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 = 0.0024286, 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, Rules used = {261} \[ -\frac{1}{12} \left (-4 x^2-9\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[-9 - 4*x^2],x]

[Out]

-(-9 - 4*x^2)^(3/2)/12

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x \sqrt{-9-4 x^2} \, dx &=-\frac{1}{12} \left (-9-4 x^2\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0016278, 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 = 0.002, size = 12, normalized size = 0.8 \begin{align*} -{\frac{1}{12} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}} \end{align*}

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 = 2.09672, size = 15, normalized size = 1. \begin{align*} -\frac{1}{12} \,{\left (-4 \, x^{2} - 9\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-4*x^2-9)^(1/2),x, algorithm="maxima")

[Out]

-1/12*(-4*x^2 - 9)^(3/2)

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Fricas [A]  time = 1.26114, size = 47, normalized size = 3.13 \begin{align*} \frac{1}{12} \,{\left (4 \, x^{2} + 9\right )} \sqrt{-4 \, x^{2} - 9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-4*x^2-9)^(1/2),x, algorithm="fricas")

[Out]

1/12*(4*x^2 + 9)*sqrt(-4*x^2 - 9)

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Sympy [B]  time = 0.203126, size = 31, normalized size = 2.07 \begin{align*} \frac{x^{2} \sqrt{- 4 x^{2} - 9}}{3} + \frac{3 \sqrt{- 4 x^{2} - 9}}{4} \end{align*}

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 [C]  time = 1.58329, size = 15, normalized size = 1. \begin{align*} \frac{1}{12} i \,{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-4*x^2-9)^(1/2),x, algorithm="giac")

[Out]

1/12*I*(4*x^2 + 9)^(3/2)