3.107 \(\int \sqrt{2+4 x-3 x^2} \, dx\)

Optimal. Leaf size=45 \[ -\frac{1}{6} \sqrt{-3 x^2+4 x+2} (2-3 x)-\frac{5 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}} \]

[Out]

-((2 - 3*x)*Sqrt[2 + 4*x - 3*x^2])/6 - (5*ArcSin[(2 - 3*x)/Sqrt[10]])/(3*Sqrt[3])

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Rubi [A]  time = 0.0159289, antiderivative size = 45, normalized size of antiderivative = 1., 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 = {612, 619, 216} \[ -\frac{1}{6} \sqrt{-3 x^2+4 x+2} (2-3 x)-\frac{5 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 + 4*x - 3*x^2],x]

[Out]

-((2 - 3*x)*Sqrt[2 + 4*x - 3*x^2])/6 - (5*ArcSin[(2 - 3*x)/Sqrt[10]])/(3*Sqrt[3])

Rule 612

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 619

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 216

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]

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{5}{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{1}{6} \sqrt{\frac{5}{6}} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{40}}} \, dx,x,4-6 x\right )\\ &=-\frac{1}{6} (2-3 x) \sqrt{2+4 x-3 x^2}-\frac{5 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0192693, size = 46, normalized size = 1.02 \[ \left (\frac{x}{2}-\frac{1}{3}\right ) \sqrt{-3 x^2+4 x+2}-\frac{5 \sin ^{-1}\left (\frac{2-3 x}{\sqrt{10}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + 4*x - 3*x^2],x]

[Out]

(-1/3 + x/2)*Sqrt[2 + 4*x - 3*x^2] - (5*ArcSin[(2 - 3*x)/Sqrt[10]])/(3*Sqrt[3])

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Maple [A]  time = 0.047, size = 35, normalized size = 0.8 \begin{align*} -{\frac{-6\,x+4}{12}\sqrt{-3\,{x}^{2}+4\,x+2}}+{\frac{5\,\sqrt{3}}{9}\arcsin \left ({\frac{3\,\sqrt{10}}{10} \left ( x-{\frac{2}{3}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(-6*x+4)*(-3*x^2+4*x+2)^(1/2)+5/9*3^(1/2)*arcsin(3/10*10^(1/2)*(x-2/3))

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Maxima [A]  time = 1.71788, size = 62, normalized size = 1.38 \begin{align*} \frac{1}{2} \, \sqrt{-3 \, x^{2} + 4 \, x + 2} x - \frac{5}{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} \end{align*}

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 - 5/9*sqrt(3)*arcsin(-1/10*sqrt(10)*(3*x - 2)) - 1/3*sqrt(-3*x^2 + 4*x + 2)

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Fricas [A]  time = 1.94372, size = 166, normalized size = 3.69 \begin{align*} \frac{1}{6} \, \sqrt{-3 \, x^{2} + 4 \, x + 2}{\left (3 \, x - 2\right )} - \frac{5}{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 ) \end{align*}

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

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

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 [A]  time = 1.26562, size = 49, normalized size = 1.09 \begin{align*} \frac{1}{6} \, \sqrt{-3 \, x^{2} + 4 \, x + 2}{\left (3 \, x - 2\right )} + \frac{5}{9} \, \sqrt{3} \arcsin \left (\frac{1}{10} \, \sqrt{10}{\left (3 \, x - 2\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sqrt(-3*x^2 + 4*x + 2)*(3*x - 2) + 5/9*sqrt(3)*arcsin(1/10*sqrt(10)*(3*x - 2))