Optimal. Leaf size=87 \[ \frac{1}{4} \sqrt{x+1} (1-x)^{7/2}+\frac{7}{12} \sqrt{x+1} (1-x)^{5/2}+\frac{35}{24} \sqrt{x+1} (1-x)^{3/2}+\frac{35}{8} \sqrt{x+1} \sqrt{1-x}+\frac{35}{8} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0661754, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{1}{4} \sqrt{x+1} (1-x)^{7/2}+\frac{7}{12} \sqrt{x+1} (1-x)^{5/2}+\frac{35}{24} \sqrt{x+1} (1-x)^{3/2}+\frac{35}{8} \sqrt{x+1} \sqrt{1-x}+\frac{35}{8} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(1 - x)^(7/2)/Sqrt[1 + x],x]
[Out]
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Rubi in Sympy [A] time = 8.40539, size = 71, normalized size = 0.82 \[ \frac{\left (- x + 1\right )^{\frac{7}{2}} \sqrt{x + 1}}{4} + \frac{7 \left (- x + 1\right )^{\frac{5}{2}} \sqrt{x + 1}}{12} + \frac{35 \left (- x + 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{24} + \frac{35 \sqrt{- x + 1} \sqrt{x + 1}}{8} + \frac{35 \operatorname{asin}{\left (x \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x)**(7/2)/(1+x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0381746, size = 49, normalized size = 0.56 \[ \frac{1}{24} \sqrt{1-x^2} \left (-6 x^3+32 x^2-81 x+160\right )+\frac{35}{4} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x)^(7/2)/Sqrt[1 + x],x]
[Out]
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Maple [A] time = 0.007, size = 85, normalized size = 1. \[{\frac{1}{4} \left ( 1-x \right ) ^{{\frac{7}{2}}}\sqrt{1+x}}+{\frac{7}{12} \left ( 1-x \right ) ^{{\frac{5}{2}}}\sqrt{1+x}}+{\frac{35}{24} \left ( 1-x \right ) ^{{\frac{3}{2}}}\sqrt{1+x}}+{\frac{35}{8}\sqrt{1-x}\sqrt{1+x}}+{\frac{35\,\arcsin \left ( x \right ) }{8}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x)^(7/2)/(1+x)^(1/2),x)
[Out]
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Maxima [A] time = 1.50939, size = 76, normalized size = 0.87 \[ -\frac{1}{4} \, \sqrt{-x^{2} + 1} x^{3} + \frac{4}{3} \, \sqrt{-x^{2} + 1} x^{2} - \frac{27}{8} \, \sqrt{-x^{2} + 1} x + \frac{20}{3} \, \sqrt{-x^{2} + 1} + \frac{35}{8} \, \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x + 1)^(7/2)/sqrt(x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.208456, size = 224, normalized size = 2.57 \[ \frac{24 \, x^{7} - 128 \, x^{6} + 252 \, x^{5} - 96 \, x^{4} - 924 \, x^{3} + 384 \, x^{2} -{\left (6 \, x^{7} - 32 \, x^{6} + 33 \, x^{5} + 96 \, x^{4} - 600 \, x^{3} + 384 \, x^{2} + 648 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 210 \,{\left (x^{4} - 8 \, x^{2} + 4 \,{\left (x^{2} - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 8\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 648 \, x}{24 \,{\left (x^{4} - 8 \, x^{2} + 4 \,{\left (x^{2} - 2\right )} \sqrt{x + 1} \sqrt{-x + 1} + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x + 1)^(7/2)/sqrt(x + 1),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-x)**(7/2)/(1+x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.226323, size = 136, normalized size = 1.56 \[ -\frac{1}{24} \,{\left ({\left (2 \,{\left (3 \, x - 10\right )}{\left (x + 1\right )} + 43\right )}{\left (x + 1\right )} - 39\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \,{\left ({\left (2 \, x - 5\right )}{\left (x + 1\right )} + 9\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{3}{2} \, \sqrt{x + 1}{\left (x - 2\right )} \sqrt{-x + 1} + \sqrt{x + 1} \sqrt{-x + 1} + \frac{35}{4} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x + 1)^(7/2)/sqrt(x + 1),x, algorithm="giac")
[Out]