3.113 \(\int \sqrt{-2+5 x-3 x^2} \, dx\)

Optimal. Leaf size=39 \[ -\frac{1}{12} \sqrt{-3 x^2+5 x-2} (5-6 x)-\frac{\sin ^{-1}(5-6 x)}{24 \sqrt{3}} \]

[Out]

-((5 - 6*x)*Sqrt[-2 + 5*x - 3*x^2])/12 - ArcSin[5 - 6*x]/(24*Sqrt[3])

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Rubi [A]  time = 0.0076886, antiderivative size = 39, 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} \[ -\frac{1}{12} \sqrt{-3 x^2+5 x-2} (5-6 x)-\frac{\sin ^{-1}(5-6 x)}{24 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-2 + 5*x - 3*x^2],x]

[Out]

-((5 - 6*x)*Sqrt[-2 + 5*x - 3*x^2])/12 - ArcSin[5 - 6*x]/(24*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+5 x-3 x^2} \, dx &=-\frac{1}{12} (5-6 x) \sqrt{-2+5 x-3 x^2}+\frac{1}{24} \int \frac{1}{\sqrt{-2+5 x-3 x^2}} \, dx\\ &=-\frac{1}{12} (5-6 x) \sqrt{-2+5 x-3 x^2}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,5-6 x\right )}{24 \sqrt{3}}\\ &=-\frac{1}{12} (5-6 x) \sqrt{-2+5 x-3 x^2}-\frac{\sin ^{-1}(5-6 x)}{24 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0185895, size = 40, normalized size = 1.03 \[ \left (\frac{x}{2}-\frac{5}{12}\right ) \sqrt{-3 x^2+5 x-2}-\frac{\sin ^{-1}(5-6 x)}{24 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-2 + 5*x - 3*x^2],x]

[Out]

(-5/12 + x/2)*Sqrt[-2 + 5*x - 3*x^2] - ArcSin[5 - 6*x]/(24*Sqrt[3])

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Maple [A]  time = 0.045, size = 32, normalized size = 0.8 \begin{align*}{\frac{\arcsin \left ( -5+6\,x \right ) \sqrt{3}}{72}}-{\frac{5-6\,x}{12}\sqrt{-3\,{x}^{2}+5\,x-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x^2+5*x-2)^(1/2),x)

[Out]

1/72*arcsin(-5+6*x)*3^(1/2)-1/12*(5-6*x)*(-3*x^2+5*x-2)^(1/2)

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Maxima [A]  time = 1.74272, size = 55, normalized size = 1.41 \begin{align*} \frac{1}{2} \, \sqrt{-3 \, x^{2} + 5 \, x - 2} x + \frac{1}{72} \, \sqrt{3} \arcsin \left (6 \, x - 5\right ) - \frac{5}{12} \, \sqrt{-3 \, x^{2} + 5 \, x - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^2+5*x-2)^(1/2),x, algorithm="maxima")

[Out]

1/2*sqrt(-3*x^2 + 5*x - 2)*x + 1/72*sqrt(3)*arcsin(6*x - 5) - 5/12*sqrt(-3*x^2 + 5*x - 2)

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Fricas [A]  time = 1.96606, size = 169, normalized size = 4.33 \begin{align*} \frac{1}{12} \, \sqrt{-3 \, x^{2} + 5 \, x - 2}{\left (6 \, x - 5\right )} - \frac{1}{72} \, \sqrt{3} \arctan \left (\frac{\sqrt{3} \sqrt{-3 \, x^{2} + 5 \, x - 2}{\left (6 \, x - 5\right )}}{6 \,{\left (3 \, x^{2} - 5 \, x + 2\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^2+5*x-2)^(1/2),x, algorithm="fricas")

[Out]

1/12*sqrt(-3*x^2 + 5*x - 2)*(6*x - 5) - 1/72*sqrt(3)*arctan(1/6*sqrt(3)*sqrt(-3*x^2 + 5*x - 2)*(6*x - 5)/(3*x^
2 - 5*x + 2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- 3 x^{2} + 5 x - 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x**2+5*x-2)**(1/2),x)

[Out]

Integral(sqrt(-3*x**2 + 5*x - 2), x)

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Giac [A]  time = 1.22334, size = 42, normalized size = 1.08 \begin{align*} \frac{1}{12} \, \sqrt{-3 \, x^{2} + 5 \, x - 2}{\left (6 \, x - 5\right )} + \frac{1}{72} \, \sqrt{3} \arcsin \left (6 \, x - 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x^2+5*x-2)^(1/2),x, algorithm="giac")

[Out]

1/12*sqrt(-3*x^2 + 5*x - 2)*(6*x - 5) + 1/72*sqrt(3)*arcsin(6*x - 5)