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

Optimal. Leaf size=35 \[ -\frac{1}{16} \sqrt{3 x-4 x^2} (3-8 x)-\frac{9}{64} \sin ^{-1}\left (1-\frac{8 x}{3}\right ) \]

[Out]

-((3 - 8*x)*Sqrt[3*x - 4*x^2])/16 - (9*ArcSin[1 - (8*x)/3])/64

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Rubi [A]  time = 0.0091721, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {612, 619, 216} \[ -\frac{1}{16} \sqrt{3 x-4 x^2} (3-8 x)-\frac{9}{64} \sin ^{-1}\left (1-\frac{8 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3*x - 4*x^2],x]

[Out]

-((3 - 8*x)*Sqrt[3*x - 4*x^2])/16 - (9*ArcSin[1 - (8*x)/3])/64

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{3 x-4 x^2} \, dx &=-\frac{1}{16} (3-8 x) \sqrt{3 x-4 x^2}+\frac{9}{32} \int \frac{1}{\sqrt{3 x-4 x^2}} \, dx\\ &=-\frac{1}{16} (3-8 x) \sqrt{3 x-4 x^2}-\frac{3}{64} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{9}}} \, dx,x,3-8 x\right )\\ &=-\frac{1}{16} (3-8 x) \sqrt{3 x-4 x^2}-\frac{9}{64} \sin ^{-1}\left (1-\frac{8 x}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0351241, size = 58, normalized size = 1.66 \[ \frac{-2 x \left (32 x^2-36 x+9\right )-9 \sqrt{3-4 x} \sqrt{x} \sin ^{-1}\left (\sqrt{1-\frac{4 x}{3}}\right )}{32 \sqrt{-x (4 x-3)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3*x - 4*x^2],x]

[Out]

(-2*x*(9 - 36*x + 32*x^2) - 9*Sqrt[3 - 4*x]*Sqrt[x]*ArcSin[Sqrt[1 - (4*x)/3]])/(32*Sqrt[-(x*(-3 + 4*x))])

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Maple [A]  time = 0.045, size = 28, normalized size = 0.8 \begin{align*}{\frac{9}{64}\arcsin \left ( -1+{\frac{8\,x}{3}} \right ) }-{\frac{3-8\,x}{16}\sqrt{-4\,{x}^{2}+3\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/64*arcsin(-1+8/3*x)-1/16*(3-8*x)*(-4*x^2+3*x)^(1/2)

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Maxima [A]  time = 1.75766, size = 49, normalized size = 1.4 \begin{align*} \frac{1}{2} \, \sqrt{-4 \, x^{2} + 3 \, x} x - \frac{3}{16} \, \sqrt{-4 \, x^{2} + 3 \, x} - \frac{9}{64} \, \arcsin \left (-\frac{8}{3} \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-4*x^2 + 3*x)*x - 3/16*sqrt(-4*x^2 + 3*x) - 9/64*arcsin(-8/3*x + 1)

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Fricas [A]  time = 2.18149, size = 101, normalized size = 2.89 \begin{align*} \frac{1}{16} \, \sqrt{-4 \, x^{2} + 3 \, x}{\left (8 \, x - 3\right )} - \frac{9}{32} \, \arctan \left (\frac{\sqrt{-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sqrt(-4*x^2 + 3*x)*(8*x - 3) - 9/32*arctan(1/2*sqrt(-4*x^2 + 3*x)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-4*x**2 + 3*x), x)

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Giac [A]  time = 1.30559, size = 36, normalized size = 1.03 \begin{align*} \frac{1}{16} \, \sqrt{-4 \, x^{2} + 3 \, x}{\left (8 \, x - 3\right )} + \frac{9}{64} \, \arcsin \left (\frac{8}{3} \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sqrt(-4*x^2 + 3*x)*(8*x - 3) + 9/64*arcsin(8/3*x - 1)