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]

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

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 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 {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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 1.01, size = 38, normalized size = 1.09 \[ \frac {1}{16} \, \sqrt {-4 \, x^{2} + 3 \, x} {\left (8 \, x - 3\right )} - \frac {9}{32} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 3 \, x}}{2 \, x}\right ) \]

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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giac [A]  time = 0.48, size = 27, normalized size = 0.77 \[ \frac {1}{16} \, \sqrt {-4 \, x^{2} + 3 \, x} {\left (8 \, x - 3\right )} + \frac {9}{64} \, \arcsin \left (\frac {8}{3} \, x - 1\right ) \]

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)

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maple [A]  time = 0.05, size = 28, normalized size = 0.80 \[ \frac {9 \arcsin \left (\frac {8 x}{3}-1\right )}{64}-\frac {\left (-8 x +3\right ) \sqrt {-4 x^{2}+3 x}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.88, size = 36, normalized size = 1.03 \[ \frac {1}{2} \, \sqrt {-4 \, x^{2} + 3 \, x} x - \frac {3}{16} \, \sqrt {-4 \, x^{2} + 3 \, x} - \frac {9}{64} \, \arcsin \left (-\frac {8}{3} \, x + 1\right ) \]

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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mupad [B]  time = 0.05, size = 26, normalized size = 0.74 \[ \frac {9\,\mathrm {asin}\left (\frac {8\,x}{3}-1\right )}{64}+\left (\frac {x}{2}-\frac {3}{16}\right )\,\sqrt {3\,x-4\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x - 4*x^2)^(1/2),x)

[Out]

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

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

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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