3.131 \(\int \frac{1}{\left (5-4 x-x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=43 \[ \frac{2 (x+2)}{243 \sqrt{-x^2-4 x+5}}+\frac{x+2}{27 \left (-x^2-4 x+5\right )^{3/2}} \]

[Out]

(2 + x)/(27*(5 - 4*x - x^2)^(3/2)) + (2*(2 + x))/(243*Sqrt[5 - 4*x - x^2])

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Rubi [A]  time = 0.0199797, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 (x+2)}{243 \sqrt{-x^2-4 x+5}}+\frac{x+2}{27 \left (-x^2-4 x+5\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(5 - 4*x - x^2)^(-5/2),x]

[Out]

(2 + x)/(27*(5 - 4*x - x^2)^(3/2)) + (2*(2 + x))/(243*Sqrt[5 - 4*x - x^2])

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Rubi in Sympy [A]  time = 1.68306, size = 36, normalized size = 0.84 \[ \frac{2 x + 4}{54 \left (- x^{2} - 4 x + 5\right )^{\frac{3}{2}}} + \frac{4 x + 8}{486 \sqrt{- x^{2} - 4 x + 5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**2-4*x+5)**(5/2),x)

[Out]

(2*x + 4)/(54*(-x**2 - 4*x + 5)**(3/2)) + (4*x + 8)/(486*sqrt(-x**2 - 4*x + 5))

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Mathematica [A]  time = 0.0269016, size = 31, normalized size = 0.72 \[ -\frac{(x+2) \left (2 x^2+8 x-19\right )}{243 \left (-x^2-4 x+5\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - 4*x - x^2)^(-5/2),x]

[Out]

-((2 + x)*(-19 + 8*x + 2*x^2))/(243*(5 - 4*x - x^2)^(3/2))

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Maple [A]  time = 0.004, size = 36, normalized size = 0.8 \[{\frac{ \left ( x+5 \right ) \left ( -1+x \right ) \left ( 2\,{x}^{3}+12\,{x}^{2}-3\,x-38 \right ) }{243} \left ( -{x}^{2}-4\,x+5 \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^2-4*x+5)^(5/2),x)

[Out]

1/243*(x+5)*(-1+x)*(2*x^3+12*x^2-3*x-38)/(-x^2-4*x+5)^(5/2)

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Maxima [A]  time = 0.749633, size = 80, normalized size = 1.86 \[ \frac{2 \, x}{243 \, \sqrt{-x^{2} - 4 \, x + 5}} + \frac{4}{243 \, \sqrt{-x^{2} - 4 \, x + 5}} + \frac{x}{27 \,{\left (-x^{2} - 4 \, x + 5\right )}^{\frac{3}{2}}} + \frac{2}{27 \,{\left (-x^{2} - 4 \, x + 5\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/243*x/sqrt(-x^2 - 4*x + 5) + 4/243/sqrt(-x^2 - 4*x + 5) + 1/27*x/(-x^2 - 4*x +
5)^(3/2) + 2/27/(-x^2 - 4*x + 5)^(3/2)

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Fricas [A]  time = 0.218417, size = 66, normalized size = 1.53 \[ -\frac{{\left (2 \, x^{3} + 12 \, x^{2} - 3 \, x - 38\right )} \sqrt{-x^{2} - 4 \, x + 5}}{243 \,{\left (x^{4} + 8 \, x^{3} + 6 \, x^{2} - 40 \, x + 25\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/243*(2*x^3 + 12*x^2 - 3*x - 38)*sqrt(-x^2 - 4*x + 5)/(x^4 + 8*x^3 + 6*x^2 - 4
0*x + 25)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- x^{2} - 4 x + 5\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**2-4*x+5)**(5/2),x)

[Out]

Integral((-x**2 - 4*x + 5)**(-5/2), x)

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GIAC/XCAS [A]  time = 0.215895, size = 49, normalized size = 1.14 \[ -\frac{{\left ({\left (2 \,{\left (x + 6\right )} x - 3\right )} x - 38\right )} \sqrt{-x^{2} - 4 \, x + 5}}{243 \,{\left (x^{2} + 4 \, x - 5\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/243*((2*(x + 6)*x - 3)*x - 38)*sqrt(-x^2 - 4*x + 5)/(x^2 + 4*x - 5)^2