\(\int \frac {x}{\sqrt {2+4 x-3 x^2}} \, dx\) [2412]

Optimal result

Integrand size = 16, antiderivative size = 40 \[ \int \frac {x}{\sqrt {2+4 x-3 x^2}} \, dx=-\frac {1}{3} \sqrt {2+4 x-3 x^2}-\frac {2 \arcsin \left (\frac {2-3 x}{\sqrt {10}}\right )}{3 \sqrt {3}} \]

[Out]

-2/9*arcsin(1/10*(2-3*x)*10^(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, 222} \[ \int \frac {x}{\sqrt {2+4 x-3 x^2}} \, dx=-\frac {2 \arcsin \left (\frac {2-3 x}{\sqrt {10}}\right )}{3 \sqrt {3}}-\frac {1}{3} \sqrt {-3 x^2+4 x+2} \]

[In]

Int[x/Sqrt[2 + 4*x - 3*x^2],x]

[Out]

-1/3*Sqrt[2 + 4*x - 3*x^2] - (2*ArcSin[(2 - 3*x)/Sqrt[10]])/(3*Sqrt[3])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[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}{40}}} \, dx,x,4-6 x\right )}{3 \sqrt {30}} \\ & = -\frac {1}{3} \sqrt {2+4 x-3 x^2}-\frac {2 \sin ^{-1}\left (\frac {2-3 x}{\sqrt {10}}\right )}{3 \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.50 \[ \int \frac {x}{\sqrt {2+4 x-3 x^2}} \, dx=-\frac {1}{3} \sqrt {2+4 x-3 x^2}-\frac {4 \arctan \left (\frac {\sqrt {3} x}{\sqrt {2}-\sqrt {2+4 x-3 x^2}}\right )}{3 \sqrt {3}} \]

[In]

Integrate[x/Sqrt[2 + 4*x - 3*x^2],x]

[Out]

-1/3*Sqrt[2 + 4*x - 3*x^2] - (4*ArcTan[(Sqrt[3]*x)/(Sqrt[2] - Sqrt[2 + 4*x - 3*x^2])])/(3*Sqrt[3])

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.75

method	result	size
default	\(-\frac {\sqrt {-3 x^{2}+4 x +2}}{3}+\frac {2 \sqrt {3}\, \arcsin \left (\frac {3 \sqrt {10}\, \left (x -\frac {2}{3}\right )}{10}\right )}{9}\)	\(30\)
risch	\(\frac {3 x^{2}-4 x -2}{3 \sqrt {-3 x^{2}+4 x +2}}+\frac {2 \sqrt {3}\, \arcsin \left (\frac {3 \sqrt {10}\, \left (x -\frac {2}{3}\right )}{10}\right )}{9}\)	\(40\)
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)*arcsin(3/10*10^(1/2)*(x-2/3))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.38 \[ \int \frac {x}{\sqrt {2+4 x-3 x^2}} \, dx=-\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x + 2} {\left (3 \, x - 2\right )}}{3 \, {\left (3 \, x^{2} - 4 \, x - 2\right )}}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} \]

[In]

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

[Out]

-2/9*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(-3*x^2 + 4*x + 2)*(3*x - 2)/(3*x^2 - 4*x - 2)) - 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 \[ \int \frac {x}{\sqrt {2+4 x-3 x^2}} \, dx=- \frac {\sqrt {- 3 x^{2} + 4 x + 2}}{3} + \frac {2 \sqrt {3} \operatorname {asin}{\left (\frac {3 \sqrt {10} \left (x - \frac {2}{3}\right )}{10} \right )}}{9} \]

[In]

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

[Out]

-sqrt(-3*x**2 + 4*x + 2)/3 + 2*sqrt(3)*asin(3*sqrt(10)*(x - 2/3)/10)/9

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78 \[ \int \frac {x}{\sqrt {2+4 x-3 x^2}} \, dx=-\frac {2}{9} \, \sqrt {3} \arcsin \left (-\frac {1}{10} \, \sqrt {10} {\left (3 \, x - 2\right )}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} \]

[In]

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

[Out]

-2/9*sqrt(3)*arcsin(-1/10*sqrt(10)*(3*x - 2)) - 1/3*sqrt(-3*x^2 + 4*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.78 \[ \int \frac {x}{\sqrt {2+4 x-3 x^2}} \, dx=\frac {2}{9} \, \sqrt {3} \arcsin \left (\frac {1}{10} \, \sqrt {10} {\left (3 \, x - 2\right )}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} \]

[In]

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

[Out]

2/9*sqrt(3)*arcsin(1/10*sqrt(10)*(3*x - 2)) - 1/3*sqrt(-3*x^2 + 4*x + 2)

Mupad [B] (verification not implemented)

Time = 10.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.15 \[ \int \frac {x}{\sqrt {2+4 x-3 x^2}} \, dx=-\frac {\sqrt {-3\,x^2+4\,x+2}}{3}-\frac {\sqrt {3}\,\ln \left (\sqrt {-3\,x^2+4\,x+2}+\frac {\sqrt {3}\,\left (3\,x-2\right )\,1{}\mathrm {i}}{3}\right )\,2{}\mathrm {i}}{9} \]

[In]

int(x/(4*x - 3*x^2 + 2)^(1/2),x)

[Out]

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