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

Optimal. Leaf size=15 \[ -\frac{1}{4} \sqrt{-4 x^2-9} \]

[Out]

-Sqrt[-9 - 4*x^2]/4

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Rubi [A]  time = 0.0022589, 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} \[ -\frac{1}{4} \sqrt{-4 x^2-9} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[-9 - 4*x^2],x]

[Out]

-Sqrt[-9 - 4*x^2]/4

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 \frac{x}{\sqrt{-9-4 x^2}} \, dx &=-\frac{1}{4} \sqrt{-9-4 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0014836, size = 15, normalized size = 1. \[ -\frac{1}{4} \sqrt{-4 x^2-9} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[-9 - 4*x^2],x]

[Out]

-Sqrt[-9 - 4*x^2]/4

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Maple [A]  time = 0.002, size = 12, normalized size = 0.8 \begin{align*} -{\frac{1}{4}\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]

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

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Maxima [A]  time = 1.89684, size = 15, normalized size = 1. \begin{align*} -\frac{1}{4} \, \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="maxima")

[Out]

-1/4*sqrt(-4*x^2 - 9)

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Fricas [A]  time = 1.20037, size = 31, normalized size = 2.07 \begin{align*} -\frac{1}{4} \, \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]

-1/4*sqrt(-4*x^2 - 9)

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Sympy [A]  time = 0.14674, size = 14, normalized size = 0.93 \begin{align*} - \frac{\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]

-sqrt(-4*x**2 - 9)/4

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Giac [C]  time = 2.40221, size = 15, normalized size = 1. \begin{align*} -\frac{1}{4} i \, \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="giac")

[Out]

-1/4*I*sqrt(4*x^2 + 9)