3.1365 \(\int \frac{(3-2 x)^{5/2}}{\sqrt{1-3 x+x^2}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{4}{5} \sqrt{x^2-3 x+1} (3-2 x)^{3/2}+\frac{6 \sqrt [4]{5} \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt{x^2-3 x+1}}-\frac{6 \sqrt [4]{5} \sqrt{-x^2+3 x-1} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt{x^2-3 x+1}} \]

[Out]

(-4*(3 - 2*x)^(3/2)*Sqrt[1 - 3*x + x^2])/5 - (6*5^(1/4)*Sqrt[-1 + 3*x - x^2]*Ell
ipticE[ArcSin[Sqrt[3 - 2*x]/5^(1/4)], -1])/Sqrt[1 - 3*x + x^2] + (6*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 = 0.210504, 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 \[ -\frac{4}{5} \sqrt{x^2-3 x+1} (3-2 x)^{3/2}+\frac{6 \sqrt [4]{5} \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt{x^2-3 x+1}}-\frac{6 \sqrt [4]{5} \sqrt{-x^2+3 x-1} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{\sqrt{x^2-3 x+1}} \]

Antiderivative was successfully verified.

[In]  Int[(3 - 2*x)^(5/2)/Sqrt[1 - 3*x + x^2],x]

[Out]

(-4*(3 - 2*x)^(3/2)*Sqrt[1 - 3*x + x^2])/5 - (6*5^(1/4)*Sqrt[-1 + 3*x - x^2]*Ell
ipticE[ArcSin[Sqrt[3 - 2*x]/5^(1/4)], -1])/Sqrt[1 - 3*x + x^2] + (6*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 = 34.5642, size = 134, normalized size = 1.05 \[ - \frac{4 \left (- 2 x + 3\right )^{\frac{3}{2}} \sqrt{x^{2} - 3 x + 1}}{5} - \frac{6 \cdot 5^{\frac{3}{4}} \sqrt{- \frac{x^{2}}{5} + \frac{3 x}{5} - \frac{1}{5}} E\left (\operatorname{asin}{\left (\frac{5^{\frac{3}{4}} \sqrt{- 2 x + 3}}{5} \right )}\middle | -1\right )}{\sqrt{x^{2} - 3 x + 1}} + \frac{6 \cdot 5^{\frac{3}{4}} \sqrt{- \frac{x^{2}}{5} + \frac{3 x}{5} - \frac{1}{5}} F\left (\operatorname{asin}{\left (\frac{5^{\frac{3}{4}} \sqrt{- 2 x + 3}}{5} \right )}\middle | -1\right )}{\sqrt{x^{2} - 3 x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((3-2*x)**(5/2)/(x**2-3*x+1)**(1/2),x)

[Out]

-4*(-2*x + 3)**(3/2)*sqrt(x**2 - 3*x + 1)/5 - 6*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) + 6*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 = 0.172064, size = 96, normalized size = 0.75 \[ \frac{2 \left (-2 \left (x^2-3 x+1\right ) (3-2 x)^{3/2}-15 \sqrt [4]{5} \sqrt{-x^2+3 x-1} \left (E\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )-F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )\right )\right )}{5 \sqrt{x^2-3 x+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 - 2*x)^(5/2)/Sqrt[1 - 3*x + x^2],x]

[Out]

(2*(-2*(3 - 2*x)^(3/2)*(1 - 3*x + x^2) - 15*5^(1/4)*Sqrt[-1 + 3*x - x^2]*(Ellipt
icE[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 = 0.056, size = 127, normalized size = 1. \[ -{\frac{1}{10\,{x}^{3}-45\,{x}^{2}+55\,x-15}\sqrt{3-2\,x}\sqrt{{x}^{2}-3\,x+1} \left ( 3\,\sqrt{ \left ( -3+2\,x \right ) \sqrt{5}}\sqrt{ \left ( 2\,x-3+\sqrt{5} \right ) \sqrt{5}}{\it EllipticE} \left ( 1/10\,\sqrt{2}\sqrt{5}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}},\sqrt{2} \right ) \sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}\sqrt{5}-16\,{x}^{4}+96\,{x}^{3}-196\,{x}^{2}+156\,x-36 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((3-2*x)^(5/2)/(x^2-3*x+1)^(1/2),x)

[Out]

-1/5*(3-2*x)^(1/2)*(x^2-3*x+1)^(1/2)*(3*((-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)-16*x^4+96*x^3-196*x^2+156*x-3
6)/(2*x^3-9*x^2+11*x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 3\right )}^{\frac{5}{2}}}{\sqrt{x^{2} - 3 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 3)^(5/2)/sqrt(x^2 - 3*x + 1),x, algorithm="maxima")

[Out]

integrate((-2*x + 3)^(5/2)/sqrt(x^2 - 3*x + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (4 \, x^{2} - 12 \, x + 9\right )} \sqrt{-2 \, x + 3}}{\sqrt{x^{2} - 3 \, x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 3)^(5/2)/sqrt(x^2 - 3*x + 1),x, algorithm="fricas")

[Out]

integral((4*x^2 - 12*x + 9)*sqrt(-2*x + 3)/sqrt(x^2 - 3*x + 1), x)

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Sympy [A]  time = 51.0124, size = 41, normalized size = 0.32 \[ \frac{\sqrt{5} i \left (- 2 x + 3\right )^{\frac{7}{2}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{\left (- 2 x + 3\right )^{2}}{5}} \right )}}{10 \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((3-2*x)**(5/2)/(x**2-3*x+1)**(1/2),x)

[Out]

sqrt(5)*I*(-2*x + 3)**(7/2)*gamma(7/4)*hyper((1/2, 7/4), (11/4,), (-2*x + 3)**2/
5)/(10*gamma(11/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-2 \, x + 3\right )}^{\frac{5}{2}}}{\sqrt{x^{2} - 3 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x + 3)^(5/2)/sqrt(x^2 - 3*x + 1),x, algorithm="giac")

[Out]

integrate((-2*x + 3)^(5/2)/sqrt(x^2 - 3*x + 1), x)