3.572 \(\int \frac{1}{x^2 \sqrt{-9-4 x^2}} \, dx\)

Optimal. Leaf size=18 \[ \frac{\sqrt{-4 x^2-9}}{9 x} \]

[Out]

Sqrt[-9 - 4*x^2]/(9*x)

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Rubi [A]  time = 0.0030486, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ \frac{\sqrt{-4 x^2-9}}{9 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[-9 - 4*x^2]),x]

[Out]

Sqrt[-9 - 4*x^2]/(9*x)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0021481, size = 18, normalized size = 1. \[ \frac{\sqrt{-4 x^2-9}}{9 x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[-9 - 4*x^2]),x]

[Out]

Sqrt[-9 - 4*x^2]/(9*x)

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Maple [A]  time = 0.003, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{9\,x}\sqrt{-4\,{x}^{2}-9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 3.66322, size = 19, normalized size = 1.06 \begin{align*} \frac{\sqrt{-4 \, x^{2} - 9}}{9 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*sqrt(-4*x^2 - 9)/x

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Fricas [A]  time = 1.18863, size = 32, normalized size = 1.78 \begin{align*} \frac{\sqrt{-4 \, x^{2} - 9}}{9 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*sqrt(-4*x^2 - 9)/x

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Sympy [C]  time = 0.7266, size = 15, normalized size = 0.83 \begin{align*} \frac{2 i \sqrt{1 + \frac{9}{4 x^{2}}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*I*sqrt(1 + 9/(4*x**2))/9

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Giac [C]  time = 2.56933, size = 50, normalized size = 2.78 \begin{align*} \frac{i \, \sqrt{4 \, x^{2} + 9} + 3 i}{18 \, x} + \frac{8 \, x}{9 \,{\left (-4 i \, \sqrt{4 \, x^{2} + 9} - 12 i\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(I*sqrt(4*x^2 + 9) + 3*I)/x + 8/9*x/(-4*I*sqrt(4*x^2 + 9) - 12*I)