Optimal. Leaf size=41 \[ \frac{(x+1)^{3/2}}{15 (1-x)^{3/2}}+\frac{(x+1)^{3/2}}{5 (1-x)^{5/2}} \]
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Rubi [A] time = 0.0241274, 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 \[ \frac{(x+1)^{3/2}}{15 (1-x)^{3/2}}+\frac{(x+1)^{3/2}}{5 (1-x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 + x]/(1 - x)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 4.01212, size = 29, normalized size = 0.71 \[ \frac{\left (x + 1\right )^{\frac{3}{2}}}{15 \left (- x + 1\right )^{\frac{3}{2}}} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{5 \left (- x + 1\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+x)**(1/2)/(1-x)**(7/2),x)
[Out]
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Mathematica [A] time = 0.0159323, size = 28, normalized size = 0.68 \[ \frac{\sqrt{1-x^2} \left (x^2-3 x-4\right )}{15 (x-1)^3} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[1 + x]/(1 - x)^(7/2),x]
[Out]
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Maple [A] time = 0.006, size = 18, normalized size = 0.4 \[ -{\frac{x-4}{15} \left ( 1+x \right ) ^{{\frac{3}{2}}} \left ( 1-x \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+x)^(1/2)/(1-x)^(7/2),x)
[Out]
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Maxima [A] time = 1.33895, size = 86, normalized size = 2.1 \[ -\frac{2 \, \sqrt{-x^{2} + 1}}{5 \,{\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac{\sqrt{-x^{2} + 1}}{15 \,{\left (x^{2} - 2 \, x + 1\right )}} + \frac{\sqrt{-x^{2} + 1}}{15 \,{\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + 1)/(-x + 1)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.207174, size = 146, normalized size = 3.56 \[ \frac{3 \, x^{5} - 20 \, x^{4} + 35 \, x^{3} + 30 \, x^{2} + 5 \,{\left (x^{4} - x^{3} - 6 \, x^{2} + 12 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 60 \, x}{15 \,{\left (x^{5} - 5 \, x^{4} + 5 \, x^{3} + 5 \, x^{2} +{\left (x^{4} - 7 \, x^{2} + 10 \, x - 4\right )} \sqrt{x + 1} \sqrt{-x + 1} - 10 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + 1)/(-x + 1)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 113.561, size = 173, normalized size = 4.22 \[ \begin{cases} \frac{i \left (x + 1\right )^{\frac{5}{2}}}{15 \sqrt{x - 1} \left (x + 1\right )^{2} - 60 \sqrt{x - 1} \left (x + 1\right ) + 60 \sqrt{x - 1}} - \frac{5 i \left (x + 1\right )^{\frac{3}{2}}}{15 \sqrt{x - 1} \left (x + 1\right )^{2} - 60 \sqrt{x - 1} \left (x + 1\right ) + 60 \sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- \frac{\left (x + 1\right )^{\frac{5}{2}}}{15 \sqrt{- x + 1} \left (x + 1\right )^{2} - 60 \sqrt{- x + 1} \left (x + 1\right ) + 60 \sqrt{- x + 1}} + \frac{5 \left (x + 1\right )^{\frac{3}{2}}}{15 \sqrt{- x + 1} \left (x + 1\right )^{2} - 60 \sqrt{- x + 1} \left (x + 1\right ) + 60 \sqrt{- x + 1}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+x)**(1/2)/(1-x)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.209052, size = 30, normalized size = 0.73 \[ \frac{{\left (x + 1\right )}^{\frac{3}{2}}{\left (x - 4\right )} \sqrt{-x + 1}}{15 \,{\left (x - 1\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x + 1)/(-x + 1)^(7/2),x, algorithm="giac")
[Out]