Optimal. Leaf size=70 \[ \frac{1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac{5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac{5}{16} \sqrt{1-x} x \sqrt{x+1}+\frac{5}{16} \sin ^{-1}(x) \]
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Rubi [A] time = 0.0450069, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac{5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac{5}{16} \sqrt{1-x} x \sqrt{x+1}+\frac{5}{16} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(1 - x)^(5/2)*(1 + x)^(5/2),x]
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Rubi in Sympy [A] time = 6.88058, size = 60, normalized size = 0.86 \[ \frac{x \left (- x + 1\right )^{\frac{5}{2}} \left (x + 1\right )^{\frac{5}{2}}}{6} + \frac{5 x \left (- x + 1\right )^{\frac{3}{2}} \left (x + 1\right )^{\frac{3}{2}}}{24} + \frac{5 x \sqrt{- x + 1} \sqrt{x + 1}}{16} + \frac{5 \operatorname{asin}{\left (x \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-x)**(5/2)*(1+x)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0293732, size = 34, normalized size = 0.49 \[ \frac{1}{48} \left (x \sqrt{1-x^2} \left (8 x^4-26 x^2+33\right )+15 \sin ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - x)^(5/2)*(1 + x)^(5/2),x]
[Out]
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Maple [B] time = 0.004, size = 113, normalized size = 1.6 \[{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{5}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{6} \left ( 1-x \right ) ^{{\frac{3}{2}}} \left ( 1+x \right ) ^{{\frac{7}{2}}}}+{\frac{1}{8}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{7}{2}}}}-{\frac{1}{24}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{5}{2}}}}-{\frac{5}{48}\sqrt{1-x} \left ( 1+x \right ) ^{{\frac{3}{2}}}}-{\frac{5}{16}\sqrt{1-x}\sqrt{1+x}}+{\frac{5\,\arcsin \left ( x \right ) }{16}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-x)^(5/2)*(1+x)^(5/2),x)
[Out]
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Maxima [A] time = 1.49399, size = 55, normalized size = 0.79 \[ \frac{1}{6} \,{\left (-x^{2} + 1\right )}^{\frac{5}{2}} x + \frac{5}{24} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x + \frac{5}{16} \, \sqrt{-x^{2} + 1} x + \frac{5}{16} \, \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(5/2)*(-x + 1)^(5/2),x, algorithm="maxima")
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Fricas [A] time = 0.207443, size = 243, normalized size = 3.47 \[ -\frac{48 \, x^{11} - 460 \, x^{9} + 1698 \, x^{7} - 3174 \, x^{5} + 2944 \, x^{3} -{\left (8 \, x^{11} - 170 \, x^{9} + 885 \, x^{7} - 2098 \, x^{5} + 2416 \, x^{3} - 1056 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} + 30 \,{\left (x^{6} - 18 \, x^{4} + 48 \, x^{2} + 2 \,{\left (3 \, x^{4} - 16 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - 32\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 1056 \, x}{48 \,{\left (x^{6} - 18 \, x^{4} + 48 \, x^{2} + 2 \,{\left (3 \, x^{4} - 16 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(5/2)*(-x + 1)^(5/2),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)**(5/2)*(1+x)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.224703, size = 138, normalized size = 1.97 \[ \frac{1}{48} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left (x + 1\right )}{\left (x - 4\right )} + 39\right )}{\left (x + 1\right )} - 37\right )}{\left (x + 1\right )} + 31\right )}{\left (x + 1\right )} - 3\right )} \sqrt{x + 1} \sqrt{-x + 1} - \frac{1}{4} \,{\left ({\left (2 \,{\left (x + 1\right )}{\left (x - 2\right )} + 5\right )}{\left (x + 1\right )} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{1}{2} \, \sqrt{x + 1} x \sqrt{-x + 1} + \frac{5}{8} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(5/2)*(-x + 1)^(5/2),x, algorithm="giac")
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