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

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

[Out]

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

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Rubi [A]  time = 0.0501769, 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 \[ -\frac{\sqrt{4 x^2+9}}{2 x^2}-\frac{2}{3} \tanh ^{-1}\left (\frac{1}{3} \sqrt{4 x^2+9}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[9 + 4*x^2]/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.88493, size = 32, normalized size = 0.82 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{4 x^{2} + 9}}{3} \right )}}{3} - \frac{\sqrt{4 x^{2} + 9}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0222228, size = 43, normalized size = 1.1 \[ -\frac{\sqrt{4 x^2+9}}{2 x^2}-\frac{2}{3} \log \left (\sqrt{4 x^2+9}+3\right )+\frac{2 \log (x)}{3} \]

Antiderivative was successfully verified.

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

[Out]

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

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

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

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Maxima [A]  time = 1.4986, size = 47, normalized size = 1.21 \[ \frac{2}{9} \, \sqrt{4 \, x^{2} + 9} - \frac{{\left (4 \, x^{2} + 9\right )}^{\frac{3}{2}}}{18 \, x^{2}} - \frac{2}{3} \, \operatorname{arsinh}\left (\frac{3}{2 \,{\left | x \right |}}\right ) \]

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*arcsinh(3/2/abs(x))

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Fricas [A]  time = 0.231006, size = 189, normalized size = 4.85 \[ \frac{48 \, x^{3} - 4 \,{\left (8 \, x^{4} - 4 \, \sqrt{4 \, x^{2} + 9} x^{3} + 9 \, x^{2}\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} + 3\right ) + 4 \,{\left (8 \, x^{4} - 4 \, \sqrt{4 \, x^{2} + 9} x^{3} + 9 \, x^{2}\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} - 3\right ) - 3 \,{\left (8 \, x^{2} + 9\right )} \sqrt{4 \, x^{2} + 9} + 108 \, x}{6 \,{\left (8 \, x^{4} - 4 \, \sqrt{4 \, x^{2} + 9} x^{3} + 9 \, x^{2}\right )}} \]

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*(48*x^3 - 4*(8*x^4 - 4*sqrt(4*x^2 + 9)*x^3 + 9*x^2)*log(-2*x + sqrt(4*x^2 +
9) + 3) + 4*(8*x^4 - 4*sqrt(4*x^2 + 9)*x^3 + 9*x^2)*log(-2*x + sqrt(4*x^2 + 9) -
3) - 3*(8*x^2 + 9)*sqrt(4*x^2 + 9) + 108*x)/(8*x^4 - 4*sqrt(4*x^2 + 9)*x^3 + 9*
x^2)

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Sympy [A]  time = 5.92481, size = 24, normalized size = 0.62 \[ - \frac{2 \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{3} - \frac{\sqrt{1 + \frac{9}{4 x^{2}}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.206722, size = 58, normalized size = 1.49 \[ -\frac{\sqrt{4 \, x^{2} + 9}}{2 \, x^{2}} - \frac{1}{3} \,{\rm ln}\left (\sqrt{4 \, x^{2} + 9} + 3\right ) + \frac{1}{3} \,{\rm ln}\left (\sqrt{4 \, x^{2} + 9} - 3\right ) \]

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