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]

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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Rubi [A]  time = 0.01, antiderivative size = 39, 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} \[ -\frac {1}{12} \sqrt {-3 x^2+5 x-2} (5-6 x)-\frac {\sin ^{-1}(5-6 x)}{24 \sqrt {3}} \]

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 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+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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.92, size = 60, normalized size = 1.54 \[ \frac {1}{12} \, \sqrt {-3 \, x^{2} + 5 \, x - 2} {\left (6 \, x - 5\right )} - \frac {1}{72} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3 \, x^{2} + 5 \, x - 2} {\left (6 \, x - 5\right )}}{6 \, {\left (3 \, x^{2} - 5 \, x + 2\right )}}\right ) \]

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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giac [A]  time = 0.53, size = 31, normalized size = 0.79 \[ \frac {1}{12} \, \sqrt {-3 \, x^{2} + 5 \, x - 2} {\left (6 \, x - 5\right )} + \frac {1}{72} \, \sqrt {3} \arcsin \left (6 \, x - 5\right ) \]

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)

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maple [A]  time = 0.04, size = 32, normalized size = 0.82 \[ \frac {\sqrt {3}\, \arcsin \left (6 x -5\right )}{72}-\frac {\left (-6 x +5\right ) \sqrt {-3 x^{2}+5 x -2}}{12} \]

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

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maxima [A]  time = 3.08, size = 41, normalized size = 1.05 \[ \frac {1}{2} \, \sqrt {-3 \, x^{2} + 5 \, x - 2} x + \frac {1}{72} \, \sqrt {3} \arcsin \left (6 \, x - 5\right ) - \frac {5}{12} \, \sqrt {-3 \, x^{2} + 5 \, x - 2} \]

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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mupad [B]  time = 0.05, size = 30, normalized size = 0.77 \[ \frac {\sqrt {3}\,\mathrm {asin}\left (6\,x-5\right )}{72}+\left (\frac {x}{2}-\frac {5}{12}\right )\,\sqrt {-3\,x^2+5\,x-2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x - 3*x^2 - 2)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- 3 x^{2} + 5 x - 2}\, dx \]

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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