\(\int \sqrt {5+x^2} \, dx\) [152]

Optimal result

Integrand size = 9, antiderivative size = 27 \[ \int \sqrt {5+x^2} \, dx=\frac {1}{2} x \sqrt {5+x^2}+\frac {5}{2} \text {arcsinh}\left (\frac {x}{\sqrt {5}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {201, 221} \[ \int \sqrt {5+x^2} \, dx=\frac {5}{2} \text {arcsinh}\left (\frac {x}{\sqrt {5}}\right )+\frac {1}{2} \sqrt {x^2+5} x \]

[In]

Int[Sqrt[5 + x^2],x]

[Out]

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

Rule 201

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 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]

Rubi steps \begin{align*} \text {integral}& = \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} \text {arcsinh}\left (\frac {x}{\sqrt {5}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \sqrt {5+x^2} \, dx=\frac {1}{2} x \sqrt {5+x^2}-\frac {5}{2} \log \left (-x+\sqrt {5+x^2}\right ) \]

[In]

Integrate[Sqrt[5 + x^2],x]

[Out]

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

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78

method	result	size
default	\(\frac {5 \,\operatorname {arcsinh}\left (\frac {x \sqrt {5}}{5}\right )}{2}+\frac {x \sqrt {x^{2}+5}}{2}\)	\(21\)
risch	\(\frac {5 \,\operatorname {arcsinh}\left (\frac {x \sqrt {5}}{5}\right )}{2}+\frac {x \sqrt {x^{2}+5}}{2}\)	\(21\)
trager	\(\frac {x \sqrt {x^{2}+5}}{2}-\frac {5 \ln \left (x -\sqrt {x^{2}+5}\right )}{2}\)	\(26\)
meijerg	\(-\frac {5 \left (-\frac {2 \sqrt {\pi }\, x \sqrt {5}\, \sqrt {1+\frac {x^{2}}{5}}}{5}-2 \sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {5}}{5}\right )\right )}{4 \sqrt {\pi }}\)	\(37\)
pseudoelliptic	\(\frac {x \sqrt {x^{2}+5}}{2}+\frac {5 \ln \left (\frac {\sqrt {x^{2}+5}+x}{x}\right )}{4}-\frac {5 \ln \left (\frac {\sqrt {x^{2}+5}-x}{x}\right )}{4}\)	\(46\)

[In]

int((x^2+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \sqrt {5+x^2} \, dx=\frac {1}{2} \, \sqrt {x^{2} + 5} x - \frac {5}{2} \, \log \left (-x + \sqrt {x^{2} + 5}\right ) \]

[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))

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \sqrt {5+x^2} \, dx=\frac {x \sqrt {x^{2} + 5}}{2} + \frac {5 \operatorname {asinh}{\left (\frac {\sqrt {5} x}{5} \right )}}{2} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \sqrt {5+x^2} \, dx=\frac {1}{2} \, \sqrt {x^{2} + 5} x + \frac {5}{2} \, \operatorname {arsinh}\left (\frac {1}{5} \, \sqrt {5} x\right ) \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \sqrt {5+x^2} \, dx=\frac {1}{2} \, \sqrt {x^{2} + 5} x - \frac {5}{2} \, \log \left (-x + \sqrt {x^{2} + 5}\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \sqrt {5+x^2} \, dx=\frac {5\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x}{5}\right )}{2}+\frac {x\,\sqrt {x^2+5}}{2} \]

[In]

int((x^2 + 5)^(1/2),x)

[Out]

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