3.609 \(\int \sqrt{8 x-8 x^2+4 x^3-x^4} \, dx\)

Optimal. Leaf size=62 \[ \frac{1}{3} \sqrt{-(x-1)^4-2 (x-1)^2+3} (x-1)-\frac{4 F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}}+\frac{2 E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}} \]

[Out]

(Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/3 + (2*EllipticE[ArcSin[1 - x], -
1/3])/Sqrt[3] - (4*EllipticF[ArcSin[1 - x], -1/3])/Sqrt[3]

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Rubi [A]  time = 0.154578, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ -\frac{1}{3} \sqrt{-(1-x)^4-2 (1-x)^2+3} (1-x)-\frac{4 F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}}+\frac{2 E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[8*x - 8*x^2 + 4*x^3 - x^4],x]

[Out]

-(Sqrt[3 - 2*(1 - x)^2 - (1 - x)^4]*(1 - x))/3 + (2*EllipticE[ArcSin[1 - x], -1/
3])/Sqrt[3] - (4*EllipticF[ArcSin[1 - x], -1/3])/Sqrt[3]

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Rubi in Sympy [A]  time = 19.9395, size = 58, normalized size = 0.94 \[ \frac{\left (x - 1\right ) \sqrt{- \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3}}{3} - \frac{2 \sqrt{3} E\left (\operatorname{asin}{\left (x - 1 \right )}\middle | - \frac{1}{3}\right )}{3} + \frac{4 \sqrt{3} F\left (\operatorname{asin}{\left (x - 1 \right )}\middle | - \frac{1}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x - 1)*sqrt(-(x - 1)**4 - 2*(x - 1)**2 + 3)/3 - 2*sqrt(3)*elliptic_e(asin(x - 1
), -1/3)/3 + 4*sqrt(3)*elliptic_f(asin(x - 1), -1/3)/3

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Mathematica [C]  time = 0.886113, size = 256, normalized size = 4.13 \[ -\frac{x^5-5 x^4+14 x^3-24 x^2+8 i \sqrt{2} \sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}} \sqrt{\frac{x^2-2 x+4}{x^2}} x^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )-\frac{2 i \sqrt{2} (x-2) \sqrt{\frac{x^2-2 x+4}{x^2}} x E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )}{\sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}}}+24 x-16}{3 \sqrt{-x \left (x^3-4 x^2+8 x-8\right )}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[8*x - 8*x^2 + 4*x^3 - x^4],x]

[Out]

-(-16 + 24*x - 24*x^2 + 14*x^3 - 5*x^4 + x^5 - ((2*I)*Sqrt[2]*(-2 + x)*x*Sqrt[(4
- 2*x + x^2)/x^2]*EllipticE[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4)
)], (2*Sqrt[3])/(-I + Sqrt[3])])/Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)] + (8*I
)*Sqrt[2]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*x^2*Sqrt[(4 - 2*x + x^2)/x^2]
*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-
I + Sqrt[3])])/(3*Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))])

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Maple [B]  time = 0.039, size = 946, normalized size = 15.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x*(-x^4+4*x^3-8*x^2+8*x)^(1/2)-1/3*(-x^4+4*x^3-8*x^2+8*x)^(1/2)+8/3*(-1+I*3^
(1/2))*((-I*3^(1/2)-1)*x/(1-I*3^(1/2))/(x-2))^(1/2)*(x-2)^2*((x-I*3^(1/2)-1)/(I*
3^(1/2)+1)/(x-2))^(1/2)*((x-1+I*3^(1/2))/(1-I*3^(1/2))/(x-2))^(1/2)/(-I*3^(1/2)-
1)/(-x*(x-2)*(x-I*3^(1/2)-1)*(x-1+I*3^(1/2)))^(1/2)*EllipticF(((-I*3^(1/2)-1)*x/
(1-I*3^(1/2))/(x-2))^(1/2),((1-I*3^(1/2))*(-1+I*3^(1/2))/(-I*3^(1/2)-1)/(I*3^(1/
2)+1))^(1/2))+8/3*(-1+I*3^(1/2))*((-I*3^(1/2)-1)*x/(1-I*3^(1/2))/(x-2))^(1/2)*(x
-2)^2*((x-I*3^(1/2)-1)/(I*3^(1/2)+1)/(x-2))^(1/2)*((x-1+I*3^(1/2))/(1-I*3^(1/2))
/(x-2))^(1/2)/(-I*3^(1/2)-1)/(-x*(x-2)*(x-I*3^(1/2)-1)*(x-1+I*3^(1/2)))^(1/2)*(2
*EllipticF(((-I*3^(1/2)-1)*x/(1-I*3^(1/2))/(x-2))^(1/2),((1-I*3^(1/2))*(-1+I*3^(
1/2))/(-I*3^(1/2)-1)/(I*3^(1/2)+1))^(1/2))-2*EllipticPi(((-I*3^(1/2)-1)*x/(1-I*3
^(1/2))/(x-2))^(1/2),(1-I*3^(1/2))/(-I*3^(1/2)-1),((1-I*3^(1/2))*(-1+I*3^(1/2))/
(-I*3^(1/2)-1)/(I*3^(1/2)+1))^(1/2)))-2/3*(x*(x-I*3^(1/2)-1)*(x-1+I*3^(1/2))+2*(
-1+I*3^(1/2))*((-I*3^(1/2)-1)*x/(1-I*3^(1/2))/(x-2))^(1/2)*(x-2)^2*((x-I*3^(1/2)
-1)/(I*3^(1/2)+1)/(x-2))^(1/2)*((x-1+I*3^(1/2))/(1-I*3^(1/2))/(x-2))^(1/2)*(1/2*
(6-2*I*3^(1/2))/(-I*3^(1/2)-1)*EllipticF(((-I*3^(1/2)-1)*x/(1-I*3^(1/2))/(x-2))^
(1/2),((1-I*3^(1/2))*(-1+I*3^(1/2))/(-I*3^(1/2)-1)/(I*3^(1/2)+1))^(1/2))+1/2*(-I
*3^(1/2)-1)*EllipticE(((-I*3^(1/2)-1)*x/(1-I*3^(1/2))/(x-2))^(1/2),((1-I*3^(1/2)
)*(-1+I*3^(1/2))/(-I*3^(1/2)-1)/(I*3^(1/2)+1))^(1/2))-4/(-I*3^(1/2)-1)*EllipticP
i(((-I*3^(1/2)-1)*x/(1-I*3^(1/2))/(x-2))^(1/2),(-1+I*3^(1/2))/(I*3^(1/2)+1),((1-
I*3^(1/2))*(-1+I*3^(1/2))/(-I*3^(1/2)-1)/(I*3^(1/2)+1))^(1/2))))/(-x*(x-2)*(x-I*
3^(1/2)-1)*(x-1+I*3^(1/2)))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x),x, algorithm="fricas")

[Out]

integral(sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- x^{4} + 4 x^{3} - 8 x^{2} + 8 x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-x**4 + 4*x**3 - 8*x**2 + 8*x), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x),x, algorithm="giac")

[Out]

integrate(sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x), x)