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

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

[Out]

-Sqrt[9 - 4*x^2]/(9*x)

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Rubi [A]  time = 0.0174813, 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 \[ -\frac{\sqrt{9-4 x^2}}{9 x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[9 - 4*x^2]),x]

[Out]

-Sqrt[9 - 4*x^2]/(9*x)

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Rubi in Sympy [A]  time = 2.99591, size = 14, normalized size = 0.78 \[ - \frac{\sqrt{- 4 x^{2} + 9}}{9 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-4*x**2 + 9)/(9*x)

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Mathematica [A]  time = 0.0106142, size = 18, normalized size = 1. \[ -\frac{\sqrt{9-4 x^2}}{9 x} \]

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 = 0.005, size = 25, normalized size = 1.4 \[{\frac{ \left ( 2\,x-3 \right ) \left ( 2\,x+3 \right ) }{9\,x}{\frac{1}{\sqrt{-4\,{x}^{2}+9}}}} \]

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

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Maxima [A]  time = 1.47881, size = 19, normalized size = 1.06 \[ -\frac{\sqrt{-4 \, x^{2} + 9}}{9 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-4*x^2 + 9)*x^2),x, algorithm="maxima")

[Out]

-1/9*sqrt(-4*x^2 + 9)/x

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Fricas [A]  time = 0.234267, size = 50, normalized size = 2.78 \[ \frac{4 \, x^{2} + 3 \, \sqrt{-4 \, x^{2} + 9} - 9}{9 \,{\left (\sqrt{-4 \, x^{2} + 9} x - 3 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-4*x^2 + 9)*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.61198, size = 37, normalized size = 2.06 \[ \begin{cases} - \frac{i \sqrt{4 x^{2} - 9}}{9 x} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{9} > 1 \\- \frac{\sqrt{- 4 x^{2} + 9}}{9 x} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-I*sqrt(4*x**2 - 9)/(9*x), 4*Abs(x**2)/9 > 1), (-sqrt(-4*x**2 + 9)/(9
*x), True))

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GIAC/XCAS [A]  time = 0.218706, size = 45, normalized size = 2.5 \[ \frac{2 \, x}{9 \,{\left (\sqrt{-4 \, x^{2} + 9} - 3\right )}} - \frac{\sqrt{-4 \, x^{2} + 9} - 3}{18 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-4*x^2 + 9)*x^2),x, algorithm="giac")

[Out]

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