3.128 \(\int \frac{1}{(2+3 x+x^2)^{3/2}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{2 (2 x+3)}{\sqrt{x^2+3 x+2}} \]

[Out]

(-2*(3 + 2*x))/Sqrt[2 + 3*x + x^2]

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Rubi [A]  time = 0.0023999, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {613} \[ -\frac{2 (2 x+3)}{\sqrt{x^2+3 x+2}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x + x^2)^(-3/2),x]

[Out]

(-2*(3 + 2*x))/Sqrt[2 + 3*x + 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 (2+3 x+x^2\right )^{3/2}} \, dx &=-\frac{2 (3+2 x)}{\sqrt{2+3 x+x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0055414, size = 19, normalized size = 1. \[ -\frac{2 (2 x+3)}{\sqrt{x^2+3 x+2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x + x^2)^(-3/2),x]

[Out]

(-2*(3 + 2*x))/Sqrt[2 + 3*x + x^2]

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Maple [A]  time = 0.049, size = 24, normalized size = 1.3 \begin{align*} -2\,{\frac{ \left ( 2+x \right ) \left ( 1+x \right ) \left ( 3+2\,x \right ) }{ \left ({x}^{2}+3\,x+2 \right ) ^{3/2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17317, size = 35, normalized size = 1.84 \begin{align*} -\frac{4 \, x}{\sqrt{x^{2} + 3 \, x + 2}} - \frac{6}{\sqrt{x^{2} + 3 \, x + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*x/sqrt(x^2 + 3*x + 2) - 6/sqrt(x^2 + 3*x + 2)

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Fricas [B]  time = 1.87356, size = 95, normalized size = 5. \begin{align*} -\frac{2 \,{\left (2 \, x^{2} + \sqrt{x^{2} + 3 \, x + 2}{\left (2 \, x + 3\right )} + 6 \, x + 4\right )}}{x^{2} + 3 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(2*x^2 + sqrt(x^2 + 3*x + 2)*(2*x + 3) + 6*x + 4)/(x^2 + 3*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+3*x+2)**(3/2),x)

[Out]

Integral((x**2 + 3*x + 2)**(-3/2), x)

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Giac [A]  time = 1.19274, size = 23, normalized size = 1.21 \begin{align*} -\frac{2 \,{\left (2 \, x + 3\right )}}{\sqrt{x^{2} + 3 \, x + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(2*x + 3)/sqrt(x^2 + 3*x + 2)