3.608 \(\int \left (8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=102 \[ \frac{1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac{2}{35} \left (13-3 (x-1)^2\right ) (x-1) \sqrt{-(x-1)^4-2 (x-1)^2+3}-\frac{176}{35} \sqrt{3} F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )+\frac{16}{5} \sqrt{3} E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right ) \]

[Out]

(2*(13 - 3*(-1 + x)^2)*Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/35 + ((3 -
2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x))/7 + (16*Sqrt[3]*EllipticE[ArcSin[1 -
x], -1/3])/5 - (176*Sqrt[3]*EllipticF[ArcSin[1 - x], -1/3])/35

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Rubi [A]  time = 0.202189, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304 \[ -\frac{1}{7} (1-x) \left (-(1-x)^4-2 (1-x)^2+3\right )^{3/2}-\frac{2}{35} \left (13-3 (1-x)^2\right ) (1-x) \sqrt{-(1-x)^4-2 (1-x)^2+3}-\frac{176}{35} \sqrt{3} F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )+\frac{16}{5} \sqrt{3} E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(13 - 3*(1 - x)^2)*Sqrt[3 - 2*(1 - x)^2 - (1 - x)^4]*(1 - x))/35 - ((3 - 2*(
1 - x)^2 - (1 - x)^4)^(3/2)*(1 - x))/7 + (16*Sqrt[3]*EllipticE[ArcSin[1 - x], -1
/3])/5 - (176*Sqrt[3]*EllipticF[ArcSin[1 - x], -1/3])/35

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Rubi in Sympy [A]  time = 22.9351, size = 88, normalized size = 0.86 \[ \frac{\left (x - 1\right ) \left (- 6 \left (x - 1\right )^{2} + 26\right ) \sqrt{- \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3}}{35} + \frac{\left (x - 1\right ) \left (- \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3\right )^{\frac{3}{2}}}{7} - \frac{16 \sqrt{3} E\left (\operatorname{asin}{\left (x - 1 \right )}\middle | - \frac{1}{3}\right )}{5} + \frac{176 \sqrt{3} F\left (\operatorname{asin}{\left (x - 1 \right )}\middle | - \frac{1}{3}\right )}{35} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x - 1)*(-6*(x - 1)**2 + 26)*sqrt(-(x - 1)**4 - 2*(x - 1)**2 + 3)/35 + (x - 1)*(
-(x - 1)**4 - 2*(x - 1)**2 + 3)**(3/2)/7 - 16*sqrt(3)*elliptic_e(asin(x - 1), -1
/3)/5 + 176*sqrt(3)*elliptic_f(asin(x - 1), -1/3)/35

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Mathematica [C]  time = 0.899038, size = 278, normalized size = 2.73 \[ \frac{5 x^9-45 x^8+206 x^7-602 x^6+1152 x^5-1420 x^4+848 x^3+352 x^2-304 i \sqrt{2} \sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}} \sqrt{\frac{x^2-2 x+4}{x^2}} x^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )+\frac{112 i \sqrt{2} (x-2) \sqrt{\frac{x^2-2 x+4}{x^2}} x E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )}{\sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}}}-1056 x+896}{35 \sqrt{-x \left (x^3-4 x^2+8 x-8\right )}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(896 - 1056*x + 352*x^2 + 848*x^3 - 1420*x^4 + 1152*x^5 - 602*x^6 + 206*x^7 - 45
*x^8 + 5*x^9 + ((112*I)*Sqrt[2]*(-2 + x)*x*Sqrt[(4 - 2*x + x^2)/x^2]*EllipticE[A
rcSin[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)] - (304*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 + S
qrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])])/(35*Sqrt[-(x*
(-8 + 8*x - 4*x^2 + x^3))])

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Maple [B]  time = 0.184, size = 1050, normalized size = 10.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7*x^5*(-x^4+4*x^3-8*x^2+8*x)^(1/2)+5/7*x^4*(-x^4+4*x^3-8*x^2+8*x)^(1/2)-66/35
*x^3*(-x^4+4*x^3-8*x^2+8*x)^(1/2)+14/5*x^2*(-x^4+4*x^3-8*x^2+8*x)^(1/2)-32/35*x*
(-x^4+4*x^3-8*x^2+8*x)^(1/2)-4/7*(-x^4+4*x^3-8*x^2+8*x)^(1/2)+32/7*(-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))+64/5*(-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*Ell
ipticF(((-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)))-16/5*(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)*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))))/(-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. \[ \int{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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