3.980 \(\int \sqrt {(4-x) x} \, dx\)

Optimal. Leaf size=33 \[ -\frac {1}{2} \sqrt {4 x-x^2} (2-x)-2 \sin ^{-1}\left (1-\frac {x}{2}\right ) \]

[Out]

2*arcsin(-1+1/2*x)-1/2*(2-x)*(-x^2+4*x)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1979, 612, 619, 216} \[ -\frac {1}{2} \sqrt {4 x-x^2} (2-x)-2 \sin ^{-1}\left (1-\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(4 - x)*x],x]

[Out]

-((2 - x)*Sqrt[4*x - x^2])/2 - 2*ArcSin[1 - x/2]

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]

Rule 1979

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 32, normalized size = 0.97 \[ \frac {1}{2} (x-2) \sqrt {-((x-4) x)}-4 \sin ^{-1}\left (\sqrt {1-\frac {x}{4}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(4 - x)*x],x]

[Out]

((-2 + x)*Sqrt[-((-4 + x)*x)])/2 - 4*ArcSin[Sqrt[1 - x/4]]

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fricas [A]  time = 0.44, size = 35, normalized size = 1.06 \[ \frac {1}{2} \, \sqrt {-x^{2} + 4 \, x} {\left (x - 2\right )} - 4 \, \arctan \left (\frac {\sqrt {-x^{2} + 4 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-x^2 + 4*x)*(x - 2) - 4*arctan(sqrt(-x^2 + 4*x)/x)

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giac [A]  time = 0.39, size = 25, normalized size = 0.76 \[ \frac {1}{2} \, \sqrt {-x^{2} + 4 \, x} {\left (x - 2\right )} + 2 \, \arcsin \left (\frac {1}{2} \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-x^2 + 4*x)*(x - 2) + 2*arcsin(1/2*x - 1)

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maple [A]  time = 0.01, size = 28, normalized size = 0.85 \[ 2 \arcsin \left (\frac {x}{2}-1\right )-\frac {\left (-2 x +4\right ) \sqrt {-x^{2}+4 x}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(-2*x+4)*(-x^2+4*x)^(1/2)+2*arcsin(1/2*x-1)

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maxima [A]  time = 1.02, size = 36, normalized size = 1.09 \[ \frac {1}{2} \, \sqrt {-x^{2} + 4 \, x} x - \sqrt {-x^{2} + 4 \, x} - 2 \, \arcsin \left (-\frac {1}{2} \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-x^2 + 4*x)*x - sqrt(-x^2 + 4*x) - 2*arcsin(-1/2*x + 1)

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mupad [B]  time = 3.47, size = 26, normalized size = 0.79 \[ 2\,\mathrm {asin}\left (\frac {x}{2}-1\right )+\left (\frac {x}{2}-1\right )\,\sqrt {4\,x-x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x*(x - 4))^(1/2),x)

[Out]

2*asin(x/2 - 1) + (x/2 - 1)*(4*x - x^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {x \left (4 - x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x*(4 - x)), x)

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