3.482 \(\int \frac{\sqrt{-9-4 x^2}}{x^3} \, dx\)

Optimal. Leaf size=39 \[ -\frac{\sqrt{-4 x^2-9}}{2 x^2}-\frac{2}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]

[Out]

-Sqrt[-9 - 4*x^2]/(2*x^2) - (2*ArcTan[Sqrt[-9 - 4*x^2]/3])/3

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Rubi [A]  time = 0.0510821, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\sqrt{-4 x^2-9}}{2 x^2}-\frac{2}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{-4 x^2-9}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-9 - 4*x^2]/x^3,x]

[Out]

-Sqrt[-9 - 4*x^2]/(2*x^2) - (2*ArcTan[Sqrt[-9 - 4*x^2]/3])/3

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Rubi in Sympy [A]  time = 5.75434, size = 36, normalized size = 0.92 \[ - \frac{2 \operatorname{atan}{\left (\frac{\sqrt{- 4 x^{2} - 9}}{3} \right )}}{3} - \frac{\sqrt{- 4 x^{2} - 9}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-4*x**2-9)**(1/2)/x**3,x)

[Out]

-2*atan(sqrt(-4*x**2 - 9)/3)/3 - sqrt(-4*x**2 - 9)/(2*x**2)

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Mathematica [A]  time = 0.0199935, size = 37, normalized size = 0.95 \[ \frac{2}{3} \tan ^{-1}\left (\frac{3}{\sqrt{-4 x^2-9}}\right )-\frac{\sqrt{-4 x^2-9}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-9 - 4*x^2]/x^3,x]

[Out]

-Sqrt[-9 - 4*x^2]/(2*x^2) + (2*ArcTan[3/Sqrt[-9 - 4*x^2]])/3

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Maple [A]  time = 0.006, size = 41, normalized size = 1.1 \[{\frac{1}{18\,{x}^{2}} \left ( -4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}+{\frac{2}{9}\sqrt{-4\,{x}^{2}-9}}+{\frac{2}{3}\arctan \left ( 3\,{\frac{1}{\sqrt{-4\,{x}^{2}-9}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-4*x^2-9)^(1/2)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.49906, size = 69, normalized size = 1.77 \[ \frac{2}{9} \, \sqrt{-4 \, x^{2} - 9} + \frac{{\left (-4 \, x^{2} - 9\right )}^{\frac{3}{2}}}{18 \, x^{2}} + \frac{2}{3} i \, \log \left (\frac{6 \, \sqrt{4 \, x^{2} + 9}}{{\left | x \right |}} + \frac{18}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-4*x^2 - 9)/x^3,x, algorithm="maxima")

[Out]

2/9*sqrt(-4*x^2 - 9) + 1/18*(-4*x^2 - 9)^(3/2)/x^2 + 2/3*I*log(6*sqrt(4*x^2 + 9)
/abs(x) + 18/abs(x))

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Fricas [A]  time = 0.235661, size = 88, normalized size = 2.26 \[ \frac{-2 i \, x^{2} \log \left (-\frac{4 \,{\left (i \, \sqrt{-4 \, x^{2} - 9} - 3\right )}}{3 \, x}\right ) + 2 i \, x^{2} \log \left (-\frac{4 \,{\left (-i \, \sqrt{-4 \, x^{2} - 9} - 3\right )}}{3 \, x}\right ) - 3 \, \sqrt{-4 \, x^{2} - 9}}{6 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-4*x^2 - 9)/x^3,x, algorithm="fricas")

[Out]

1/6*(-2*I*x^2*log(-4/3*(I*sqrt(-4*x^2 - 9) - 3)/x) + 2*I*x^2*log(-4/3*(-I*sqrt(-
4*x^2 - 9) - 3)/x) - 3*sqrt(-4*x^2 - 9))/x^2

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Sympy [A]  time = 6.0263, size = 27, normalized size = 0.69 \[ - \frac{2 i \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{3} - \frac{i \sqrt{1 + \frac{9}{4 x^{2}}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*x**2-9)**(1/2)/x**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.20471, size = 42, normalized size = 1.08 \[ -\frac{\sqrt{4 \, x^{2} + 9} i}{2 \, x^{2}} - \frac{2}{3} \, \arctan \left (\frac{1}{3} \, \sqrt{4 \, x^{2} + 9} i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-4*x^2 - 9)/x^3,x, algorithm="giac")

[Out]

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