\(\int \frac {x}{\sqrt {1+x+x^2}} \, dx\) [153]

Optimal result

Integrand size = 12, antiderivative size = 27 \[ \int \frac {x}{\sqrt {1+x+x^2}} \, dx=\sqrt {1+x+x^2}-\frac {1}{2} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {654, 633, 221} \[ \int \frac {x}{\sqrt {1+x+x^2}} \, dx=\sqrt {x^2+x+1}-\frac {1}{2} \text {arcsinh}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]

[In]

Int[x/Sqrt[1 + x + x^2],x]

[Out]

Sqrt[1 + x + x^2] - ArcSinh[(1 + 2*x)/Sqrt[3]]/2

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 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \sqrt {1+x+x^2}-\frac {1}{2} \int \frac {1}{\sqrt {1+x+x^2}} \, dx \\ & = \sqrt {1+x+x^2}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right )}{2 \sqrt {3}} \\ & = \sqrt {1+x+x^2}-\frac {1}{2} \text {arcsinh}\left (\frac {1+2 x}{\sqrt {3}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {x}{\sqrt {1+x+x^2}} \, dx=\sqrt {1+x+x^2}+\frac {1}{2} \log \left (-1-2 x+2 \sqrt {1+x+x^2}\right ) \]

[In]

Integrate[x/Sqrt[1 + x + x^2],x]

[Out]

Sqrt[1 + x + x^2] + Log[-1 - 2*x + 2*Sqrt[1 + x + x^2]]/2

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78

method	result	size
default	\(\sqrt {x^{2}+x +1}-\frac {\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{2}\)	\(21\)
risch	\(\sqrt {x^{2}+x +1}-\frac {\operatorname {arcsinh}\left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{2}\)	\(21\)
trager	\(\sqrt {x^{2}+x +1}-\frac {\ln \left (2 x +1+2 \sqrt {x^{2}+x +1}\right )}{2}\)	\(28\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {1+x+x^2}} \, dx=\sqrt {x^{2} + x + 1} + \frac {1}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {x}{\sqrt {1+x+x^2}} \, dx=\sqrt {x^{2} + x + 1} - \frac {\operatorname {asinh}{\left (\frac {2 \sqrt {3} \left (x + \frac {1}{2}\right )}{3} \right )}}{2} \]

[In]

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

[Out]

sqrt(x**2 + x + 1) - asinh(2*sqrt(3)*(x + 1/2)/3)/2

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {x}{\sqrt {1+x+x^2}} \, dx=\sqrt {x^{2} + x + 1} - \frac {1}{2} \, \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) \]

[In]

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

[Out]

sqrt(x^2 + x + 1) - 1/2*arcsinh(1/3*sqrt(3)*(2*x + 1))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {1+x+x^2}} \, dx=\sqrt {x^{2} + x + 1} + \frac {1}{2} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {x}{\sqrt {1+x+x^2}} \, dx=\sqrt {x^2+x+1}-\frac {\ln \left (x+\sqrt {x^2+x+1}+\frac {1}{2}\right )}{2} \]

[In]

int(x/(x + x^2 + 1)^(1/2),x)

[Out]

(x + x^2 + 1)^(1/2) - log(x + (x + x^2 + 1)^(1/2) + 1/2)/2