3.1.67 \(\int \frac {1}{(4+12 x+9 x^2)^{3/2}} \, dx\) [67]

Optimal. Leaf size=25 \[ -\frac {1}{6 (2+3 x) \sqrt {4+12 x+9 x^2}} \]

[Out]

-1/6/(2+3*x)/((2+3*x)^2)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 25, 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 = {621} \begin {gather*} -\frac {1}{6 (3 x+2) \sqrt {9 x^2+12 x+4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + 12*x + 9*x^2)^(-3/2),x]

[Out]

-1/6*1/((2 + 3*x)*Sqrt[4 + 12*x + 9*x^2])

Rule 621

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[2*((a + b*x + c*x^2)^(p + 1)/((2*p + 1)*(b + 2*
c*x))), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {1}{\left (4+12 x+9 x^2\right )^{3/2}} \, dx &=-\frac {1}{6 (2+3 x) \sqrt {4+12 x+9 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 20, normalized size = 0.80 \begin {gather*} -\frac {2+3 x}{6 \left ((2+3 x)^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + 12*x + 9*x^2)^(-3/2),x]

[Out]

-1/6*(2 + 3*x)/((2 + 3*x)^2)^(3/2)

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Maple [A]
time = 0.43, size = 17, normalized size = 0.68

method	result	size
meijerg	\(\frac {x \left (2+\frac {3 x}{2}\right )}{16 \left (1+\frac {3 x}{2}\right )^{2}}\)	\(16\)
gosper	\(-\frac {2+3 x}{6 \left (\left (2+3 x \right )^{2}\right )^{\frac {3}{2}}}\)	\(17\)
default	\(-\frac {2+3 x}{6 \left (\left (2+3 x \right )^{2}\right )^{\frac {3}{2}}}\)	\(17\)
risch	\(-\frac {\sqrt {\left (2+3 x \right )^{2}}}{6 \left (2+3 x \right )^{3}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(9*x^2+12*x+4)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/6*(2+3*x)/((2+3*x)^2)^(3/2)

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Maxima [A]
time = 0.51, size = 9, normalized size = 0.36 \begin {gather*} -\frac {1}{6 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+12*x+4)^(3/2),x, algorithm="maxima")

[Out]

-1/6/(3*x + 2)^2

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Fricas [A]
time = 2.01, size = 14, normalized size = 0.56 \begin {gather*} -\frac {1}{6 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+12*x+4)^(3/2),x, algorithm="fricas")

[Out]

-1/6/(9*x^2 + 12*x + 4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (9 x^{2} + 12 x + 4\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x**2+12*x+4)**(3/2),x)

[Out]

Integral((9*x**2 + 12*x + 4)**(-3/2), x)

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Giac [A]
time = 1.49, size = 17, normalized size = 0.68 \begin {gather*} -\frac {1}{6 \, {\left (3 \, x + 2\right )}^{2} \mathrm {sgn}\left (3 \, x + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(9*x^2+12*x+4)^(3/2),x, algorithm="giac")

[Out]

-1/6/((3*x + 2)^2*sgn(3*x + 2))

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Mupad [B]
time = 0.17, size = 21, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {9\,x^2+12\,x+4}}{6\,{\left (3\,x+2\right )}^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(12*x + 9*x^2 + 4)^(3/2),x)

[Out]

-(12*x + 9*x^2 + 4)^(1/2)/(6*(3*x + 2)^3)

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