3.152 \(\int \sqrt{5+x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{2} \sqrt{x^2+5} x+\frac{5}{2} \sinh ^{-1}\left (\frac{x}{\sqrt{5}}\right ) \]

[Out]

(x*Sqrt[5 + x^2])/2 + (5*ArcSinh[x/Sqrt[5]])/2

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Rubi [A]  time = 0.0032337, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {195, 215} \[ \frac{1}{2} \sqrt{x^2+5} x+\frac{5}{2} \sinh ^{-1}\left (\frac{x}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[5 + x^2],x]

[Out]

(x*Sqrt[5 + x^2])/2 + (5*ArcSinh[x/Sqrt[5]])/2

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 \sqrt{5+x^2} \, dx &=\frac{1}{2} x \sqrt{5+x^2}+\frac{5}{2} \int \frac{1}{\sqrt{5+x^2}} \, dx\\ &=\frac{1}{2} x \sqrt{5+x^2}+\frac{5}{2} \sinh ^{-1}\left (\frac{x}{\sqrt{5}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0063963, size = 27, normalized size = 1. \[ \frac{1}{2} \sqrt{x^2+5} x+\frac{5}{2} \sinh ^{-1}\left (\frac{x}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[5 + x^2],x]

[Out]

(x*Sqrt[5 + x^2])/2 + (5*ArcSinh[x/Sqrt[5]])/2

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Maple [A]  time = 0.003, size = 21, normalized size = 0.8 \begin{align*}{\frac{5}{2}{\it Arcsinh} \left ({\frac{x\sqrt{5}}{5}} \right ) }+{\frac{x}{2}\sqrt{{x}^{2}+5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+5)^(1/2),x)

[Out]

5/2*arcsinh(1/5*x*5^(1/2))+1/2*x*(x^2+5)^(1/2)

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Maxima [A]  time = 1.41488, size = 27, normalized size = 1. \begin{align*} \frac{1}{2} \, \sqrt{x^{2} + 5} x + \frac{5}{2} \, \operatorname{arsinh}\left (\frac{1}{5} \, \sqrt{5} x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+5)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(x^2 + 5)*x + 5/2*arcsinh(1/5*sqrt(5)*x)

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Fricas [A]  time = 1.05863, size = 69, normalized size = 2.56 \begin{align*} \frac{1}{2} \, \sqrt{x^{2} + 5} x - \frac{5}{2} \, \log \left (-x + \sqrt{x^{2} + 5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+5)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(x^2 + 5)*x - 5/2*log(-x + sqrt(x^2 + 5))

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Sympy [A]  time = 0.190459, size = 24, normalized size = 0.89 \begin{align*} \frac{x \sqrt{x^{2} + 5}}{2} + \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{5} x}{5} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+5)**(1/2),x)

[Out]

x*sqrt(x**2 + 5)/2 + 5*asinh(sqrt(5)*x/5)/2

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Giac [A]  time = 1.09194, size = 34, normalized size = 1.26 \begin{align*} \frac{1}{2} \, \sqrt{x^{2} + 5} x - \frac{5}{2} \, \log \left (-x + \sqrt{x^{2} + 5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+5)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(x^2 + 5)*x - 5/2*log(-x + sqrt(x^2 + 5))