c^7)))*log((a*b^4 - 3*a^2*b^2*c + a^3*c^2)*x - 1/2*(b^6 - 7*a*b^4
*c + 13*a^2*b^2*c^2 - 4*a^3*c^3 - (b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*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)/(b^6*c^10 - 12*a*b^4*
c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c +
5*a^2*b*c^2 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*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)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12
- 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))) + c*sqrt(sqrt(1/2)*sqr
t(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 - (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*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)/(b^6*c^10 - 12*a*b^4*
c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))*l
og((a*b^4 - 3*a^2*b^2*c + a^3*c^2)*x + 1/2*(b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4
*a^3*c^3 + (b^5*c^5 - 8*a*b^3*c^6 + 16*a^2*b*c^7)*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)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12
- 64*a^3*c^13)))*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 - (b^4*c^5
- 8*a*b^2*c^6 + 16*a^2*c^7)*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)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*
c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))) - c*sqrt(sqrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5
*a^2*b*c^2 - (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*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)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12
- 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)))*log((a*b^4 - 3*a^2*b^2*c
+ a^3*c^2)*x - 1/2*(b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3 + (b^5*c^5 - 8
*a*b^3*c^6 + 16*a^2*b*c^7)*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)/(b^6*c^10 - 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))*sqrt(s
qrt(1/2)*sqrt(-(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 - (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*
c^7)*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)/(b^6*c^10
- 12*a*b^4*c^11 + 48*a^2*b^2*c^12 - 64*a^3*c^13)))/(b^4*c^5 - 8*a*b^2*c^6 + 16*
a^2*c^7)))) + 4*x)/c

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Sympy [A]  time = 52.0304, size = 218, normalized size = 0.58 \[ \operatorname{RootSum}{\left (t^{8} \left (16777216 a^{4} c^{9} - 16777216 a^{3} b^{2} c^{8} + 6291456 a^{2} b^{4} c^{7} - 1048576 a b^{6} c^{6} + 65536 b^{8} c^{5}\right ) + t^{4} \left (20480 a^{4} b c^{4} - 30720 a^{3} b^{3} c^{3} + 15616 a^{2} b^{5} c^{2} - 3328 a b^{7} c + 256 b^{9}\right ) + a^{5}, \left ( t \mapsto t \log{\left (x + \frac{16384 t^{5} a^{2} b c^{7} - 8192 t^{5} a b^{3} c^{6} + 1024 t^{5} b^{5} c^{5} - 8 t a^{3} c^{3} + 36 t a^{2} b^{2} c^{2} - 24 t a b^{4} c + 4 t b^{6}}{a^{3} c^{2} - 3 a^{2} b^{2} c + a b^{4}} \right )} \right )\right )} + \frac{x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(_t**8*(16777216*a**4*c**9 - 16777216*a**3*b**2*c**8 + 6291456*a**2*b**4*
c**7 - 1048576*a*b**6*c**6 + 65536*b**8*c**5) + _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) + a**5, Lambd
a(_t, _t*log(x + (16384*_t**5*a**2*b*c**7 - 8192*_t**5*a*b**3*c**6 + 1024*_t**5*
b**5*c**5 - 8*_t*a**3*c**3 + 36*_t*a**2*b**2*c**2 - 24*_t*a*b**4*c + 4*_t*b**6)/
(a**3*c**2 - 3*a**2*b**2*c + a*b**4)))) + x/c

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(c + b/x^4 + a/x^8), x)