3.21.69 \(\int \frac {3+x}{\sqrt {5-4 x-x^2}} \, dx\)

Optimal. Leaf size=29 \[ -\sqrt {-x^2-4 x+5}-\sin ^{-1}\left (\frac {1}{3} (-x-2)\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {640, 619, 216} \begin {gather*} -\sqrt {-x^2-4 x+5}-\sin ^{-1}\left (\frac {1}{3} (-x-2)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + x)/Sqrt[5 - 4*x - x^2],x]

[Out]

-Sqrt[5 - 4*x - x^2] - ArcSin[(-2 - x)/3]

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 {3+x}{\sqrt {5-4 x-x^2}} \, dx &=-\sqrt {5-4 x-x^2}+\int \frac {1}{\sqrt {5-4 x-x^2}} \, dx\\ &=-\sqrt {5-4 x-x^2}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{36}}} \, dx,x,-4-2 x\right )\\ &=-\sqrt {5-4 x-x^2}-\sin ^{-1}\left (\frac {1}{3} (-2-x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 25, normalized size = 0.86 \begin {gather*} \sin ^{-1}\left (\frac {x+2}{3}\right )-\sqrt {-x^2-4 x+5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + x)/Sqrt[5 - 4*x - x^2],x]

[Out]

-Sqrt[5 - 4*x - x^2] + ArcSin[(2 + x)/3]

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IntegrateAlgebraic [A]  time = 0.18, size = 40, normalized size = 1.38 \begin {gather*} -\sqrt {-x^2-4 x+5}-2 \tan ^{-1}\left (\frac {\sqrt {-x^2-4 x+5}}{x+5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(3 + x)/Sqrt[5 - 4*x - x^2],x]

[Out]

-Sqrt[5 - 4*x - x^2] - 2*ArcTan[Sqrt[5 - 4*x - x^2]/(5 + x)]

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fricas [B]  time = 0.41, size = 44, normalized size = 1.52 \begin {gather*} -\sqrt {-x^{2} - 4 \, x + 5} - \arctan \left (\frac {\sqrt {-x^{2} - 4 \, x + 5} {\left (x + 2\right )}}{x^{2} + 4 \, x - 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(-x^2-4*x+5)^(1/2),x, algorithm="fricas")

[Out]

-sqrt(-x^2 - 4*x + 5) - arctan(sqrt(-x^2 - 4*x + 5)*(x + 2)/(x^2 + 4*x - 5))

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giac [A]  time = 0.19, size = 21, normalized size = 0.72 \begin {gather*} -\sqrt {-x^{2} - 4 \, x + 5} + \arcsin \left (\frac {1}{3} \, x + \frac {2}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(-x^2-4*x+5)^(1/2),x, algorithm="giac")

[Out]

-sqrt(-x^2 - 4*x + 5) + arcsin(1/3*x + 2/3)

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maple [A]  time = 0.06, size = 22, normalized size = 0.76 \begin {gather*} \arcsin \left (\frac {x}{3}+\frac {2}{3}\right )-\sqrt {-x^{2}-4 x +5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+3)/(-x^2-4*x+5)^(1/2),x)

[Out]

arcsin(1/3*x+2/3)-(-x^2-4*x+5)^(1/2)

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maxima [A]  time = 2.00, size = 23, normalized size = 0.79 \begin {gather*} -\sqrt {-x^{2} - 4 \, x + 5} - \arcsin \left (-\frac {1}{3} \, x - \frac {2}{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(-x^2-4*x+5)^(1/2),x, algorithm="maxima")

[Out]

-sqrt(-x^2 - 4*x + 5) - arcsin(-1/3*x - 2/3)

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mupad [B]  time = 1.20, size = 46, normalized size = 1.59 \begin {gather*} 3\,\mathrm {asin}\left (\frac {x}{3}+\frac {2}{3}\right )-\sqrt {-x^2-4\,x+5}+\ln \left (x\,1{}\mathrm {i}+\sqrt {-x^2-4\,x+5}+2{}\mathrm {i}\right )\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 3)/(5 - x^2 - 4*x)^(1/2),x)

[Out]

log(x*1i + (5 - x^2 - 4*x)^(1/2) + 2i)*2i + 3*asin(x/3 + 2/3) - (5 - x^2 - 4*x)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 3}{\sqrt {- \left (x - 1\right ) \left (x + 5\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+x)/(-x**2-4*x+5)**(1/2),x)

[Out]

Integral((x + 3)/sqrt(-(x - 1)*(x + 5)), x)

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