3.2.29 \(\int \frac {1}{(27-24 x+4 x^2)^{3/2}} \, dx\) [129]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {627} \begin {gather*} \frac {3-x}{9 \sqrt {4 x^2-24 x+27}} \end {gather*}

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 627

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*}

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Mathematica [A]
time = 0.09, size = 23, normalized size = 1.00 \begin {gather*} \frac {3-x}{9 \sqrt {27-24 x+4 x^2}} \end {gather*}

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 = 0.52, size = 20, normalized size = 0.87

method	result	size
trager	\(-\frac {x -3}{9 \sqrt {4 x^{2}-24 x +27}}\)	\(18\)
risch	\(-\frac {x -3}{9 \sqrt {4 x^{2}-24 x +27}}\)	\(18\)
default	\(-\frac {8 x -24}{72 \sqrt {4 x^{2}-24 x +27}}\)	\(20\)
gosper	\(-\frac {\left (-3+2 x \right ) \left (2 x -9\right ) \left (x -3\right )}{9 \left (4 x^{2}-24 x +27\right )^{\frac {3}{2}}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x^2-24*x+27)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/72*(8*x-24)/(4*x^2-24*x+27)^(1/2)

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Maxima [A]
time = 0.29, size = 30, normalized size = 1.30 \begin {gather*} -\frac {x}{9 \, \sqrt {4 \, x^{2} - 24 \, x + 27}} + \frac {1}{3 \, \sqrt {4 \, x^{2} - 24 \, x + 27}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (17) = 34\).
time = 2.18, size = 41, normalized size = 1.78 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (4 x^{2} - 24 x + 27\right )^{\frac {3}{2}}}\, dx \end {gather*}

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 = 1.39, size = 17, normalized size = 0.74 \begin {gather*} -\frac {x - 3}{9 \, \sqrt {4 \, x^{2} - 24 \, x + 27}} \end {gather*}

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)

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Mupad [B]
time = 0.06, size = 17, normalized size = 0.74 \begin {gather*} -\frac {x-3}{9\,\sqrt {4\,x^2-24\,x+27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4*x^2 - 24*x + 27)^(3/2),x)

[Out]

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

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