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

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

[Out]

-(9 - 4*x^2)^(3/2)/12

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Rubi [A]  time = 0.0023749, 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} \[ -\frac{1}{12} \left (9-4 x^2\right )^{3/2} \]

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 = 0.0017305, size = 15, normalized size = 1. \[ -\frac{1}{12} \left (9-4 x^2\right )^{3/2} \]

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 = 0.001, size = 22, normalized size = 1.5 \begin{align*}{\frac{ \left ( -3+2\,x \right ) \left ( 3+2\,x \right ) }{12}\sqrt{-4\,{x}^{2}+9}} \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*(-3+2*x)*(3+2*x)*(-4*x^2+9)^(1/2)

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Maxima [A]  time = 2.80043, 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 = 1.49731, 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 = 0.187359, size = 27, normalized size = 1.8 \begin{align*} \frac{x^{2} \sqrt{9 - 4 x^{2}}}{3} - \frac{3 \sqrt{9 - 4 x^{2}}}{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(9 - 4*x**2)/3 - 3*sqrt(9 - 4*x**2)/4

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Giac [A]  time = 2.69025, 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="giac")

[Out]

-1/12*(-4*x^2 + 9)^(3/2)