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

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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} \begin {gather*} \sqrt {x^2+9}+2 \sinh ^{-1}\left (\frac {x}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + x)/Sqrt[9 + x^2],x]

[Out]

Sqrt[9 + x^2] + 2*ArcSinh[x/3]

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]

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]

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*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \sqrt {x^2+9}+2 \sinh ^{-1}\left (\frac {x}{3}\right ) \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.13, size = 26, normalized size = 1.44 \begin {gather*} \sqrt {x^2+9}-2 \log \left (\sqrt {x^2+9}-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(2 + x)/Sqrt[9 + x^2],x]

[Out]

Sqrt[9 + x^2] - 2*Log[-x + Sqrt[9 + x^2]]

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fricas [A]  time = 0.40, size = 22, normalized size = 1.22 \begin {gather*} \sqrt {x^{2} + 9} - 2 \, \log \left (-x + \sqrt {x^{2} + 9}\right ) \end {gather*}

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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giac [A]  time = 0.21, size = 22, normalized size = 1.22 \begin {gather*} \sqrt {x^{2} + 9} - 2 \, \log \left (-x + \sqrt {x^{2} + 9}\right ) \end {gather*}

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))

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maple [A]  time = 0.04, size = 15, normalized size = 0.83 \begin {gather*} 2 \arcsinh \left (\frac {x}{3}\right )+\sqrt {x^{2}+9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.92, size = 14, normalized size = 0.78 \begin {gather*} \sqrt {x^{2} + 9} + 2 \, \operatorname {arsinh}\left (\frac {1}{3} \, x\right ) \end {gather*}

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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mupad [B]  time = 0.04, size = 14, normalized size = 0.78 \begin {gather*} 2\,\mathrm {asinh}\left (\frac {x}{3}\right )+\sqrt {x^2+9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2)/(x^2 + 9)^(1/2),x)

[Out]

2*asinh(x/3) + (x^2 + 9)^(1/2)

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sympy [A]  time = 0.15, size = 14, normalized size = 0.78 \begin {gather*} \sqrt {x^{2} + 9} + 2 \operatorname {asinh}{\left (\frac {x}{3} \right )} \end {gather*}

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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