3.2.23 \(\int \frac {x}{\sqrt {3-2 x-x^2}} \, dx\) [123]

Optimal. Leaf size=27 \[ -\sqrt {3-2 x-x^2}+\sin ^{-1}\left (\frac {1}{2} (-1-x)\right ) \]

[Out]

-arcsin(1/2+1/2*x)-(-x^2-2*x+3)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {654, 633, 222} \begin {gather*} \sin ^{-1}\left (\frac {1}{2} (-x-1)\right )-\sqrt {-x^2-2 x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[3 - 2*x - x^2],x]

[Out]

-Sqrt[3 - 2*x - x^2] + ArcSin[(-1 - x)/2]

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 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]

Rule 654

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

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Mathematica [A]
time = 0.06, size = 40, normalized size = 1.48 \begin {gather*} -\sqrt {3-2 x-x^2}+2 \tan ^{-1}\left (\frac {\sqrt {3-2 x-x^2}}{3+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[3 - 2*x - x^2],x]

[Out]

-Sqrt[3 - 2*x - x^2] + 2*ArcTan[Sqrt[3 - 2*x - x^2]/(3 + x)]

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x/Sqrt[3 - 2*x - x^2],x]')

[Out]

cought exception: maximum recursion depth exceeded while calling a Python object

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Maple [A]
time = 0.10, size = 24, normalized size = 0.89

method	result	size
default	\(-\arcsin \left (\frac {1}{2}+\frac {x}{2}\right )-\sqrt {-x^{2}-2 x +3}\)	\(24\)
risch	\(\frac {x^{2}+2 x -3}{\sqrt {-x^{2}-2 x +3}}-\arcsin \left (\frac {1}{2}+\frac {x}{2}\right )\)	\(31\)
trager	\(-\sqrt {-x^{2}-2 x +3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}-2 x +3}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsin(1/2+1/2*x)-(-x^2-2*x+3)^(1/2)

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Maxima [A]
time = 0.38, size = 21, normalized size = 0.78 \begin {gather*} -\sqrt {-x^{2} - 2 \, x + 3} + \arcsin \left (-\frac {1}{2} \, x - \frac {1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 - 2*x + 3) + arcsin(-1/2*x - 1/2)

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Fricas [A]
time = 0.34, size = 42, normalized size = 1.56 \begin {gather*} -\sqrt {-x^{2} - 2 \, x + 3} + \arctan \left (\frac {\sqrt {-x^{2} - 2 \, x + 3} {\left (x + 1\right )}}{x^{2} + 2 \, x - 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 - 2*x + 3) + arctan(sqrt(-x^2 - 2*x + 3)*(x + 1)/(x^2 + 2*x - 3))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {- \left (x - 1\right ) \left (x + 3\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(-(x - 1)*(x + 3)), x)

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Giac [A]
time = 0.00, size = 24, normalized size = 0.89 \begin {gather*} -\sqrt {-x^{2}-2 x+3}-\arcsin \left (\frac {x+1}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 - 2*x + 3) - arcsin(1/2*x + 1/2)

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Mupad [B]
time = 0.22, size = 38, normalized size = 1.41 \begin {gather*} -\sqrt {-x^2-2\,x+3}+\ln \left (x\,1{}\mathrm {i}+\sqrt {-x^2-2\,x+3}+1{}\mathrm {i}\right )\,1{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(3 - x^2 - 2*x)^(1/2),x)

[Out]

log(x*1i + (3 - x^2 - 2*x)^(1/2) + 1i)*1i - (3 - x^2 - 2*x)^(1/2)

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