3.1366 \(\int \frac{\sqrt{3-2 x}}{\sqrt{1-3 x+x^2}} \, dx\)

Optimal. Leaf size=103 \[ \frac{2 \sqrt [4]{5} \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt{x^2-3 x+1}}-\frac{2 \sqrt [4]{5} \sqrt{-x^2+3 x-1} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt{x^2-3 x+1}} \]

[Out]

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

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Rubi [A]  time = 0.165479, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{2 \sqrt [4]{5} \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt{x^2-3 x+1}}-\frac{2 \sqrt [4]{5} \sqrt{-x^2+3 x-1} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt{x^2-3 x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.4698, size = 110, normalized size = 1.07 \[ - \frac{2 \cdot 5^{\frac{3}{4}} \sqrt{- \frac{x^{2}}{5} + \frac{3 x}{5} - \frac{1}{5}} E\left (\operatorname{asin}{\left (\frac{5^{\frac{3}{4}} \sqrt{- 2 x + 3}}{5} \right )}\middle | -1\right )}{\sqrt{x^{2} - 3 x + 1}} + \frac{2 \cdot 5^{\frac{3}{4}} \sqrt{- \frac{x^{2}}{5} + \frac{3 x}{5} - \frac{1}{5}} F\left (\operatorname{asin}{\left (\frac{5^{\frac{3}{4}} \sqrt{- 2 x + 3}}{5} \right )}\middle | -1\right )}{\sqrt{x^{2} - 3 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*5**(3/4)*sqrt(-x**2/5 + 3*x/5 - 1/5)*elliptic_e(asin(5**(3/4)*sqrt(-2*x + 3)/
5), -1)/sqrt(x**2 - 3*x + 1) + 2*5**(3/4)*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 = 0.065424, size = 74, normalized size = 0.72 \[ -\frac{2 \sqrt [4]{5} \sqrt{5-(3-2 x)^2} \left (E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )-F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )\right )}{\sqrt{(3-2 x)^2-5}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 105, normalized size = 1. \[ -{\frac{\sqrt{5}}{10\,{x}^{3}-45\,{x}^{2}+55\,x-15}\sqrt{3-2\,x}\sqrt{{x}^{2}-3\,x+1}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}\sqrt{ \left ( -3+2\,x \right ) \sqrt{5}}\sqrt{ \left ( 2\,x-3+\sqrt{5} \right ) \sqrt{5}}{\it EllipticE} \left ({\frac{\sqrt{2}\sqrt{5}}{10}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}},\sqrt{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((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)*5^(1/2)*EllipticE(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. \[ \int \frac{\sqrt{-2 \, x + 3}}{\sqrt{x^{2} - 3 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.12065, size = 41, normalized size = 0.4 \[ \frac{\sqrt{5} i \left (- 2 x + 3\right )^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{\left (- 2 x + 3\right )^{2}}{5}} \right )}}{10 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(5)*I*(-2*x + 3)**(3/2)*gamma(3/4)*hyper((1/2, 3/4), (7/4,), (-2*x + 3)**2/5
)/(10*gamma(7/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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