3.616 \(\int \frac{1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{3/2}} \, dx\)

Optimal. Leaf size=73 \[ \frac{\left ((x-1)^2+5\right ) (x-1)}{24 \sqrt{-(x-1)^4-2 (x-1)^2+3}}-\frac{F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{4 \sqrt{3}}+\frac{E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{8 \sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.154776, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{\left ((x-1)^2+5\right ) (1-x)}{24 \sqrt{-(1-x)^4-2 (1-x)^2+3}}-\frac{F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{4 \sqrt{3}}+\frac{E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{8 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.9846, size = 63, normalized size = 0.86 \[ \frac{\left (x - 1\right ) \left (2 \left (x - 1\right )^{2} + 10\right )}{48 \sqrt{- \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3}} - \frac{\sqrt{3} E\left (\operatorname{asin}{\left (x - 1 \right )}\middle | - \frac{1}{3}\right )}{24} + \frac{\sqrt{3} F\left (\operatorname{asin}{\left (x - 1 \right )}\middle | - \frac{1}{3}\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x - 1)*(2*(x - 1)**2 + 10)/(48*sqrt(-(x - 1)**4 - 2*(x - 1)**2 + 3)) - sqrt(3)*
elliptic_e(asin(x - 1), -1/3)/24 + sqrt(3)*elliptic_f(asin(x - 1), -1/3)/12

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Mathematica [C]  time = 1.27464, size = 298, normalized size = 4.08 \[ \frac{(x-2)^2 x \left (x^2-2 x+4\right ) \left (-\frac{3 \left (x^2-2 x+4\right ) x}{x-2}-3 \left (x^2-2 x+4\right )-4 (2-x) \sqrt{\frac{x^2-2 x+4}{(x-2)^2}} \left (\sqrt{\frac{x^2-2 x+4}{(x-2)^2}} x+4 i \sqrt{2} \sqrt{\frac{i x}{\left (\sqrt{3}+i\right ) (x-2)}} F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}-i-\frac{4 i}{x-2}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{i+\sqrt{3}}\right )-\sqrt{2} \left (\sqrt{3}+i\right ) \sqrt{\frac{i x}{\left (\sqrt{3}+i\right ) (x-2)}} E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}-i-\frac{4 i}{x-2}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{i+\sqrt{3}}\right )\right )+2 (x-1) x\right )}{96 \left (-x \left (x^3-4 x^2+8 x-8\right )\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((-2 + x)^2*x*(4 - 2*x + x^2)*(2*(-1 + x)*x - 3*(4 - 2*x + x^2) - (3*x*(4 - 2*x
+ x^2))/(-2 + x) - 4*(2 - x)*Sqrt[(4 - 2*x + x^2)/(-2 + x)^2]*(x*Sqrt[(4 - 2*x +
x^2)/(-2 + x)^2] - Sqrt[2]*(I + Sqrt[3])*Sqrt[(I*x)/((I + Sqrt[3])*(-2 + x))]*E
llipticE[ArcSin[Sqrt[-I + Sqrt[3] - (4*I)/(-2 + x)]/(Sqrt[2]*3^(1/4))], (2*Sqrt[
3])/(I + Sqrt[3])] + (4*I)*Sqrt[2]*Sqrt[(I*x)/((I + Sqrt[3])*(-2 + x))]*Elliptic
F[ArcSin[Sqrt[-I + Sqrt[3] - (4*I)/(-2 + x)]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(I
+ Sqrt[3])])))/(96*(-(x*(-8 + 8*x - 4*x^2 + x^3)))^(3/2))

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Maple [B]  time = 0.045, size = 963, normalized size = 13.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32*(-x^3+4*x^2-8*x+8)/(x*(-x^3+4*x^2-8*x+8))^(1/2)+2*x*(1/24+1/192*x^2)/(-x*(
x^3-4*x^2+8*x-8))^(1/2)+1/6*(-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))+1/6*(-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)))-1/24*(
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. \[ \int \frac{1}{\left (-{\left (x^{2} - 2 \, x + 4\right )}{\left (x - 2\right )} x\right )^{\frac{3}{2}}}\,{d x} \]

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. \[{\rm integral}\left (-\frac{1}{{\left (x^{4} - 4 \, x^{3} + 8 \, x^{2} - 8 \, x\right )} \sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-1/((x^4 - 4*x^3 + 8*x^2 - 8*x)*sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x)), 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(1/((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. \[ \int \frac{1}{\left (-{\left (x^{2} - 2 \, x + 4\right )}{\left (x - 2\right )} x\right )^{\frac{3}{2}}}\,{d x} \]

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)