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]

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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 1.33, size = 18, normalized size = 1.06 \[ \frac {x + \sqrt {x^{2} + 2 \, x} + 2}{x + 2} \]

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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giac [A]  time = 0.19, size = 18, normalized size = 1.06 \[ \frac {2}{x - \sqrt {x^{2} + 2 \, x} + 2} \]

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)

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maple [A]  time = 0.04, size = 12, normalized size = 0.71 \[ \frac {x}{\sqrt {x^{2}+2 x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 15, normalized size = 0.88 \[ \frac {\sqrt {x^{2} + 2 \, x}}{x + 2} \]

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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mupad [B]  time = 0.22, size = 15, normalized size = 0.88 \[ \frac {\sqrt {x^2+2\,x}}{x+2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((2*x + x^2)^(1/2)*(x + 2)),x)

[Out]

(2*x + x^2)^(1/2)/(x + 2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {x \left (x + 2\right )} \left (x + 2\right )}\, dx \]

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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