3.24.100 \(\int \frac {3+x}{\sqrt {5-4 x-x^2}} \, dx\) [2400]

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

[Out]

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

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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 = {654, 633, 222} \begin {gather*} -\text {ArcSin}\left (\frac {1}{3} (-x-2)\right )-\sqrt {-x^2-4 x+5} \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 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 {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} \text {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 = 0.09, size = 40, normalized size = 1.38 \begin {gather*} -\sqrt {5-4 x-x^2}-2 \tan ^{-1}\left (\frac {\sqrt {5-4 x-x^2}}{5+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(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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Maple [A]
time = 0.70, size = 22, normalized size = 0.76

method	result	size
default	\(\arcsin \left (\frac {2}{3}+\frac {x}{3}\right )-\sqrt {-x^{2}-4 x +5}\)	\(22\)
risch	\(\frac {x^{2}+4 x -5}{\sqrt {-x^{2}-4 x +5}}+\arcsin \left (\frac {2}{3}+\frac {x}{3}\right )\)	\(29\)
trager	\(-\sqrt {-x^{2}-4 x +5}-\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (x \RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-x^{2}-4 x +5}+2 \RootOf \left (\textit {\_Z}^{2}+1\right )\right )\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (21) = 42\).
time = 2.63, 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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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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Giac [A]
time = 1.31, 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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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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