3.617 \(\int \frac{1}{\left ((2-x) x \left (4-2 x+x^2\right )\right )^{5/2}} \, dx\)

Optimal. Leaf size=109 \[ \frac{\left (7 (x-1)^2+26\right ) (x-1)}{432 \sqrt{-(x-1)^4-2 (x-1)^2+3}}+\frac{\left ((x-1)^2+5\right ) (x-1)}{72 \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}-\frac{11 F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}}+\frac{7 E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}} \]

[Out]

((5 + (-1 + x)^2)*(-1 + x))/(72*(3 - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)) + ((26 +
7*(-1 + x)^2)*(-1 + x))/(432*Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]) + (7*EllipticE
[ArcSin[1 - x], -1/3])/(144*Sqrt[3]) - (11*EllipticF[ArcSin[1 - x], -1/3])/(144*
Sqrt[3])

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Rubi [A]  time = 0.201044, antiderivative size = 109, 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 \[ -\frac{\left ((x-1)^2+5\right ) (1-x)}{72 \left (-(1-x)^4-2 (1-x)^2+3\right )^{3/2}}-\frac{\left (7 (1-x)^2+26\right ) (1-x)}{432 \sqrt{-(1-x)^4-2 (1-x)^2+3}}-\frac{11 F\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}}+\frac{7 E\left (\sin ^{-1}(1-x)|-\frac{1}{3}\right )}{144 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[((2 - x)*x*(4 - 2*x + x^2))^(-5/2),x]

[Out]

-((26 + 7*(1 - x)^2)*(1 - x))/(432*Sqrt[3 - 2*(1 - x)^2 - (1 - x)^4]) - ((5 + (-
1 + x)^2)*(1 - x))/(72*(3 - 2*(1 - x)^2 - (1 - x)^4)^(3/2)) + (7*EllipticE[ArcSi
n[1 - x], -1/3])/(144*Sqrt[3]) - (11*EllipticF[ArcSin[1 - x], -1/3])/(144*Sqrt[3
])

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Rubi in Sympy [A]  time = 14.6772, size = 97, normalized size = 0.89 \[ \frac{\left (x - 1\right ) \left (2 \left (x - 1\right )^{2} + 10\right )}{144 \left (- \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3\right )^{\frac{3}{2}}} + \frac{\left (x - 1\right ) \left (112 \left (x - 1\right )^{2} + 416\right )}{6912 \sqrt{- \left (x - 1\right )^{4} - 2 \left (x - 1\right )^{2} + 3}} - \frac{7 \sqrt{3} E\left (\operatorname{asin}{\left (x - 1 \right )}\middle | - \frac{1}{3}\right )}{432} + \frac{11 \sqrt{3} F\left (\operatorname{asin}{\left (x - 1 \right )}\middle | - \frac{1}{3}\right )}{432} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((2-x)*x*(x**2-2*x+4))**(5/2),x)

[Out]

(x - 1)*(2*(x - 1)**2 + 10)/(144*(-(x - 1)**4 - 2*(x - 1)**2 + 3)**(3/2)) + (x -
1)*(112*(x - 1)**2 + 416)/(6912*sqrt(-(x - 1)**4 - 2*(x - 1)**2 + 3)) - 7*sqrt(
3)*elliptic_e(asin(x - 1), -1/3)/432 + 11*sqrt(3)*elliptic_f(asin(x - 1), -1/3)/
432

