3.130 \(\int \frac{x}{(5-4 x-x^2)^{3/2}} \, dx\)

Optimal. Leaf size=23 \[ \frac{5-2 x}{9 \sqrt{-x^2-4 x+5}} \]

[Out]

(5 - 2*x)/(9*Sqrt[5 - 4*x - x^2])

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Rubi [A]  time = 0.0045149, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {636} \[ \frac{5-2 x}{9 \sqrt{-x^2-4 x+5}} \]

Antiderivative was successfully verified.

[In]

Int[x/(5 - 4*x - x^2)^(3/2),x]

[Out]

(5 - 2*x)/(9*Sqrt[5 - 4*x - x^2])

Rule 636

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-2*(b*d - 2*a*e + (2*c*
d - b*e)*x))/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] &&
NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{x}{\left (5-4 x-x^2\right )^{3/2}} \, dx &=\frac{5-2 x}{9 \sqrt{5-4 x-x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0296655, size = 23, normalized size = 1. \[ \frac{5-2 x}{9 \sqrt{-x^2-4 x+5}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(5 - 4*x - x^2)^(3/2),x]

[Out]

(5 - 2*x)/(9*Sqrt[5 - 4*x - x^2])

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Maple [A]  time = 0.045, size = 26, normalized size = 1.1 \begin{align*}{\frac{ \left ( x+5 \right ) \left ( -1+x \right ) \left ( 2\,x-5 \right ) }{9} \left ( -{x}^{2}-4\,x+5 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(x+5)*(-1+x)*(2*x-5)/(-x^2-4*x+5)^(3/2)

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Maxima [A]  time = 1.18211, size = 41, normalized size = 1.78 \begin{align*} -\frac{2 \, x}{9 \, \sqrt{-x^{2} - 4 \, x + 5}} + \frac{5}{9 \, \sqrt{-x^{2} - 4 \, x + 5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/9*x/sqrt(-x^2 - 4*x + 5) + 5/9/sqrt(-x^2 - 4*x + 5)

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Fricas [A]  time = 1.92854, size = 70, normalized size = 3.04 \begin{align*} \frac{\sqrt{-x^{2} - 4 \, x + 5}{\left (2 \, x - 5\right )}}{9 \,{\left (x^{2} + 4 \, x - 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*sqrt(-x^2 - 4*x + 5)*(2*x - 5)/(x^2 + 4*x - 5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (- \left (x - 1\right ) \left (x + 5\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.32979, size = 39, normalized size = 1.7 \begin{align*} \frac{\sqrt{-x^{2} - 4 \, x + 5}{\left (2 \, x - 5\right )}}{9 \,{\left (x^{2} + 4 \, x - 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*sqrt(-x^2 - 4*x + 5)*(2*x - 5)/(x^2 + 4*x - 5)