\(\int \frac {x}{\sqrt {2+4 x+3 x^2}} \, dx\) [2411]

Optimal result

Integrand size = 16, antiderivative size = 40 \[ \int \frac {x}{\sqrt {2+4 x+3 x^2}} \, dx=\frac {1}{3} \sqrt {2+4 x+3 x^2}-\frac {2 \text {arcsinh}\left (\frac {2+3 x}{\sqrt {2}}\right )}{3 \sqrt {3}} \]

[Out]

-2/9*arcsinh(1/2*(2+3*x)*2^(1/2))*3^(1/2)+1/3*(3*x^2+4*x+2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {654, 633, 221} \[ \int \frac {x}{\sqrt {2+4 x+3 x^2}} \, dx=\frac {1}{3} \sqrt {3 x^2+4 x+2}-\frac {2 \text {arcsinh}\left (\frac {3 x+2}{\sqrt {2}}\right )}{3 \sqrt {3}} \]

[In]

Int[x/Sqrt[2 + 4*x + 3*x^2],x]

[Out]

Sqrt[2 + 4*x + 3*x^2]/3 - (2*ArcSinh[(2 + 3*x)/Sqrt[2]])/(3*Sqrt[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 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}& = \frac {1}{3} \sqrt {2+4 x+3 x^2}-\frac {2}{3} \int \frac {1}{\sqrt {2+4 x+3 x^2}} \, dx \\ & = \frac {1}{3} \sqrt {2+4 x+3 x^2}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{8}}} \, dx,x,4+6 x\right )}{3 \sqrt {6}} \\ & = \frac {1}{3} \sqrt {2+4 x+3 x^2}-\frac {2 \sinh ^{-1}\left (\frac {2+3 x}{\sqrt {2}}\right )}{3 \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.20 \[ \int \frac {x}{\sqrt {2+4 x+3 x^2}} \, dx=\frac {1}{9} \left (3 \sqrt {2+4 x+3 x^2}+2 \sqrt {3} \log \left (-2-3 x+\sqrt {6+12 x+9 x^2}\right )\right ) \]

[In]

Integrate[x/Sqrt[2 + 4*x + 3*x^2],x]

[Out]

(3*Sqrt[2 + 4*x + 3*x^2] + 2*Sqrt[3]*Log[-2 - 3*x + Sqrt[6 + 12*x + 9*x^2]])/9

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.75

method	result	size
default	\(\frac {\sqrt {3 x^{2}+4 x +2}}{3}-\frac {2 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {3 \sqrt {2}\, \left (x +\frac {2}{3}\right )}{2}\right )}{9}\)	\(30\)
risch	\(\frac {\sqrt {3 x^{2}+4 x +2}}{3}-\frac {2 \sqrt {3}\, \operatorname {arcsinh}\left (\frac {3 \sqrt {2}\, \left (x +\frac {2}{3}\right )}{2}\right )}{9}\)	\(30\)
trager	\(\frac {\sqrt {3 x^{2}+4 x +2}}{3}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +3 \sqrt {3 x^{2}+4 x +2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{9}\)	\(57\)

[In]

int(x/(3*x^2+4*x+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(3*x^2+4*x+2)^(1/2)-2/9*3^(1/2)*arcsinh(3/2*2^(1/2)*(x+2/3))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.30 \[ \int \frac {x}{\sqrt {2+4 x+3 x^2}} \, dx=\frac {1}{9} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 4 \, x + 2} {\left (3 \, x + 2\right )} - 9 \, x^{2} - 12 \, x - 5\right ) + \frac {1}{3} \, \sqrt {3 \, x^{2} + 4 \, x + 2} \]

[In]

integrate(x/(3*x^2+4*x+2)^(1/2),x, algorithm="fricas")

[Out]

1/9*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 4*x + 2)*(3*x + 2) - 9*x^2 - 12*x - 5) + 1/3*sqrt(3*x^2 + 4*x + 2)

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92 \[ \int \frac {x}{\sqrt {2+4 x+3 x^2}} \, dx=\frac {\sqrt {3 x^{2} + 4 x + 2}}{3} - \frac {2 \sqrt {3} \operatorname {asinh}{\left (\frac {3 \sqrt {2} \left (x + \frac {2}{3}\right )}{2} \right )}}{9} \]

[In]

integrate(x/(3*x**2+4*x+2)**(1/2),x)

[Out]

sqrt(3*x**2 + 4*x + 2)/3 - 2*sqrt(3)*asinh(3*sqrt(2)*(x + 2/3)/2)/9

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78 \[ \int \frac {x}{\sqrt {2+4 x+3 x^2}} \, dx=-\frac {2}{9} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 2\right )}\right ) + \frac {1}{3} \, \sqrt {3 \, x^{2} + 4 \, x + 2} \]

[In]

integrate(x/(3*x^2+4*x+2)^(1/2),x, algorithm="maxima")

[Out]

-2/9*sqrt(3)*arcsinh(1/2*sqrt(2)*(3*x + 2)) + 1/3*sqrt(3*x^2 + 4*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.20 \[ \int \frac {x}{\sqrt {2+4 x+3 x^2}} \, dx=\frac {2}{9} \, \sqrt {3} \log \left (-\sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 4 \, x + 2}\right )} - 2\right ) + \frac {1}{3} \, \sqrt {3 \, x^{2} + 4 \, x + 2} \]

[In]

integrate(x/(3*x^2+4*x+2)^(1/2),x, algorithm="giac")

[Out]

2/9*sqrt(3)*log(-sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 4*x + 2)) - 2) + 1/3*sqrt(3*x^2 + 4*x + 2)

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.10 \[ \int \frac {x}{\sqrt {2+4 x+3 x^2}} \, dx=\frac {\sqrt {3\,x^2+4\,x+2}}{3}-\frac {2\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+4\,x+2}+\frac {\sqrt {3}\,\left (3\,x+2\right )}{3}\right )}{9} \]

[In]

int(x/(4*x + 3*x^2 + 2)^(1/2),x)

[Out]

(4*x + 3*x^2 + 2)^(1/2)/3 - (2*3^(1/2)*log((4*x + 3*x^2 + 2)^(1/2) + (3^(1/2)*(3*x + 2))/3))/9