3.40 \(\int x \sqrt{3 x-4 x^2} \, dx\)

Optimal. Leaf size=52 \[ -\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}-\frac{3}{128} (3-8 x) \sqrt{3 x-4 x^2}-\frac{27}{512} \sin ^{-1}\left (1-\frac{8 x}{3}\right ) \]

[Out]

(-3*(3 - 8*x)*Sqrt[3*x - 4*x^2])/128 - (3*x - 4*x^2)^(3/2)/12 - (27*ArcSin[1 - (
8*x)/3])/512

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Rubi [A]  time = 0.0452581, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}-\frac{3}{128} (3-8 x) \sqrt{3 x-4 x^2}-\frac{27}{512} \sin ^{-1}\left (1-\frac{8 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[3*x - 4*x^2],x]

[Out]

(-3*(3 - 8*x)*Sqrt[3*x - 4*x^2])/128 - (3*x - 4*x^2)^(3/2)/12 - (27*ArcSin[1 - (
8*x)/3])/512

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Rubi in Sympy [A]  time = 4.19979, size = 44, normalized size = 0.85 \[ - \frac{3 \left (- 8 x + 3\right ) \sqrt{- 4 x^{2} + 3 x}}{128} - \frac{\left (- 4 x^{2} + 3 x\right )^{\frac{3}{2}}}{12} + \frac{27 \operatorname{asin}{\left (\frac{8 x}{3} - 1 \right )}}{512} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(-4*x**2+3*x)**(1/2),x)

[Out]

-3*(-8*x + 3)*sqrt(-4*x**2 + 3*x)/128 - (-4*x**2 + 3*x)**(3/2)/12 + 27*asin(8*x/
3 - 1)/512

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Mathematica [A]  time = 0.0448095, size = 77, normalized size = 1.48 \[ \frac{\sqrt{-x (4 x-3)} \left (2 \sqrt{x} \sqrt{4 x-3} \left (128 x^2-24 x-27\right )-81 \log \left (2 \sqrt{x}+\sqrt{4 x-3}\right )\right )}{768 \sqrt{x} \sqrt{4 x-3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[3*x - 4*x^2],x]

[Out]

(Sqrt[-(x*(-3 + 4*x))]*(2*Sqrt[x]*Sqrt[-3 + 4*x]*(-27 - 24*x + 128*x^2) - 81*Log
[2*Sqrt[x] + Sqrt[-3 + 4*x]]))/(768*Sqrt[x]*Sqrt[-3 + 4*x])

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Maple [A]  time = 0.007, size = 41, normalized size = 0.8 \[ -{\frac{1}{12} \left ( -4\,{x}^{2}+3\,x \right ) ^{{\frac{3}{2}}}}+{\frac{27}{512}\arcsin \left ( -1+{\frac{8\,x}{3}} \right ) }-{\frac{9-24\,x}{128}\sqrt{-4\,{x}^{2}+3\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(-4*x^2+3*x)^(1/2),x)

[Out]

-1/12*(-4*x^2+3*x)^(3/2)+27/512*arcsin(-1+8/3*x)-3/128*(3-8*x)*(-4*x^2+3*x)^(1/2
)

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Maxima [A]  time = 0.801115, size = 66, normalized size = 1.27 \[ -\frac{1}{12} \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}} + \frac{3}{16} \, \sqrt{-4 \, x^{2} + 3 \, x} x - \frac{9}{128} \, \sqrt{-4 \, x^{2} + 3 \, x} - \frac{27}{512} \, \arcsin \left (-\frac{8}{3} \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-4*x^2 + 3*x)*x,x, algorithm="maxima")

[Out]

-1/12*(-4*x^2 + 3*x)^(3/2) + 3/16*sqrt(-4*x^2 + 3*x)*x - 9/128*sqrt(-4*x^2 + 3*x
) - 27/512*arcsin(-8/3*x + 1)

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Fricas [A]  time = 0.219261, size = 58, normalized size = 1.12 \[ \frac{1}{384} \,{\left (128 \, x^{2} - 24 \, x - 27\right )} \sqrt{-4 \, x^{2} + 3 \, x} - \frac{27}{256} \, \arctan \left (\frac{\sqrt{-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-4*x^2 + 3*x)*x,x, algorithm="fricas")

[Out]

1/384*(128*x^2 - 24*x - 27)*sqrt(-4*x^2 + 3*x) - 27/256*arctan(1/2*sqrt(-4*x^2 +
3*x)/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{- x \left (4 x - 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(-4*x**2+3*x)**(1/2),x)

[Out]

Integral(x*sqrt(-x*(4*x - 3)), x)

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GIAC/XCAS [A]  time = 0.211824, size = 43, normalized size = 0.83 \[ \frac{1}{384} \,{\left (8 \,{\left (16 \, x - 3\right )} x - 27\right )} \sqrt{-4 \, x^{2} + 3 \, x} + \frac{27}{512} \, \arcsin \left (\frac{8}{3} \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-4*x^2 + 3*x)*x,x, algorithm="giac")

[Out]

1/384*(8*(16*x - 3)*x - 27)*sqrt(-4*x^2 + 3*x) + 27/512*arcsin(8/3*x - 1)