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Mathematica [C]  time = 1.4368, size = 327, normalized size = 3. \[ \frac{(x-2)^3 x^2 \left (x^2-2 x+4\right )^2 \left (-\frac{7 x \left (x^2-2 x+4\right )}{x-2}-19 i \sqrt{2} (x-2) \sqrt{\frac{i x}{\left (\sqrt{3}+i\right ) (x-2)}} \sqrt{\frac{x^2-2 x+4}{(x-2)^2}} F\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}-i-\frac{4 i}{x-2}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{i+\sqrt{3}}\right )+\frac{7 i \sqrt{2} x \sqrt{\frac{x^2-2 x+4}{(x-2)^2}} E\left (\sin ^{-1}\left (\frac{\sqrt{\sqrt{3}-i-\frac{4 i}{x-2}}}{\sqrt{2} \sqrt [4]{3}}\right )|\frac{2 \sqrt{3}}{i+\sqrt{3}}\right )}{\sqrt{\frac{i x}{\left (\sqrt{3}+i\right ) (x-2)}}}+\frac{7 x^7-49 x^6+187 x^5-445 x^4+670 x^3-622 x^2+216 x+36}{(x-2)^2 x \left (x^2-2 x+4\right )}\right )}{432 \left (-x \left (x^3-4 x^2+8 x-8\right )\right )^{5/2}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((2 - x)*x*(4 - 2*x + x^2))^(-5/2),x]

[Out]

((-2 + x)^3*x^2*(4 - 2*x + x^2)^2*((-7*x*(4 - 2*x + x^2))/(-2 + x) + (36 + 216*x
- 622*x^2 + 670*x^3 - 445*x^4 + 187*x^5 - 49*x^6 + 7*x^7)/((-2 + x)^2*x*(4 - 2*
x + x^2)) + ((7*I)*Sqrt[2]*x*Sqrt[(4 - 2*x + x^2)/(-2 + x)^2]*EllipticE[ArcSin[S
qrt[-I + Sqrt[3] - (4*I)/(-2 + x)]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(I + Sqrt[3])
])/Sqrt[(I*x)/((I + Sqrt[3])*(-2 + x))] - (19*I)*Sqrt[2]*(-2 + x)*Sqrt[(I*x)/((I
+ Sqrt[3])*(-2 + x))]*Sqrt[(4 - 2*x + x^2)/(-2 + x)^2]*EllipticF[ArcSin[Sqrt[-I
+ Sqrt[3] - (4*I)/(-2 + x)]/(Sqrt[2]*3^(1/4))], (2*Sqrt[3])/(I + Sqrt[3])]))/(4
32*(-(x*(-8 + 8*x - 4*x^2 + x^3)))^(5/2))

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Maple [B]  time = 0.051, size = 1039, normalized size = 9.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((2-x)*x*(x^2-2*x+4))^(5/2),x)

[Out]

-1/768*(-x^4+4*x^3-8*x^2+8*x)^(1/2)/x^2-1/96*(-x^3+4*x^2-8*x+8)/(x*(-x^3+4*x^2-8
*x+8))^(1/2)+(1/36+1/288*x^2-1/96*x)*(-x^4+4*x^3-8*x^2+8*x)^(1/2)/(x^3-4*x^2+8*x
-8)^2+2*x*(53/3456+5/1728*x^2-19/4608*x)/(-x*(x^3-4*x^2+8*x-8))^(1/2)+5/216*(-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))+7/108*(-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)))-7/432*(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)*E
llipticPi(((-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. \[ \int \frac{1}{\left (-{\left (x^{2} - 2 \, x + 4\right )}{\left (x - 2\right )} x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-(x^2 - 2*x + 4)*(x - 2)*x)^(-5/2),x, algorithm="maxima")

[Out]

integrate((-(x^2 - 2*x + 4)*(x - 2)*x)^(-5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (x^{8} - 8 \, x^{7} + 32 \, x^{6} - 80 \, x^{5} + 128 \, x^{4} - 128 \, x^{3} + 64 \, x^{2}\right )} \sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-(x^2 - 2*x + 4)*(x - 2)*x)^(-5/2),x, algorithm="fricas")

[Out]

integral(1/((x^8 - 8*x^7 + 32*x^6 - 80*x^5 + 128*x^4 - 128*x^3 + 64*x^2)*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))**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (-{\left (x^{2} - 2 \, x + 4\right )}{\left (x - 2\right )} x\right )^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-(x^2 - 2*x + 4)*(x - 2)*x)^(-5/2),x, algorithm="giac")

[Out]

integrate((-(x^2 - 2*x + 4)*(x - 2)*x)^(-5/2), x)