3.1367 \(\int \frac{1}{(3-2 x)^{3/2} \sqrt{1-3 x+x^2}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{4 \sqrt{x^2-3 x+1}}{5 \sqrt{3-2 x}}-\frac{2 \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt{x^2-3 x+1}}+\frac{2 \sqrt{-x^2+3 x-1} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt{x^2-3 x+1}} \]

[Out]

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

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Rubi [A]  time = 0.200133, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{4 \sqrt{x^2-3 x+1}}{5 \sqrt{3-2 x}}-\frac{2 \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt{x^2-3 x+1}}+\frac{2 \sqrt{-x^2+3 x-1} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{5^{3/4} \sqrt{x^2-3 x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.6283, size = 138, normalized size = 1.08 \[ \frac{2 \cdot 5^{\frac{3}{4}} \sqrt{- \frac{x^{2}}{5} + \frac{3 x}{5} - \frac{1}{5}} E\left (\operatorname{asin}{\left (\frac{5^{\frac{3}{4}} \sqrt{- 2 x + 3}}{5} \right )}\middle | -1\right )}{5 \sqrt{x^{2} - 3 x + 1}} - \frac{2 \cdot 5^{\frac{3}{4}} \sqrt{- \frac{x^{2}}{5} + \frac{3 x}{5} - \frac{1}{5}} F\left (\operatorname{asin}{\left (\frac{5^{\frac{3}{4}} \sqrt{- 2 x + 3}}{5} \right )}\middle | -1\right )}{5 \sqrt{x^{2} - 3 x + 1}} - \frac{4 \sqrt{x^{2} - 3 x + 1}}{5 \sqrt{- 2 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3-2*x)**(3/2)/(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)/(5*sqrt(x**2 - 3*x + 1)) - 2*5**(3/4)*sqrt(-x**2/5 + 3*x/5 - 1/5)*ellipti
c_f(asin(5**(3/4)*sqrt(-2*x + 3)/5), -1)/(5*sqrt(x**2 - 3*x + 1)) - 4*sqrt(x**2
- 3*x + 1)/(5*sqrt(-2*x + 3))

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Mathematica [A]  time = 0.115346, size = 95, normalized size = 0.74 \[ \frac{2 \left (\sqrt [4]{5} \sqrt{-x^2+3 x-1} \left (E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )-F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )\right )-\frac{2 \left (x^2-3 x+1\right )}{\sqrt{3-2 x}}\right )}{5 \sqrt{x^2-3 x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.055, size = 116, normalized size = 0.9 \[{\frac{1}{50\,{x}^{3}-225\,{x}^{2}+275\,x-75}\sqrt{3-2\,x}\sqrt{{x}^{2}-3\,x+1} \left ( \sqrt{ \left ( -3+2\,x \right ) \sqrt{5}}\sqrt{ \left ( 2\,x-3+\sqrt{5} \right ) \sqrt{5}}{\it EllipticE} \left ({\frac{\sqrt{2}\sqrt{5}}{10}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}},\sqrt{2} \right ) \sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}\sqrt{5}+20\,{x}^{2}-60\,x+20 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25*(3-2*x)^(1/2)*(x^2-3*x+1)^(1/2)*(((-3+2*x)*5^(1/2))^(1/2)*((2*x-3+5^(1/2))*
5^(1/2))^(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+5^(1/2))*5^(1/2))^(1/2)*5^(1/2)+20*x^2-60*x+20)/(2*x^3-9*x^2+11
*x-3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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