3.1363 \(\int \frac{1}{\sqrt{3-2 x} \sqrt{1-3 x+x^2}} \, dx\)

Optimal. Leaf size=51 \[ -\frac{2 \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt [4]{5} \sqrt{x^2-3 x+1}} \]

[Out]

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

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Rubi [A]  time = 0.0733167, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{2 \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt [4]{5} \sqrt{x^2-3 x+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[3 - 2*x]*Sqrt[1 - 3*x + x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.019, size = 56, normalized size = 1.1 \[ - \frac{2 \sqrt [4]{5} \sqrt{- \frac{x^{2}}{5} + \frac{3 x}{5} - \frac{1}{5}} F\left (\operatorname{asin}{\left (\frac{5^{\frac{3}{4}} \sqrt{- 2 x + 3}}{5} \right )}\middle | -1\right )}{\sqrt{x^{2} - 3 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*5**(1/4)*sqrt(-x**2/5 + 3*x/5 - 1/5)*elliptic_f(asin(5**(3/4)*sqrt(-2*x + 3)/
5), -1)/sqrt(x**2 - 3*x + 1)

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Mathematica [A]  time = 0.0919378, size = 60, normalized size = 1.18 \[ \frac{2 \sqrt{(3-2 x)^2-5} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{5}}{\sqrt{3-2 x}}\right )\right |-1\right )}{\sqrt [4]{5} \sqrt{1-\frac{5}{(3-2 x)^2}} (3-2 x)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[3 - 2*x]*Sqrt[1 - 3*x + x^2]),x]

[Out]

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

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Maple [B]  time = 0.046, size = 102, normalized size = 2. \[{\frac{1}{10\,{x}^{3}-45\,{x}^{2}+55\,x-15}\sqrt{3-2\,x}\sqrt{{x}^{2}-3\,x+1}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}\sqrt{ \left ( -3+2\,x \right ) \sqrt{5}}\sqrt{ \left ( 2\,x-3+\sqrt{5} \right ) \sqrt{5}}{\it EllipticF} \left ({\frac{\sqrt{2}\sqrt{5}}{10}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}},\sqrt{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(3-2*x)^(1/2)*(x^2-3*x+1)^(1/2)*((-2*x+3+5^(1/2))*5^(1/2))^(1/2)*((-3+2*x)*5
^(1/2))^(1/2)*((2*x-3+5^(1/2))*5^(1/2))^(1/2)*EllipticF(1/10*2^(1/2)*5^(1/2)*((-
2*x+3+5^(1/2))*5^(1/2))^(1/2),2^(1/2))/(2*x^3-9*x^2+11*x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} - 3 \, x + 1} \sqrt{-2 \, x + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 3*x + 1)*sqrt(-2*x + 3)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^2 - 3*x + 1)*sqrt(-2*x + 3)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{2} - 3 \, x + 1} \sqrt{-2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 3*x + 1)*sqrt(-2*x + 3)),x, algorithm="fricas")

[Out]

integral(1/(sqrt(x^2 - 3*x + 1)*sqrt(-2*x + 3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 2 x + 3} \sqrt{x^{2} - 3 x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(-2*x + 3)*sqrt(x**2 - 3*x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} - 3 \, x + 1} \sqrt{-2 \, x + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 3*x + 1)*sqrt(-2*x + 3)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^2 - 3*x + 1)*sqrt(-2*x + 3)), x)