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]
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Rubi [A] time = 0.0175299, 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]
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Rubi in Sympy [A] time = 0.802721, 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]
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Mathematica [A] time = 0.0258725, 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]
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Maple [A] time = 0.004, size = 36, normalized size = 0.8 \[{\frac{ \left ( 5+x \right ) \left ( -1+x \right ) \left ( 2\,{x}^{3}+12\,{x}^{2}-3\,x-38 \right ) }{243} \left ( -{x}^{2}-4\,x+5 \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-x^2-4*x+5)^(5/2),x)
[Out]
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Maxima [A] time = 1.43765, 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]
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Fricas [A] time = 0.206224, 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]
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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]
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GIAC/XCAS [A] time = 0.214396, 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]