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]

-5/9*arcsin(1/10*(2-3*x)*10^(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 = 0.02, antiderivative size = 45, 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 = {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 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]

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]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.76, size = 60, normalized size = 1.33 \[ \frac {1}{6} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} {\left (3 \, x - 2\right )} - \frac {5}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3 \, x^{2} + 4 \, x + 2} {\left (3 \, x - 2\right )}}{3 \, {\left (3 \, x^{2} - 4 \, x - 2\right )}}\right ) \]

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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giac [A]  time = 0.47, size = 36, normalized size = 0.80 \[ \frac {1}{6} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} {\left (3 \, x - 2\right )} + \frac {5}{9} \, \sqrt {3} \arcsin \left (\frac {1}{10} \, \sqrt {10} {\left (3 \, x - 2\right )}\right ) \]

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))

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maple [A]  time = 0.05, size = 35, normalized size = 0.78 \[ \frac {5 \sqrt {3}\, \arcsin \left (\frac {3 \sqrt {10}\, \left (x -\frac {2}{3}\right )}{10}\right )}{9}-\frac {\left (-6 x +4\right ) \sqrt {-3 x^{2}+4 x +2}}{12} \]

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 = 2.97, size = 46, normalized size = 1.02 \[ \frac {1}{2} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} x - \frac {5}{9} \, \sqrt {3} \arcsin \left (-\frac {1}{10} \, \sqrt {10} {\left (3 \, x - 2\right )}\right ) - \frac {1}{3} \, \sqrt {-3 \, x^{2} + 4 \, x + 2} \]

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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mupad [B]  time = 0.05, size = 35, normalized size = 0.78 \[ \frac {5\,\sqrt {3}\,\mathrm {asin}\left (\frac {\sqrt {10}\,\left (3\,x-2\right )}{10}\right )}{9}+\left (\frac {x}{2}-\frac {1}{3}\right )\,\sqrt {-3\,x^2+4\,x+2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - 3*x^2 + 2)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- 3 x^{2} + 4 x + 2}\, dx \]

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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