Optimal. Leaf size=42 \[ \frac{2 x}{3 \sqrt{1-x} \sqrt{x+1}}-\frac{1}{3 \sqrt{1-x} (x+1)^{3/2}} \]
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Rubi [A] time = 0.0247833, antiderivative size = 42, 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 \[ \frac{2 x}{3 \sqrt{1-x} \sqrt{x+1}}-\frac{1}{3 \sqrt{1-x} (x+1)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 - x)^(3/2)*(1 + x)^(5/2)),x]
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Rubi in Sympy [A] time = 3.41403, size = 34, normalized size = 0.81 \[ \frac{2 x}{3 \sqrt{- x + 1} \sqrt{x + 1}} - \frac{1}{3 \sqrt{- x + 1} \left (x + 1\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-x)**(3/2)/(1+x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0224186, size = 30, normalized size = 0.71 \[ \frac{2 x^2+2 x-1}{3 \sqrt{1-x} (x+1)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - x)^(3/2)*(1 + x)^(5/2)),x]
[Out]
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Maple [A] time = 0.005, size = 25, normalized size = 0.6 \[{\frac{2\,{x}^{2}+2\,x-1}{3}{\frac{1}{\sqrt{1-x}}} \left ( 1+x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-x)^(3/2)/(1+x)^(5/2),x)
[Out]
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Maxima [A] time = 1.33862, size = 51, normalized size = 1.21 \[ \frac{2 \, x}{3 \, \sqrt{-x^{2} + 1}} - \frac{1}{3 \,{\left (\sqrt{-x^{2} + 1} x + \sqrt{-x^{2} + 1}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + 1)^(5/2)*(-x + 1)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206081, size = 120, normalized size = 2.86 \[ -\frac{2 \, x^{4} + 4 \, x^{3} - 3 \, x^{2} -{\left (x^{3} - 3 \, x^{2} - 6 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 6 \, x}{3 \,{\left (2 \, x^{3} + 2 \, x^{2} -{\left (x^{3} + x^{2} - 2 \, x - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} - 2 \, x - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + 1)^(5/2)*(-x + 1)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 86.1169, size = 167, normalized size = 3.98 \[ \begin{cases} - \frac{2 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )^{2}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac{2 \sqrt{-1 + \frac{2}{x + 1}} \left (x + 1\right )}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac{\sqrt{-1 + \frac{2}{x + 1}}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} & \text{for}\: 2 \left |{\frac{1}{x + 1}}\right | > 1 \\- \frac{2 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )^{2}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac{2 i \sqrt{1 - \frac{2}{x + 1}} \left (x + 1\right )}{- 6 x + 3 \left (x + 1\right )^{2} - 6} + \frac{i \sqrt{1 - \frac{2}{x + 1}}}{- 6 x + 3 \left (x + 1\right )^{2} - 6} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1-x)**(3/2)/(1+x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.209701, size = 146, normalized size = 3.48 \[ \frac{{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}}{96 \,{\left (x + 1\right )}^{\frac{3}{2}}} + \frac{7 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}}{32 \, \sqrt{x + 1}} - \frac{\sqrt{x + 1} \sqrt{-x + 1}}{4 \,{\left (x - 1\right )}} - \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (\frac{21 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{2}}{x + 1} + 1\right )}}{96 \,{\left (\sqrt{2} - \sqrt{-x + 1}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x + 1)^(5/2)*(-x + 1)^(3/2)),x, algorithm="giac")
[Out]