\(\int \frac {2+x}{\sqrt {9+x^2}} \, dx\) [587]

Optimal result

Integrand size = 13, antiderivative size = 18 \[ \int \frac {2+x}{\sqrt {9+x^2}} \, dx=\sqrt {9+x^2}+2 \text {arcsinh}\left (\frac {x}{3}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {655, 221} \[ \int \frac {2+x}{\sqrt {9+x^2}} \, dx=2 \text {arcsinh}\left (\frac {x}{3}\right )+\sqrt {x^2+9} \]

[In]

Int[(2 + x)/Sqrt[9 + x^2],x]

[Out]

Sqrt[9 + x^2] + 2*ArcSinh[x/3]

Rule 221

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 655

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*} \text {integral}& = \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] (verified)

Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int \frac {2+x}{\sqrt {9+x^2}} \, dx=\sqrt {9+x^2}-2 \log \left (-x+\sqrt {9+x^2}\right ) \]

[In]

Integrate[(2 + x)/Sqrt[9 + x^2],x]

[Out]

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

Maple [A] (verified)

Time = 2.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
default	\(2 \,\operatorname {arcsinh}\left (\frac {x}{3}\right )+\sqrt {x^{2}+9}\)	\(15\)
risch	\(2 \,\operatorname {arcsinh}\left (\frac {x}{3}\right )+\sqrt {x^{2}+9}\)	\(15\)
trager	\(\sqrt {x^{2}+9}+2 \ln \left (x +\sqrt {x^{2}+9}\right )\)	\(21\)
meijerg	\(2 \,\operatorname {arcsinh}\left (\frac {x}{3}\right )+\frac {-3 \sqrt {\pi }+3 \sqrt {\pi }\, \sqrt {\frac {x^{2}}{9}+1}}{\sqrt {\pi }}\)	\(33\)

[In]

int((2+x)/(x^2+9)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {2+x}{\sqrt {9+x^2}} \, dx=\sqrt {x^{2} + 9} - 2 \, \log \left (-x + \sqrt {x^{2} + 9}\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {2+x}{\sqrt {9+x^2}} \, dx=\sqrt {x^{2} + 9} + 2 \operatorname {asinh}{\left (\frac {x}{3} \right )} \]

[In]

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

[Out]

sqrt(x**2 + 9) + 2*asinh(x/3)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {2+x}{\sqrt {9+x^2}} \, dx=\sqrt {x^{2} + 9} + 2 \, \operatorname {arsinh}\left (\frac {1}{3} \, x\right ) \]

[In]

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

[Out]

sqrt(x^2 + 9) + 2*arcsinh(1/3*x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {2+x}{\sqrt {9+x^2}} \, dx=\sqrt {x^{2} + 9} - 2 \, \log \left (-x + \sqrt {x^{2} + 9}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.34 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {2+x}{\sqrt {9+x^2}} \, dx=2\,\mathrm {asinh}\left (\frac {x}{3}\right )+\sqrt {x^2+9} \]

[In]

int((x + 2)/(x^2 + 9)^(1/2),x)

[Out]

2*asinh(x/3) + (x^2 + 9)^(1/2)