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

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

[Out]

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

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Rubi [A]  time = 0.0674758, 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 \[ \frac{\sqrt{4 x^2-9}}{18 x^2}+\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right )-\frac{\sqrt{4 x^2-9}}{4 x^4} \]

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*ArcTan[Sqrt[-9 + 4*x^
2]/3])/27

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Rubi in Sympy [A]  time = 6.70903, size = 46, normalized size = 0.81 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{4 x^{2} - 9}}{3} \right )}}{27} + \frac{\sqrt{4 x^{2} - 9}}{18 x^{2}} - \frac{\sqrt{4 x^{2} - 9}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atan(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 = 0.0309164, size = 46, normalized size = 0.81 \[ \left (\frac{1}{18 x^2}-\frac{1}{4 x^4}\right ) \sqrt{4 x^2-9}-\frac{2}{27} \tan ^{-1}\left (\frac{3}{\sqrt{4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-1/(4*x^4) + 1/(18*x^2))*Sqrt[-9 + 4*x^2] - (2*ArcTan[3/Sqrt[-9 + 4*x^2]])/27

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Maple [A]  time = 0.007, size = 55, normalized size = 1. \[{\frac{1}{36\,{x}^{4}} \left ( 4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}+{\frac{1}{162\,{x}^{2}} \left ( 4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}-{\frac{2}{81}\sqrt{4\,{x}^{2}-9}}-{\frac{2}{27}\arctan \left ( 3\,{\frac{1}{\sqrt{4\,{x}^{2}-9}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4*x^2-9)^(1/2)/x^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*arc
tan(3/(4*x^2-9)^(1/2))

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Maxima [A]  time = 1.49376, size = 66, normalized size = 1.16 \[ -\frac{2}{81} \, \sqrt{4 \, x^{2} - 9} + \frac{{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}}}{162 \, x^{2}} + \frac{{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}}}{36 \, x^{4}} - \frac{2}{27} \, \arcsin \left (\frac{3}{2 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 - 9)/x^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*arcsin(3/2/abs(x))

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Fricas [A]  time = 0.225739, size = 196, normalized size = 3.44 \[ -\frac{1536 \, x^{7} - 12096 \, x^{5} + 27216 \, x^{3} - 16 \,{\left (128 \, x^{8} - 288 \, x^{6} + 81 \, x^{4} - 8 \,{\left (8 \, x^{7} - 9 \, x^{5}\right )} \sqrt{4 \, x^{2} - 9}\right )} \arctan \left (-\frac{2}{3} \, x + \frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\right ) - 3 \,{\left (256 \, x^{6} - 1728 \, x^{4} + 2754 \, x^{2} - 729\right )} \sqrt{4 \, x^{2} - 9} - 17496 \, x}{108 \,{\left (128 \, x^{8} - 288 \, x^{6} + 81 \, x^{4} - 8 \,{\left (8 \, x^{7} - 9 \, x^{5}\right )} \sqrt{4 \, x^{2} - 9}\right )}} \]

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*(1536*x^7 - 12096*x^5 + 27216*x^3 - 16*(128*x^8 - 288*x^6 + 81*x^4 - 8*(8
*x^7 - 9*x^5)*sqrt(4*x^2 - 9))*arctan(-2/3*x + 1/3*sqrt(4*x^2 - 9)) - 3*(256*x^6
- 1728*x^4 + 2754*x^2 - 729)*sqrt(4*x^2 - 9) - 17496*x)/(128*x^8 - 288*x^6 + 81
*x^4 - 8*(8*x^7 - 9*x^5)*sqrt(4*x^2 - 9))

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Sympy [A]  time = 12.5172, size = 141, normalized size = 2.47 \[ \begin{cases} \frac{2 i \operatorname{acosh}{\left (\frac{3}{2 x} \right )}}{27} - \frac{i}{9 x \sqrt{-1 + \frac{9}{4 x^{2}}}} + \frac{3 i}{4 x^{3} \sqrt{-1 + \frac{9}{4 x^{2}}}} - \frac{9 i}{8 x^{5} \sqrt{-1 + \frac{9}{4 x^{2}}}} & \text{for}\: \frac{9 \left |{\frac{1}{x^{2}}}\right |}{4} > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{3}{2 x} \right )}}{27} + \frac{1}{9 x \sqrt{1 - \frac{9}{4 x^{2}}}} - \frac{3}{4 x^{3} \sqrt{1 - \frac{9}{4 x^{2}}}} + \frac{9}{8 x^{5} \sqrt{1 - \frac{9}{4 x^{2}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.203685, size = 55, normalized size = 0.96 \[ \frac{{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}} - 9 \, \sqrt{4 \, x^{2} - 9}}{72 \, x^{4}} + \frac{2}{27} \, \arctan \left (\frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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