3.1362 \(\int \frac{(3-2 x)^{3/2}}{\sqrt{1-3 x+x^2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{4}{3} \sqrt{3-2 x} \sqrt{x^2-3 x+1}-\frac{2\ 5^{3/4} \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{3 \sqrt{x^2-3 x+1}} \]

[Out]

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

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Rubi [A]  time = 0.113038, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{4}{3} \sqrt{3-2 x} \sqrt{x^2-3 x+1}-\frac{2\ 5^{3/4} \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{3 \sqrt{x^2-3 x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.3058, size = 82, normalized size = 1.04 \[ - \frac{4 \sqrt{- 2 x + 3} \sqrt{x^{2} - 3 x + 1}}{3} - \frac{10 \sqrt [4]{5} \sqrt{- \frac{x^{2}}{5} + \frac{3 x}{5} - \frac{1}{5}} F\left (\operatorname{asin}{\left (\frac{5^{\frac{3}{4}} \sqrt{- 2 x + 3}}{5} \right )}\middle | -1\right )}{3 \sqrt{x^{2} - 3 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*sqrt(-2*x + 3)*sqrt(x**2 - 3*x + 1)/3 - 10*5**(1/4)*sqrt(-x**2/5 + 3*x/5 - 1/
5)*elliptic_f(asin(5**(3/4)*sqrt(-2*x + 3)/5), -1)/(3*sqrt(x**2 - 3*x + 1))

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Mathematica [A]  time = 0.306393, size = 79, normalized size = 1. \[ \frac{2}{3} \sqrt{x^2-3 x+1} \left (\frac{5^{3/4} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{5}}{\sqrt{3-2 x}}\right )\right |-1\right )}{(3-2 x) \sqrt{\frac{x^2-3 x+1}{(3-2 x)^2}}}-2 \sqrt{3-2 x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.143, size = 118, normalized size = 1.5 \[{\frac{1}{6\,{x}^{3}-27\,{x}^{2}+33\,x-9}\sqrt{3-2\,x}\sqrt{{x}^{2}-3\,x+1} \left ( \sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}\sqrt{ \left ( -3+2\,x \right ) \sqrt{5}}\sqrt{ \left ( 2\,x-3+\sqrt{5} \right ) \sqrt{5}}{\it EllipticF} \left ({\frac{\sqrt{2}\sqrt{5}}{10}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}},\sqrt{2} \right ) -8\,{x}^{3}+36\,{x}^{2}-44\,x+12 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(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))-8*x^3+36*x^2-44*x+12)/(2*x^3-9*x^2+11*x-
3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.3898, size = 41, normalized size = 0.52 \[ \frac{\sqrt{5} i \left (- 2 x + 3\right )^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{\left (- 2 x + 3\right )^{2}}{5}} \right )}}{10 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(5)*I*(-2*x + 3)**(5/2)*gamma(5/4)*hyper((1/2, 5/4), (9/4,), (-2*x + 3)**2/5
)/(10*gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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