3.478 \(\int x \sqrt{-9-4 x^2} \, dx\)

Optimal. Leaf size=15 \[ -\frac{1}{12} \left (-4 x^2-9\right )^{3/2} \]

[Out]

-(-9 - 4*x^2)^(3/2)/12

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Rubi [A]  time = 0.00875601, 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 \[ -\frac{1}{12} \left (-4 x^2-9\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[-9 - 4*x^2],x]

[Out]

-(-9 - 4*x^2)^(3/2)/12

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Rubi in Sympy [A]  time = 1.97019, size = 14, normalized size = 0.93 \[ - \frac{\left (- 4 x^{2} - 9\right )^{\frac{3}{2}}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(-4*x**2-9)**(1/2),x)

[Out]

-(-4*x**2 - 9)**(3/2)/12

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Mathematica [A]  time = 0.00321487, size = 15, normalized size = 1. \[ -\frac{1}{12} \left (-4 x^2-9\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[-9 - 4*x^2],x]

[Out]

-(-9 - 4*x^2)^(3/2)/12

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Maple [A]  time = 0.003, size = 12, normalized size = 0.8 \[ -{\frac{1}{12} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(-4*x^2-9)^(1/2),x)

[Out]

-1/12*(-4*x^2-9)^(3/2)

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Maxima [A]  time = 1.34706, size = 15, normalized size = 1. \[ -\frac{1}{12} \,{\left (-4 \, x^{2} - 9\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-4*x^2 - 9)*x,x, algorithm="maxima")

[Out]

-1/12*(-4*x^2 - 9)^(3/2)

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Fricas [A]  time = 0.230497, size = 24, normalized size = 1.6 \[ \frac{1}{12} \,{\left (4 \, x^{2} + 9\right )} \sqrt{-4 \, x^{2} - 9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-4*x^2 - 9)*x,x, algorithm="fricas")

[Out]

1/12*(4*x^2 + 9)*sqrt(-4*x^2 - 9)

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Sympy [A]  time = 0.511571, size = 31, normalized size = 2.07 \[ \frac{x^{2} \sqrt{- 4 x^{2} - 9}}{3} + \frac{3 \sqrt{- 4 x^{2} - 9}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(-4*x**2-9)**(1/2),x)

[Out]

x**2*sqrt(-4*x**2 - 9)/3 + 3*sqrt(-4*x**2 - 9)/4

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GIAC/XCAS [A]  time = 0.20191, size = 16, normalized size = 1.07 \[ \frac{1}{12} \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} i \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-4*x^2 - 9)*x,x, algorithm="giac")

[Out]

1/12*(4*x^2 + 9)^(3/2)*i