Optimal. Leaf size=20 \[ -\frac{(1-x)^{3/2}}{3 (x+1)^{3/2}} \]
[Out]
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Rubi [A] time = 0.0117837, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{(1-x)^{3/2}}{3 (x+1)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - x]/(1 + x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 2.38087, size = 15, normalized size = 0.75 \[ - \frac{\left (- x + 1\right )^{\frac{3}{2}}}{3 \left (x + 1\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x)**(1/2)/(1+x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0123056, size = 20, normalized size = 1. \[ -\frac{(1-x)^{3/2}}{3 (x+1)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - x]/(1 + x)^(5/2),x]
[Out]
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Maple [A] time = 0.004, size = 15, normalized size = 0.8 \[ -{\frac{1}{3} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x)^(1/2)/(1+x)^(5/2),x)
[Out]
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Maxima [A] time = 1.34178, size = 51, normalized size = 2.55 \[ -\frac{2 \, \sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x + 1)/(x + 1)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205144, size = 74, normalized size = 3.7 \[ -\frac{2 \,{\left (x^{3} + 3 \, \sqrt{x + 1} x \sqrt{-x + 1} - 3 \, x\right )}}{3 \,{\left (x^{3} +{\left (x^{2} + 3 \, x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \, x - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x + 1)/(x + 1)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.7706, size = 66, normalized size = 3.3 \[ \begin{cases} \frac{\sqrt{-1 + \frac{2}{x + 1}}}{3} - \frac{2 \sqrt{-1 + \frac{2}{x + 1}}}{3 \left (x + 1\right )} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\\frac{i \sqrt{1 - \frac{2}{x + 1}}}{3} - \frac{2 i \sqrt{1 - \frac{2}{x + 1}}}{3 \left (x + 1\right )} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-x)**(1/2)/(1+x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.214232, size = 120, normalized size = 6. \[ \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{24 \,{\left (x + 1\right )}^{\frac{3}{2}}} - \frac{\sqrt{2} - \sqrt{-x + 1}}{8 \, \sqrt{x + 1}} + \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{3 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{24 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x + 1)/(x + 1)^(5/2),x, algorithm="giac")
[Out]