3.1.9 \(\int \sqrt {3 x-4 x^2} \, dx\) [9]

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

[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 = {626, 633, 222} \begin {gather*} -\frac {9}{64} \text {ArcSin}\left (1-\frac {8 x}{3}\right )-\frac {1}{16} \sqrt {3 x-4 x^2} (3-8 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[3*x - 4*x^2],x]

[Out]

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

Rule 222

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 626

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 633

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} \text {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 = 0.05, size = 55, normalized size = 1.57 \begin {gather*} \frac {1}{32} \sqrt {-x (-3+4 x)} \left (-6+16 x+\frac {9 \log \left (-2 \sqrt {x}+\sqrt {-3+4 x}\right )}{\sqrt {x} \sqrt {-3+4 x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3*x - 4*x^2],x]

[Out]

(Sqrt[-(x*(-3 + 4*x))]*(-6 + 16*x + (9*Log[-2*Sqrt[x] + Sqrt[-3 + 4*x]])/(Sqrt[x]*Sqrt[-3 + 4*x])))/32

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Maple [A]
time = 0.41, size = 28, normalized size = 0.80

method	result	size
default	\(\frac {9 \arcsin \left (-1+\frac {8 x}{3}\right )}{64}-\frac {\left (3-8 x \right ) \sqrt {-4 x^{2}+3 x}}{16}\)	\(28\)
risch	\(-\frac {\left (-3+8 x \right ) x \left (-3+4 x \right )}{16 \sqrt {-x \left (-3+4 x \right )}}+\frac {9 \arcsin \left (-1+\frac {8 x}{3}\right )}{64}\)	\(33\)
meijerg	\(-\frac {9 i \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {3}\, \left (3-8 x \right ) \sqrt {-\frac {4 x}{3}+1}}{9}+\frac {i \sqrt {\pi }\, \arcsin \left (\frac {2 \sqrt {3}\, \sqrt {x}}{3}\right )}{2}\right )}{16 \sqrt {\pi }}\)	\(47\)
trager	\(\left (-\frac {3}{16}+\frac {x}{2}\right ) \sqrt {-4 x^{2}+3 x}+\frac {9 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-8 x \RootOf \left (\textit {\_Z}^{2}+1\right )+4 \sqrt {-4 x^{2}+3 x}+3 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{64}\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.52, size = 36, normalized size = 1.03 \begin {gather*} \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 {gather*}

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 = 1.60, size = 38, normalized size = 1.09 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- 4 x^{2} + 3 x}\, dx \end {gather*}

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 = 2.93, size = 27, normalized size = 0.77 \begin {gather*} \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 {gather*}

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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Mupad [B]
time = 0.05, size = 26, normalized size = 0.74 \begin {gather*} \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} \end {gather*}

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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