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

Optimal. Leaf size=15 \[ -\frac{1}{3} \left (4-x^2\right )^{3/2} \]

[Out]

-(4 - x^2)^(3/2)/3

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Rubi [A]  time = 0.00602848, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{1}{3} \left (4-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[4 - x^2],x]

[Out]

-(4 - x^2)^(3/2)/3

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Rubi in Sympy [A]  time = 0.898091, size = 10, normalized size = 0.67 \[ - \frac{\left (- x^{2} + 4\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-x**2 + 4)**(3/2)/3

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Mathematica [A]  time = 0.00255922, size = 15, normalized size = 1. \[ -\frac{1}{3} \left (4-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[4 - x^2],x]

[Out]

-(4 - x^2)^(3/2)/3

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Maple [A]  time = 0.004, size = 18, normalized size = 1.2 \[{\frac{ \left ( -2+x \right ) \left ( 2+x \right ) }{3}\sqrt{-{x}^{2}+4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(-x^2+4)^(1/2),x)

[Out]

1/3*(-2+x)*(2+x)*(-x^2+4)^(1/2)

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Maxima [A]  time = 1.33639, size = 15, normalized size = 1. \[ -\frac{1}{3} \,{\left (-x^{2} + 4\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-x^2 + 4)^(3/2)

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Fricas [A]  time = 0.20129, size = 82, normalized size = 5.47 \[ \frac{x^{6} - 24 \, x^{4} + 96 \, x^{2} + 6 \,{\left (x^{4} - 8 \, x^{2}\right )} \sqrt{-x^{2} + 4}}{3 \,{\left (6 \, x^{2} -{\left (x^{2} - 16\right )} \sqrt{-x^{2} + 4} - 32\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x^6 - 24*x^4 + 96*x^2 + 6*(x^4 - 8*x^2)*sqrt(-x^2 + 4))/(6*x^2 - (x^2 - 16)
*sqrt(-x^2 + 4) - 32)

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Sympy [A]  time = 0.231288, size = 24, normalized size = 1.6 \[ \frac{x^{2} \sqrt{- x^{2} + 4}}{3} - \frac{4 \sqrt{- x^{2} + 4}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(-x**2+4)**(1/2),x)

[Out]

x**2*sqrt(-x**2 + 4)/3 - 4*sqrt(-x**2 + 4)/3

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GIAC/XCAS [A]  time = 0.200366, size = 15, normalized size = 1. \[ -\frac{1}{3} \,{\left (-x^{2} + 4\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(-x^2 + 4)^(3/2)