3.462 \(\int \frac{\sqrt{9-4 x^2}}{x^5} \, dx\)

Optimal. Leaf size=57 \[ \frac{\sqrt{9-4 x^2}}{18 x^2}+\frac{2}{27} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right )-\frac{\sqrt{9-4 x^2}}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.0670115, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt{9-4 x^2}}{18 x^2}+\frac{2}{27} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right )-\frac{\sqrt{9-4 x^2}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[9 - 4*x^2]/x^5,x]

[Out]

-Sqrt[9 - 4*x^2]/(4*x^4) + 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 = 6.61916, size = 46, normalized size = 0.81 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{- 4 x^{2} + 9}}{3} \right )}}{27} + \frac{\sqrt{- 4 x^{2} + 9}}{18 x^{2}} - \frac{\sqrt{- 4 x^{2} + 9}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-4*x**2+9)**(1/2)/x**5,x)

[Out]

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

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Mathematica [A]  time = 0.0423702, size = 48, normalized size = 0.84 \[ \frac{1}{108} \left (8 \log \left (\sqrt{9-4 x^2}+3\right )+\frac{3 \sqrt{9-4 x^2} \left (2 x^2-9\right )}{x^4}-8 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[9 - 4*x^2]/x^5,x]

[Out]

((3*Sqrt[9 - 4*x^2]*(-9 + 2*x^2))/x^4 - 8*Log[x] + 8*Log[3 + Sqrt[9 - 4*x^2]])/1
08

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Maple [A]  time = 0.008, size = 55, normalized size = 1. \[ -{\frac{1}{36\,{x}^{4}} \left ( -4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{162\,{x}^{2}} \left ( -4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}-{\frac{2}{81}\sqrt{-4\,{x}^{2}+9}}+{\frac{2}{27}{\it Artanh} \left ( 3\,{\frac{1}{\sqrt{-4\,{x}^{2}+9}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-4*x^2+9)^(1/2)/x^5,x)

[Out]

-1/36/x^4*(-4*x^2+9)^(3/2)-1/162/x^2*(-4*x^2+9)^(3/2)-2/81*(-4*x^2+9)^(1/2)+2/27
*arctanh(3/(-4*x^2+9)^(1/2))

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Maxima [A]  time = 1.51034, size = 88, normalized size = 1.54 \[ -\frac{2}{81} \, \sqrt{-4 \, x^{2} + 9} - \frac{{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}}}{162 \, x^{2}} - \frac{{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}}}{36 \, x^{4}} + \frac{2}{27} \, \log \left (\frac{6 \, \sqrt{-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac{18}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-4*x^2 + 9)/x^5,x, algorithm="maxima")

[Out]

-2/81*sqrt(-4*x^2 + 9) - 1/162*(-4*x^2 + 9)^(3/2)/x^2 - 1/36*(-4*x^2 + 9)^(3/2)/
x^4 + 2/27*log(6*sqrt(-4*x^2 + 9)/abs(x) + 18/abs(x))

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Fricas [A]  time = 0.22692, size = 193, normalized size = 3.39 \[ -\frac{144 \, x^{6} - 1620 \, x^{4} + 5832 \, x^{2} + 8 \,{\left (2 \, x^{8} - 36 \, x^{6} + 81 \, x^{4} + 3 \,{\left (2 \, x^{6} - 9 \, x^{4}\right )} \sqrt{-4 \, x^{2} + 9}\right )} \log \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{x}\right ) - 3 \,{\left (4 \, x^{6} - 90 \, x^{4} + 486 \, x^{2} - 729\right )} \sqrt{-4 \, x^{2} + 9} - 6561}{108 \,{\left (2 \, x^{8} - 36 \, x^{6} + 81 \, x^{4} + 3 \,{\left (2 \, x^{6} - 9 \, x^{4}\right )} \sqrt{-4 \, x^{2} + 9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-4*x^2 + 9)/x^5,x, algorithm="fricas")

[Out]

-1/108*(144*x^6 - 1620*x^4 + 5832*x^2 + 8*(2*x^8 - 36*x^6 + 81*x^4 + 3*(2*x^6 -
9*x^4)*sqrt(-4*x^2 + 9))*log((sqrt(-4*x^2 + 9) - 3)/x) - 3*(4*x^6 - 90*x^4 + 486
*x^2 - 729)*sqrt(-4*x^2 + 9) - 6561)/(2*x^8 - 36*x^6 + 81*x^4 + 3*(2*x^6 - 9*x^4
)*sqrt(-4*x^2 + 9))

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Sympy [A]  time = 12.6888, size = 141, normalized size = 2.47 \[ \begin{cases} \frac{2 \operatorname{acosh}{\left (\frac{3}{2 x} \right )}}{27} - \frac{1}{9 x \sqrt{-1 + \frac{9}{4 x^{2}}}} + \frac{3}{4 x^{3} \sqrt{-1 + \frac{9}{4 x^{2}}}} - \frac{9}{8 x^{5} \sqrt{-1 + \frac{9}{4 x^{2}}}} & \text{for}\: \frac{9 \left |{\frac{1}{x^{2}}}\right |}{4} > 1 \\- \frac{2 i \operatorname{asin}{\left (\frac{3}{2 x} \right )}}{27} + \frac{i}{9 x \sqrt{1 - \frac{9}{4 x^{2}}}} - \frac{3 i}{4 x^{3} \sqrt{1 - \frac{9}{4 x^{2}}}} + \frac{9 i}{8 x^{5} \sqrt{1 - \frac{9}{4 x^{2}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*x**2+9)**(1/2)/x**5,x)

[Out]

Piecewise((2*acosh(3/(2*x))/27 - 1/(9*x*sqrt(-1 + 9/(4*x**2))) + 3/(4*x**3*sqrt(
-1 + 9/(4*x**2))) - 9/(8*x**5*sqrt(-1 + 9/(4*x**2))), 9*Abs(x**(-2))/4 > 1), (-2
*I*asin(3/(2*x))/27 + I/(9*x*sqrt(1 - 9/(4*x**2))) - 3*I/(4*x**3*sqrt(1 - 9/(4*x
**2))) + 9*I/(8*x**5*sqrt(1 - 9/(4*x**2))), True))

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GIAC/XCAS [A]  time = 0.205775, size = 77, normalized size = 1.35 \[ -\frac{{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}} + 9 \, \sqrt{-4 \, x^{2} + 9}}{72 \, x^{4}} + \frac{1}{27} \,{\rm ln}\left (\sqrt{-4 \, x^{2} + 9} + 3\right ) - \frac{1}{27} \,{\rm ln}\left (-\sqrt{-4 \, x^{2} + 9} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-4*x^2 + 9)/x^5,x, algorithm="giac")

[Out]

-1/72*((-4*x^2 + 9)^(3/2) + 9*sqrt(-4*x^2 + 9))/x^4 + 1/27*ln(sqrt(-4*x^2 + 9) +
3) - 1/27*ln(-sqrt(-4*x^2 + 9) + 3)