3.540 \(\int \frac{1}{x^3 \sqrt{9+4 x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{2}{27} \tanh ^{-1}\left (\frac{1}{3} \sqrt{4 x^2+9}\right )-\frac{\sqrt{4 x^2+9}}{18 x^2} \]

[Out]

-Sqrt[9 + 4*x^2]/(18*x^2) + (2*ArcTanh[Sqrt[9 + 4*x^2]/3])/27

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Rubi [A]  time = 0.0508805, 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 \[ \frac{2}{27} \tanh ^{-1}\left (\frac{1}{3} \sqrt{4 x^2+9}\right )-\frac{\sqrt{4 x^2+9}}{18 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[9 + 4*x^2]),x]

[Out]

-Sqrt[9 + 4*x^2]/(18*x^2) + (2*ArcTanh[Sqrt[9 + 4*x^2]/3])/27

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Rubi in Sympy [A]  time = 5.47355, size = 31, normalized size = 0.79 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{4 x^{2} + 9}}{3} \right )}}{27} - \frac{\sqrt{4 x^{2} + 9}}{18 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atanh(sqrt(4*x**2 + 9)/3)/27 - sqrt(4*x**2 + 9)/(18*x**2)

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Mathematica [A]  time = 0.0259705, size = 43, normalized size = 1.1 \[ -\frac{\sqrt{4 x^2+9}}{18 x^2}+\frac{2}{27} \log \left (\sqrt{4 x^2+9}+3\right )-\frac{2 \log (x)}{27} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Sqrt[9 + 4*x^2]),x]

[Out]

-Sqrt[9 + 4*x^2]/(18*x^2) - (2*Log[x])/27 + (2*Log[3 + Sqrt[9 + 4*x^2]])/27

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Maple [A]  time = 0.006, size = 30, normalized size = 0.8 \[ -{\frac{1}{18\,{x}^{2}}\sqrt{4\,{x}^{2}+9}}+{\frac{2}{27}{\it Artanh} \left ( 3\,{\frac{1}{\sqrt{4\,{x}^{2}+9}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(4*x^2+9)^(1/2),x)

[Out]

-1/18*(4*x^2+9)^(1/2)/x^2+2/27*arctanh(3/(4*x^2+9)^(1/2))

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Maxima [A]  time = 1.49394, size = 32, normalized size = 0.82 \[ -\frac{\sqrt{4 \, x^{2} + 9}}{18 \, x^{2}} + \frac{2}{27} \, \operatorname{arsinh}\left (\frac{3}{2 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(4*x^2 + 9)*x^3),x, algorithm="maxima")

[Out]

-1/18*sqrt(4*x^2 + 9)/x^2 + 2/27*arcsinh(3/2/abs(x))

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Fricas [A]  time = 0.2346, size = 189, normalized size = 4.85 \[ \frac{48 \, x^{3} + 4 \,{\left (8 \, x^{4} - 4 \, \sqrt{4 \, x^{2} + 9} x^{3} + 9 \, x^{2}\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} + 3\right ) - 4 \,{\left (8 \, x^{4} - 4 \, \sqrt{4 \, x^{2} + 9} x^{3} + 9 \, x^{2}\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} + 9} - 3\right ) - 3 \,{\left (8 \, x^{2} + 9\right )} \sqrt{4 \, x^{2} + 9} + 108 \, x}{54 \,{\left (8 \, x^{4} - 4 \, \sqrt{4 \, x^{2} + 9} x^{3} + 9 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(4*x^2 + 9)*x^3),x, algorithm="fricas")

[Out]

1/54*(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 = 7.92405, size = 44, normalized size = 1.13 \[ \frac{2 \operatorname{asinh}{\left (\frac{3}{2 x} \right )}}{27} - \frac{1}{9 x \sqrt{1 + \frac{9}{4 x^{2}}}} - \frac{1}{4 x^{3} \sqrt{1 + \frac{9}{4 x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*asinh(3/(2*x))/27 - 1/(9*x*sqrt(1 + 9/(4*x**2))) - 1/(4*x**3*sqrt(1 + 9/(4*x**
2)))

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GIAC/XCAS [A]  time = 0.211226, size = 58, normalized size = 1.49 \[ -\frac{\sqrt{4 \, x^{2} + 9}}{18 \, x^{2}} + \frac{1}{27} \,{\rm ln}\left (\sqrt{4 \, x^{2} + 9} + 3\right ) - \frac{1}{27} \,{\rm ln}\left (\sqrt{4 \, x^{2} + 9} - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(4*x^2 + 9)*x^3),x, algorithm="giac")

[Out]

-1/18*sqrt(4*x^2 + 9)/x^2 + 1/27*ln(sqrt(4*x^2 + 9) + 3) - 1/27*ln(sqrt(4*x^2 +
9) - 3)