3.129 \(\int \frac{1}{\left (27-24 x+4 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=23 \[ \frac{3-x}{9 \sqrt{4 x^2-24 x+27}} \]

[Out]

(3 - x)/(9*Sqrt[27 - 24*x + 4*x^2])

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Rubi [A]  time = 0.0102923, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{3-x}{9 \sqrt{4 x^2-24 x+27}} \]

Antiderivative was successfully verified.

[In]  Int[(27 - 24*x + 4*x^2)^(-3/2),x]

[Out]

(3 - x)/(9*Sqrt[27 - 24*x + 4*x^2])

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Rubi in Sympy [A]  time = 1.28625, size = 19, normalized size = 0.83 \[ \frac{- 16 x + 48}{144 \sqrt{4 x^{2} - 24 x + 27}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(4*x**2-24*x+27)**(3/2),x)

[Out]

(-16*x + 48)/(144*sqrt(4*x**2 - 24*x + 27))

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Mathematica [A]  time = 0.015467, size = 23, normalized size = 1. \[ \frac{3-x}{9 \sqrt{4 x^2-24 x+27}} \]

Antiderivative was successfully verified.

[In]  Integrate[(27 - 24*x + 4*x^2)^(-3/2),x]

[Out]

(3 - x)/(9*Sqrt[27 - 24*x + 4*x^2])

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Maple [A]  time = 0.004, size = 28, normalized size = 1.2 \[ -{\frac{ \left ( -3+2\,x \right ) \left ( 2\,x-9 \right ) \left ( x-3 \right ) }{9} \left ( 4\,{x}^{2}-24\,x+27 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(4*x^2-24*x+27)^(3/2),x)

[Out]

-1/9*(-3+2*x)*(2*x-9)*(x-3)/(4*x^2-24*x+27)^(3/2)

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Maxima [A]  time = 0.763091, size = 41, normalized size = 1.78 \[ -\frac{x}{9 \, \sqrt{4 \, x^{2} - 24 \, x + 27}} + \frac{1}{3 \, \sqrt{4 \, x^{2} - 24 \, x + 27}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 - 24*x + 27)^(-3/2),x, algorithm="maxima")

[Out]

-1/9*x/sqrt(4*x^2 - 24*x + 27) + 1/3/sqrt(4*x^2 - 24*x + 27)

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Fricas [A]  time = 0.209984, size = 42, normalized size = 1.83 \[ \frac{1}{2 \,{\left (4 \, x^{2} - 2 \, \sqrt{4 \, x^{2} - 24 \, x + 27}{\left (x - 3\right )} - 24 \, x + 27\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 - 24*x + 27)^(-3/2),x, algorithm="fricas")

[Out]

1/2/(4*x^2 - 2*sqrt(4*x^2 - 24*x + 27)*(x - 3) - 24*x + 27)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (4 x^{2} - 24 x + 27\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(4*x**2-24*x+27)**(3/2),x)

[Out]

Integral((4*x**2 - 24*x + 27)**(-3/2), x)

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GIAC/XCAS [A]  time = 0.212368, size = 23, normalized size = 1. \[ -\frac{x - 3}{9 \, \sqrt{4 \, x^{2} - 24 \, x + 27}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^2 - 24*x + 27)^(-3/2),x, algorithm="giac")

[Out]

-1/9*(x - 3)/sqrt(4*x^2 - 24*x + 27)