3.923 \(\int \frac {2+x}{\sqrt {4 x-x^2}} \, dx\)

Optimal. Leaf size=26 \[ -\sqrt {4 x-x^2}-4 \sin ^{-1}\left (1-\frac {x}{2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {640, 619, 216} \[ -\sqrt {4 x-x^2}-4 \sin ^{-1}\left (1-\frac {x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(2 + x)/Sqrt[4*x - x^2],x]

[Out]

-Sqrt[4*x - x^2] - 4*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 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 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 27, normalized size = 1.04 \[ -\sqrt {-((x-4) x)}-8 \sin ^{-1}\left (\sqrt {1-\frac {x}{4}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + x)/Sqrt[4*x - x^2],x]

[Out]

-Sqrt[-((-4 + x)*x)] - 8*ArcSin[Sqrt[1 - x/4]]

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fricas [A]  time = 0.75, size = 32, normalized size = 1.23 \[ -\sqrt {-x^{2} + 4 \, x} - 8 \, \arctan \left (\frac {\sqrt {-x^{2} + 4 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 4*x) - 8*arctan(sqrt(-x^2 + 4*x)/x)

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giac [A]  time = 0.61, size = 22, normalized size = 0.85 \[ -\sqrt {-x^{2} + 4 \, x} + 4 \, \arcsin \left (\frac {1}{2} \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 23, normalized size = 0.88 \[ 4 \arcsin \left (\frac {x}{2}-1\right )-\sqrt {-x^{2}+4 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.93, size = 22, normalized size = 0.85 \[ -\sqrt {-x^{2} + 4 \, x} - 4 \, \arcsin \left (-\frac {1}{2} \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.53, size = 22, normalized size = 0.85 \[ 4\,\mathrm {asin}\left (\frac {x}{2}-1\right )-\sqrt {4\,x-x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 2)/(4*x - x^2)^(1/2),x)

[Out]

4*asin(x/2 - 1) - (4*x - x^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 2}{\sqrt {- x \left (x - 4\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x + 2)/sqrt(-x*(x - 4)), x)

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