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

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 = 0.0026535, 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, Rules used = {613} \[ \frac{3-x}{9 \sqrt{4 x^2-24 x+27}} \]

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])

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (27-24 x+4 x^2\right )^{3/2}} \, dx &=\frac{3-x}{9 \sqrt{27-24 x+4 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0065783, 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 = 0.042, size = 28, normalized size = 1.2 \begin{align*} -{\frac{ \left ( -3+2\,x \right ) \left ( 2\,x-9 \right ) \left ( -3+x \right ) }{9} \left ( 4\,{x}^{2}-24\,x+27 \right ) ^{-{\frac{3}{2}}}} \end{align*}

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

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Maxima [A]  time = 1.18078, size = 41, normalized size = 1.78 \begin{align*} -\frac{x}{9 \, \sqrt{4 \, x^{2} - 24 \, x + 27}} + \frac{1}{3 \, \sqrt{4 \, x^{2} - 24 \, x + 27}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 [B]  time = 1.87474, size = 112, normalized size = 4.87 \begin{align*} -\frac{4 \, x^{2} + 2 \, \sqrt{4 \, x^{2} - 24 \, x + 27}{\left (x - 3\right )} - 24 \, x + 27}{18 \,{\left (4 \, x^{2} - 24 \, x + 27\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*x^2-24*x+27)^(3/2),x, algorithm="fricas")

[Out]

-1/18*(4*x^2 + 2*sqrt(4*x^2 - 24*x + 27)*(x - 3) - 24*x + 27)/(4*x^2 - 24*x + 27)

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

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 [A]  time = 1.28112, size = 23, normalized size = 1. \begin{align*} -\frac{x - 3}{9 \, \sqrt{4 \, x^{2} - 24 \, x + 27}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4*x^2-24*x+27)^(3/2),x, algorithm="giac")

[Out]

-1/9*(x - 3)/sqrt(4*x^2 - 24*x + 27)