### 3.2400 $$\int \frac{3+x}{\sqrt{5-4 x-x^2}} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0129247, antiderivative size = 29, normalized size of antiderivative = 1., 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} $-\sqrt{-x^2-4 x+5}-\sin ^{-1}\left (\frac{1}{3} (-x-2)\right )$

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 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]

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 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]

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*}

Mathematica [A]  time = 0.0081645, size = 25, normalized size = 0.86 $\sin ^{-1}\left (\frac{x+2}{3}\right )-\sqrt{-x^2-4 x+5}$

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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Maple [A]  time = 0.044, size = 22, normalized size = 0.8 \begin{align*} \arcsin \left ({\frac{2}{3}}+{\frac{x}{3}} \right ) -\sqrt{-{x}^{2}-4\,x+5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.48641, size = 31, normalized size = 1.07 \begin{align*} -\sqrt{-x^{2} - 4 \, x + 5} - \arcsin \left (-\frac{1}{3} \, x - \frac{2}{3}\right ) \end{align*}

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]  time = 2.34188, size = 105, normalized size = 3.62 \begin{align*} -\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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 3}{\sqrt{- \left (x - 1\right ) \left (x + 5\right )}}\, dx \end{align*}

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 = 1.51581, size = 28, normalized size = 0.97 \begin{align*} -\sqrt{-x^{2} - 4 \, x + 5} + \arcsin \left (\frac{1}{3} \, x + \frac{2}{3}\right ) \end{align*}

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)