3.143 \(\int \frac{1}{\left (-25+4 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{x}{25 \sqrt{4 x^2-25}} \]

[Out]

-x/(25*Sqrt[-25 + 4*x^2])

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Rubi [A]  time = 0.0056477, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{x}{25 \sqrt{4 x^2-25}} \]

Antiderivative was successfully verified.

[In]  Int[(-25 + 4*x^2)^(-3/2),x]

[Out]

-x/(25*Sqrt[-25 + 4*x^2])

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Rubi in Sympy [A]  time = 0.516835, size = 14, normalized size = 0.88 \[ - \frac{x}{25 \sqrt{4 x^{2} - 25}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(4*x**2-25)**(3/2),x)

[Out]

-x/(25*sqrt(4*x**2 - 25))

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Mathematica [A]  time = 0.00889073, size = 16, normalized size = 1. \[ -\frac{x}{25 \sqrt{4 x^2-25}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-25 + 4*x^2)^(-3/2),x]

[Out]

-x/(25*Sqrt[-25 + 4*x^2])

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Maple [A]  time = 0.003, size = 23, normalized size = 1.4 \[ -{\frac{ \left ( 2\,x-5 \right ) \left ( 5+2\,x \right ) x}{25} \left ( 4\,{x}^{2}-25 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(4*x^2-25)^(3/2),x)

[Out]

-1/25*(2*x-5)*(5+2*x)*x/(4*x^2-25)^(3/2)

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Maxima [A]  time = 1.38729, size = 16, normalized size = 1. \[ -\frac{x}{25 \, \sqrt{4 \, x^{2} - 25}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 - 25)^(-3/2),x, algorithm="maxima")

[Out]

-1/25*x/sqrt(4*x^2 - 25)

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Fricas [A]  time = 0.202002, size = 31, normalized size = 1.94 \[ \frac{1}{2 \,{\left (4 \, x^{2} - 2 \, \sqrt{4 \, x^{2} - 25} x - 25\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 - 25)^(-3/2),x, algorithm="fricas")

[Out]

1/2/(4*x^2 - 2*sqrt(4*x^2 - 25)*x - 25)

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Sympy [A]  time = 1.22245, size = 36, normalized size = 2.25 \[ \begin{cases} - \frac{x}{25 \sqrt{4 x^{2} - 25}} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{25} > 1 \\\frac{i x}{25 \sqrt{- 4 x^{2} + 25}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4*x**2-25)**(3/2),x)

[Out]

Piecewise((-x/(25*sqrt(4*x**2 - 25)), 4*Abs(x**2)/25 > 1), (I*x/(25*sqrt(-4*x**2
+ 25)), True))

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GIAC/XCAS [A]  time = 0.20469, size = 16, normalized size = 1. \[ -\frac{x}{25 \, \sqrt{4 \, x^{2} - 25}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 - 25)^(-3/2),x, algorithm="giac")

[Out]

-1/25*x/sqrt(4*x^2 - 25)