3.587 \(\int \frac{2+x}{\sqrt{9+x^2}} \, dx\)

Optimal. Leaf size=18 \[ \sqrt{x^2+9}+2 \sinh ^{-1}\left (\frac{x}{3}\right ) \]

[Out]

Sqrt[9 + x^2] + 2*ArcSinh[x/3]

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Rubi [A]  time = 0.0044369, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {641, 215} \[ \sqrt{x^2+9}+2 \sinh ^{-1}\left (\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(2 + x)/Sqrt[9 + x^2],x]

[Out]

Sqrt[9 + x^2] + 2*ArcSinh[x/3]

Rule 641

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*(a + c*x^2)^(p + 1))/(2*c*(p + 1)),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0103785, size = 18, normalized size = 1. \[ \sqrt{x^2+9}+2 \sinh ^{-1}\left (\frac{x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + x)/Sqrt[9 + x^2],x]

[Out]

Sqrt[9 + x^2] + 2*ArcSinh[x/3]

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Maple [A]  time = 0.042, size = 15, normalized size = 0.8 \begin{align*} 2\,{\it Arcsinh} \left ( x/3 \right ) +\sqrt{{x}^{2}+9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+x)/(x^2+9)^(1/2),x)

[Out]

2*arcsinh(1/3*x)+(x^2+9)^(1/2)

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Maxima [A]  time = 1.98458, size = 19, normalized size = 1.06 \begin{align*} \sqrt{x^{2} + 9} + 2 \, \operatorname{arsinh}\left (\frac{1}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(x^2+9)^(1/2),x, algorithm="maxima")

[Out]

sqrt(x^2 + 9) + 2*arcsinh(1/3*x)

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Fricas [A]  time = 1.73403, size = 58, normalized size = 3.22 \begin{align*} \sqrt{x^{2} + 9} - 2 \, \log \left (-x + \sqrt{x^{2} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(x^2+9)^(1/2),x, algorithm="fricas")

[Out]

sqrt(x^2 + 9) - 2*log(-x + sqrt(x^2 + 9))

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Sympy [A]  time = 0.143122, size = 14, normalized size = 0.78 \begin{align*} \sqrt{x^{2} + 9} + 2 \operatorname{asinh}{\left (\frac{x}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(x**2+9)**(1/2),x)

[Out]

sqrt(x**2 + 9) + 2*asinh(x/3)

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Giac [A]  time = 2.184, size = 30, normalized size = 1.67 \begin{align*} \sqrt{x^{2} + 9} - 2 \, \log \left (-x + \sqrt{x^{2} + 9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+x)/(x^2+9)^(1/2),x, algorithm="giac")

[Out]

sqrt(x^2 + 9) - 2*log(-x + sqrt(x^2 + 9))