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3.640 \(\int \frac{1}{\sqrt{8+8 x-x^3+8 x^4}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{x^2 \sqrt{\frac{\left (\frac{4}{x}+1\right )^4-6 \left (\frac{4}{x}+1\right )^2+261}{\left (\frac{\sqrt{29} (x+4)^2}{x^2}+87\right )^2}} \left (\frac{\sqrt{29} (x+4)^2}{x^2}+87\right ) F\left (2 \tan ^{-1}\left (\frac{x+4}{\sqrt{3} \sqrt [4]{29} x}\right )|\frac{1}{58} \left (29+\sqrt{29}\right )\right )}{8 \sqrt{3} \sqrt [4]{29} \sqrt{8 x^4-x^3+8 x+8}} \]

[Out]

-(x^2*Sqrt[(261 - 6*(1 + 4/x)^2 + (1 + 4/x)^4)/(87 + (Sqrt[29]*(4 + x)^2)/x^2)^2
]*(87 + (Sqrt[29]*(4 + x)^2)/x^2)*EllipticF[2*ArcTan[(4 + x)/(Sqrt[3]*29^(1/4)*x
)], (29 + Sqrt[29])/58])/(8*Sqrt[3]*29^(1/4)*Sqrt[8 + 8*x - x^3 + 8*x^4])

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Rubi [A]  time = 0.533162, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{x^2 \sqrt{\frac{\left (\frac{4}{x}+1\right )^4-6 \left (\frac{4}{x}+1\right )^2+261}{\left (\frac{\sqrt{29} (x+4)^2}{x^2}+87\right )^2}} \left (\frac{\sqrt{29} (x+4)^2}{x^2}+87\right ) F\left (2 \tan ^{-1}\left (\frac{x+4}{\sqrt{3} \sqrt [4]{29} x}\right )|\frac{1}{58} \left (29+\sqrt{29}\right )\right )}{8 \sqrt{3} \sqrt [4]{29} \sqrt{8 x^4-x^3+8 x+8}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[8 + 8*x - x^3 + 8*x^4],x]

[Out]

-(x^2*Sqrt[(261 - 6*(1 + 4/x)^2 + (1 + 4/x)^4)/(87 + (Sqrt[29]*(4 + x)^2)/x^2)^2
]*(87 + (Sqrt[29]*(4 + x)^2)/x^2)*EllipticF[2*ArcTan[(4 + x)/(Sqrt[3]*29^(1/4)*x
)], (29 + Sqrt[29])/58])/(8*Sqrt[3]*29^(1/4)*Sqrt[8 + 8*x - x^3 + 8*x^4])

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 1024 \int ^{\frac{1}{4} + \frac{1}{x}} \frac{1}{\sqrt{\frac{8388608 x^{4} - 3145728 x^{2} + 8552448}{\left (- 32 x + 8\right )^{4}}} \left (- 32 x + 8\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(8*x**4-x**3+8*x+8)**(1/2),x)

[Out]

-1024*Integral(1/(sqrt((8388608*x**4 - 3145728*x**2 + 8552448)/(-32*x + 8)**4)*(
-32*x + 8)**2), (x, 1/4 + 1/x))

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Mathematica [C]  time = 0.554979, size = 927, normalized size = 7.19 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[8 + 8*x - x^3 + 8*x^4],x]

[Out]

(-2*EllipticF[ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8
 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((
x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & ,
1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], ((Root[8 + 8*#1 - #1^3 + 8*
#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3
+ 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((Root[8 + 8*#1 -
 #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*
#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]*(x - R
oot[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])^2*Sqrt[((Root[8 + 8*#1 - #1^3 + 8*#1^4 &
 , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 +
8*#1^4 & , 3, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1
 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0]))]*(Root[8
+ 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])*Sqrt
[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 &
 , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 8*#1 - #1^3 +
8*#1^4 & , 4, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^
3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])^2*(Root[8
+ 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0])^2)])
/(Sqrt[8 + 8*x - x^3 + 8*x^4]*(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8
 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Ro
ot[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))

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Maple [C]  time = 1.776, size = 965, normalized size = 7.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(8*x^4-x^3+8*x+8)^(1/2),x)

[Out]

1/2*(-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*((R
ootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))*(x-RootOf(
8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_
Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2)*(x-RootOf(8*_
Z^4-_Z^3+8*_Z+8,index=2))^2*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_
Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))/(RootOf(8*_Z^4-_Z^3+
8*_Z+8,index=3)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8
,index=2)))^(1/2)*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8
,index=1))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,ind
ex=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))
)^(1/2)/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))/
(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*2^(1/2)/
((x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))*(
x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)))^(1
/2)*EllipticF(((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,ind
ex=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4
)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1
/2),((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))*(-R
ootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(-RootOf(8
*_Z^4-_Z^3+8*_Z+8,index=3)+RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z
^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{8 \, x^{4} - x^{3} + 8 \, x + 8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(8*x^4 - x^3 + 8*x + 8),x, algorithm="maxima")

[Out]

integrate(1/sqrt(8*x^4 - x^3 + 8*x + 8), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{8 \, x^{4} - x^{3} + 8 \, x + 8}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(8*x^4 - x^3 + 8*x + 8),x, algorithm="fricas")

[Out]

integral(1/sqrt(8*x^4 - x^3 + 8*x + 8), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{8 x^{4} - x^{3} + 8 x + 8}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(8*x**4-x**3+8*x+8)**(1/2),x)

[Out]

Integral(1/sqrt(8*x**4 - x**3 + 8*x + 8), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{8 \, x^{4} - x^{3} + 8 \, x + 8}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(8*x^4 - x^3 + 8*x + 8),x, algorithm="giac")

[Out]

integrate(1/sqrt(8*x^4 - x^3 + 8*x + 8), x)

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