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

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 = 0.197234, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ -\frac{1}{7} (1-x) \left (-(1-x)^4-2 (1-x)^2+3\right )^{3/2}-\frac{2}{35} \left (13-3 (1-x)^2\right ) (1-x) \sqrt{-(1-x)^4-2 (1-x)^2+3}-\frac{176}{35} \sqrt{3} F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )+\frac{16}{5} \sqrt{3} E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((2 - x)*x*(4 - 2*x + x^2))^(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 = 15.8143, 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(((2-x)*x*(x**2-2*x+4))**(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 = 1.31495, size = 278, normalized size = 2.73 \[ \frac{\sqrt{-x \left (x^3-4 x^2+8 x-8\right )} \left (\sqrt{\frac{x^2-2 x+4}{x^2}} \left (-5 x^7+35 x^6-116 x^5+230 x^4-228 x^3+44 x^2+152 x-224\right )+304 i \sqrt{2} \sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}} F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )+112 \sqrt{2} \left (\sqrt{3}-i\right ) \sqrt{-\frac{i (x-2)}{\left (\sqrt{3}-i\right ) x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}+i-\frac{4 i}{x}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{-i+\sqrt{3}}\right )\right )}{35 (x-2) x \sqrt{\frac{x^2-2 x+4}{x^2}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))]*(Sqrt[(4 - 2*x + x^2)/x^2]*(-224 + 152*x +
44*x^2 - 228*x^3 + 230*x^4 - 116*x^5 + 35*x^6 - 5*x^7) + 112*Sqrt[2]*(-I + Sqrt[
3])*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*EllipticE[ArcSin[Sqrt[I + Sqrt[3] -
(4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])] + (304*I)*Sqrt[2]*Sqrt
[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x
]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(-I + Sqrt[3])]))/(35*(-2 + x)*x*Sqrt[(4 - 2*x
+ x^2)/x^2])

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Maple [B]  time = 0.047, size = 1050, normalized size = 10.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((2-x)*x*(x^2-2*x+4))^(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. \[ \int \left (-{\left (x^{2} - 2 \, x + 4\right )}{\left (x - 2\right )} x\right )^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-(x^2 - 2*x + 4)*(x - 2)*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^2 - 2*x + 4)*(x - 2)*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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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