3.336 \(\int \frac{1}{(2+x) \sqrt{2 x+x^2}} \, dx\)

Optimal. Leaf size=17 \[ \frac{\sqrt{x^2+2 x}}{x+2} \]

[Out]

Sqrt[2*x + x^2]/(2 + x)

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Rubi [A]  time = 0.0061051, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {650} \[ \frac{\sqrt{x^2+2 x}}{x+2} \]

Antiderivative was successfully verified.

[In]

Int[1/((2 + x)*Sqrt[2*x + x^2]),x]

[Out]

Sqrt[2*x + x^2]/(2 + x)

Rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a +
b*x + c*x^2)^(p + 1))/((p + 1)*(2*c*d - b*e)), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] &&
EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p + 2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(2+x) \sqrt{2 x+x^2}} \, dx &=\frac{\sqrt{2 x+x^2}}{2+x}\\ \end{align*}

Mathematica [A]  time = 0.0050184, size = 11, normalized size = 0.65 \[ \frac{x}{\sqrt{x (x+2)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((2 + x)*Sqrt[2*x + x^2]),x]

[Out]

x/Sqrt[x*(2 + x)]

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Maple [A]  time = 0.046, size = 12, normalized size = 0.7 \begin{align*}{x{\frac{1}{\sqrt{{x}^{2}+2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.11167, size = 20, normalized size = 1.18 \begin{align*} \frac{\sqrt{x^{2} + 2 \, x}}{x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^2 + 2*x)/(x + 2)

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Fricas [A]  time = 1.86742, size = 47, normalized size = 2.76 \begin{align*} \frac{x + \sqrt{x^{2} + 2 \, x} + 2}{x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + sqrt(x^2 + 2*x) + 2)/(x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(x*(x + 2))*(x + 2)), x)

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Giac [A]  time = 1.61689, size = 24, normalized size = 1.41 \begin{align*} \frac{2}{x - \sqrt{x^{2} + 2 \, x} + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(x - sqrt(x^2 + 2*x) + 2)