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

Optimal. Leaf size=37 \[ \frac{8 \sqrt{4 x^2-9}}{243 x}+\frac{\sqrt{4 x^2-9}}{27 x^3} \]

[Out]

Sqrt[-9 + 4*x^2]/(27*x^3) + (8*Sqrt[-9 + 4*x^2])/(243*x)

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Rubi [A]  time = 0.0316089, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{8 \sqrt{4 x^2-9}}{243 x}+\frac{\sqrt{4 x^2-9}}{27 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*Sqrt[-9 + 4*x^2]),x]

[Out]

Sqrt[-9 + 4*x^2]/(27*x^3) + (8*Sqrt[-9 + 4*x^2])/(243*x)

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Rubi in Sympy [A]  time = 4.50453, size = 29, normalized size = 0.78 \[ \frac{8 \sqrt{4 x^{2} - 9}}{243 x} + \frac{\sqrt{4 x^{2} - 9}}{27 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*sqrt(4*x**2 - 9)/(243*x) + sqrt(4*x**2 - 9)/(27*x**3)

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Mathematica [A]  time = 0.0118691, size = 27, normalized size = 0.73 \[ \left (\frac{1}{27 x^3}+\frac{8}{243 x}\right ) \sqrt{4 x^2-9} \]

Antiderivative was successfully verified.

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

[Out]

(1/(27*x^3) + 8/(243*x))*Sqrt[-9 + 4*x^2]

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Maple [A]  time = 0.005, size = 32, normalized size = 0.9 \[{\frac{ \left ( 2\,x-3 \right ) \left ( 2\,x+3 \right ) \left ( 8\,{x}^{2}+9 \right ) }{243\,{x}^{3}}{\frac{1}{\sqrt{4\,{x}^{2}-9}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(4*x^2-9)^(1/2),x)

[Out]

1/243*(2*x-3)*(2*x+3)*(8*x^2+9)/x^3/(4*x^2-9)^(1/2)

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Maxima [A]  time = 1.50597, size = 39, normalized size = 1.05 \[ \frac{8 \, \sqrt{4 \, x^{2} - 9}}{243 \, x} + \frac{\sqrt{4 \, x^{2} - 9}}{27 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/243*sqrt(4*x^2 - 9)/x + 1/27*sqrt(4*x^2 - 9)/x^3

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Fricas [A]  time = 0.222752, size = 74, normalized size = 2. \[ \frac{4 \, x^{2} - 2 \, \sqrt{4 \, x^{2} - 9} x - 3}{32 \, x^{6} - 54 \, x^{4} -{\left (16 \, x^{5} - 9 \, x^{3}\right )} \sqrt{4 \, x^{2} - 9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(4*x^2 - 2*sqrt(4*x^2 - 9)*x - 3)/(32*x^6 - 54*x^4 - (16*x^5 - 9*x^3)*sqrt(4*x^2
- 9))

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Sympy [A]  time = 5.81184, size = 70, normalized size = 1.89 \[ \begin{cases} \frac{8 \sqrt{4 x^{2} - 9}}{243 x} + \frac{\sqrt{4 x^{2} - 9}}{27 x^{3}} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{9} > 1 \\\frac{8 i \sqrt{- 4 x^{2} + 9}}{243 x} + \frac{i \sqrt{- 4 x^{2} + 9}}{27 x^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((8*sqrt(4*x**2 - 9)/(243*x) + sqrt(4*x**2 - 9)/(27*x**3), 4*Abs(x**2)/
9 > 1), (8*I*sqrt(-4*x**2 + 9)/(243*x) + I*sqrt(-4*x**2 + 9)/(27*x**3), True))

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GIAC/XCAS [A]  time = 0.220725, size = 57, normalized size = 1.54 \[ \frac{32 \,{\left ({\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}^{2} + 3\right )}}{{\left ({\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}^{2} + 9\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32*((2*x - sqrt(4*x^2 - 9))^2 + 3)/((2*x - sqrt(4*x^2 - 9))^2 + 9)^3