3.67 \(\int \frac{1}{(4+12 x+9 x^2)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0026011, antiderivative size = 25, 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 = {607} \[ -\frac{1}{6 (3 x+2) \sqrt{9 x^2+12 x+4}} \]

Antiderivative was successfully verified.

[In]

Int[(4 + 12*x + 9*x^2)^(-3/2),x]

[Out]

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

Rule 607

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*}

Mathematica [A]  time = 0.0056801, size = 20, normalized size = 0.8 \[ -\frac{3 x+2}{6 \left ((3 x+2)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2 + 3*x)/(6*((2 + 3*x)^2)^(3/2))

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Maple [A]  time = 0.076, size = 17, normalized size = 0.7 \begin{align*} -{\frac{2+3\,x}{6} \left ( \left ( 2+3\,x \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.6986, size = 12, normalized size = 0.48 \begin{align*} -\frac{1}{6 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}

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 = 2.05615, size = 34, normalized size = 1.36 \begin{align*} -\frac{1}{6 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (9 x^{2} + 12 x + 4\right )^{\frac{3}{2}}}\, dx \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

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]

sage0*x