Optimal. Leaf size=41 \[ -\frac{\sqrt{1-x}}{3 \sqrt{x+1}}-\frac{\sqrt{1-x}}{3 (x+1)^{3/2}} \]
[Out]
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Rubi [A] time = 0.0246339, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{\sqrt{1-x}}{3 \sqrt{x+1}}-\frac{\sqrt{1-x}}{3 (x+1)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[1 - x]*(1 + x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 3.32363, size = 31, normalized size = 0.76 \[ - \frac{\sqrt{- x + 1}}{3 \sqrt{x + 1}} - \frac{\sqrt{- x + 1}}{3 \left (x + 1\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-x)**(1/2)/(1+x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0122937, size = 23, normalized size = 0.56 \[ -\frac{\sqrt{1-x} (x+2)}{3 (x+1)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[1 - x]*(1 + x)^(5/2)),x]
[Out]
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Maple [A] time = 0.004, size = 18, normalized size = 0.4 \[ -{\frac{2+x}{3}\sqrt{1-x} \left ( 1+x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-x)^(1/2)/(1+x)^(5/2),x)
[Out]
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Maxima [A] time = 1.48436, size = 51, normalized size = 1.24 \[ -\frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x^{2} + 2 \, x + 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{3 \,{\left (x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + 1)^(5/2)*sqrt(-x + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205187, size = 89, normalized size = 2.17 \[ -\frac{x^{3} - 3 \, x^{2} + 3 \,{\left (x^{2} + 2 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 6 \, x}{3 \,{\left (x^{3} +{\left (x^{2} + 3 \, x + 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 3 \, x - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + 1)^(5/2)*sqrt(-x + 1)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 18.8231, size = 66, normalized size = 1.61 \[ \begin{cases} - \frac{\sqrt{-1 + \frac{2}{x + 1}}}{3} - \frac{\sqrt{-1 + \frac{2}{x + 1}}}{3 \left (x + 1\right )} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\- \frac{i \sqrt{1 - \frac{2}{x + 1}}}{3} - \frac{i \sqrt{1 - \frac{2}{x + 1}}}{3 \left (x + 1\right )} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1-x)**(1/2)/(1+x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.208558, size = 120, normalized size = 2.93 \[ \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{48 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \frac{3 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{16 \, \sqrt{x + 1}} - \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{9 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{48 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + 1)^(5/2)*sqrt(-x + 1)),x, algorithm="giac")
[Out]