3.2.11 \(\int \sqrt {-2+4 x-3 x^2} \, dx\) [111]

Optimal. Leaf size=59 \[ -\frac {1}{6} (2-3 x) \sqrt {-2+4 x-3 x^2}+\frac {\tan ^{-1}\left (\frac {2-3 x}{\sqrt {3} \sqrt {-2+4 x-3 x^2}}\right )}{3 \sqrt {3}} \]

[Out]

1/9*arctan(1/3*(2-3*x)*3^(1/2)/(-3*x^2+4*x-2)^(1/2))*3^(1/2)-1/6*(2-3*x)*(-3*x^2+4*x-2)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {626, 635, 210} \begin {gather*} \frac {\text {ArcTan}\left (\frac {2-3 x}{\sqrt {3} \sqrt {-3 x^2+4 x-2}}\right )}{3 \sqrt {3}}-\frac {1}{6} (2-3 x) \sqrt {-3 x^2+4 x-2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-2 + 4*x - 3*x^2],x]

[Out]

-1/6*((2 - 3*x)*Sqrt[-2 + 4*x - 3*x^2]) + ArcTan[(2 - 3*x)/(Sqrt[3]*Sqrt[-2 + 4*x - 3*x^2])]/(3*Sqrt[3])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 626

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

Rule 635

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

Rubi steps

\begin {align*} \int \sqrt {-2+4 x-3 x^2} \, dx &=-\frac {1}{6} (2-3 x) \sqrt {-2+4 x-3 x^2}-\frac {1}{3} \int \frac {1}{\sqrt {-2+4 x-3 x^2}} \, dx\\ &=-\frac {1}{6} (2-3 x) \sqrt {-2+4 x-3 x^2}-\frac {2}{3} \text {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,\frac {4-6 x}{\sqrt {-2+4 x-3 x^2}}\right )\\ &=-\frac {1}{6} (2-3 x) \sqrt {-2+4 x-3 x^2}+\frac {\tan ^{-1}\left (\frac {2-3 x}{\sqrt {3} \sqrt {-2+4 x-3 x^2}}\right )}{3 \sqrt {3}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.10, size = 59, normalized size = 1.00 \begin {gather*} \frac {1}{6} (-2+3 x) \sqrt {-2+4 x-3 x^2}-\frac {i \log \left (2 i-3 i x+\sqrt {-6+12 x-9 x^2}\right )}{3 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-2 + 4*x - 3*x^2],x]

[Out]

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

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Maple [A]
time = 0.79, size = 46, normalized size = 0.78

method	result	size
default	\(-\frac {\left (-6 x +4\right ) \sqrt {-3 x^{2}+4 x -2}}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -\frac {2}{3}\right )}{\sqrt {-3 x^{2}+4 x -2}}\right )}{9}\)	\(46\)
risch	\(-\frac {\left (3 x^{2}-4 x +2\right ) \left (-2+3 x \right )}{6 \sqrt {-3 x^{2}+4 x -2}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x -\frac {2}{3}\right )}{\sqrt {-3 x^{2}+4 x -2}}\right )}{9}\)	\(56\)
trager	\(\left (-\frac {1}{3}+\frac {x}{2}\right ) \sqrt {-3 x^{2}+4 x -2}+\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (3 x \RootOf \left (\textit {\_Z}^{2}+3\right )+3 \sqrt {-3 x^{2}+4 x -2}-2 \RootOf \left (\textit {\_Z}^{2}+3\right )\right )}{9}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(-6*x+4)*(-3*x^2+4*x-2)^(1/2)-1/9*3^(1/2)*arctan(3^(1/2)*(x-2/3)/(-3*x^2+4*x-2)^(1/2))

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Maxima [C] Result contains complex when optimal does not.
time = 0.54, size = 46, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, \sqrt {-3 \, x^{2} + 4 \, x - 2} x + \frac {1}{9} i \, \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains complex when optimal does not.
time = 2.45, size = 86, normalized size = 1.46 \begin {gather*} \frac {1}{6} \, \sqrt {-3 \, x^{2} + 4 \, x - 2} {\left (3 \, x - 2\right )} - \frac {1}{18} i \, \sqrt {3} \log \left (-\frac {2 \, {\left (i \, \sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x - 2} + 3 \, x - 2\right )}}{x}\right ) + \frac {1}{18} i \, \sqrt {3} \log \left (-\frac {2 \, {\left (-i \, \sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x - 2} + 3 \, x - 2\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(-3*x^2 + 4*x - 2)*(3*x - 2) - 1/18*I*sqrt(3)*log(-2*(I*sqrt(3)*sqrt(-3*x^2 + 4*x - 2) + 3*x - 2)/x) +
1/18*I*sqrt(3)*log(-2*(-I*sqrt(3)*sqrt(-3*x^2 + 4*x - 2) + 3*x - 2)/x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- 3 x^{2} + 4 x - 2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-3*x**2 + 4*x - 2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-3*x^2 + 4*x - 2), x)

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Mupad [B]
time = 0.05, size = 36, normalized size = 0.61 \begin {gather*} \frac {\sqrt {3}\,\mathrm {asin}\left (\frac {\sqrt {2}\,\left (3\,x-2\right )\,1{}\mathrm {i}}{2}\right )}{9}+\left (\frac {x}{2}-\frac {1}{3}\right )\,\sqrt {-3\,x^2+4\,x-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - 3*x^2 - 2)^(1/2),x)

[Out]

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

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