Optimal. Leaf size=118 \[ -\frac{1}{6} \sqrt{1-x} (x+1)^{5/2} x^3-\frac{1}{15} \sqrt{1-x} (x+1)^{5/2} x^2-\frac{11}{48} \sqrt{1-x} (x+1)^{3/2}-\frac{1}{120} \sqrt{1-x} (x+1)^{5/2} (19 x+18)-\frac{11}{16} \sqrt{1-x} \sqrt{x+1}+\frac{11}{16} \sin ^{-1}(x) \]
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Rubi [A] time = 0.157265, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{1}{6} \sqrt{1-x} (x+1)^{5/2} x^3-\frac{1}{15} \sqrt{1-x} (x+1)^{5/2} x^2-\frac{11}{48} \sqrt{1-x} (x+1)^{3/2}-\frac{1}{120} \sqrt{1-x} (x+1)^{5/2} (19 x+18)-\frac{11}{16} \sqrt{1-x} \sqrt{x+1}+\frac{11}{16} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(x^4*(1 + x)^(3/2))/Sqrt[1 - x],x]
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Rubi in Sympy [A] time = 12.9892, size = 97, normalized size = 0.82 \[ - \frac{x^{3} \sqrt{- x + 1} \left (x + 1\right )^{\frac{5}{2}}}{6} - \frac{x^{2} \sqrt{- x + 1} \left (x + 1\right )^{\frac{5}{2}}}{15} - \frac{\sqrt{- x + 1} \left (x + 1\right )^{\frac{5}{2}} \left (57 x + 54\right )}{360} - \frac{11 \sqrt{- x + 1} \left (x + 1\right )^{\frac{3}{2}}}{48} - \frac{11 \sqrt{- x + 1} \sqrt{x + 1}}{16} + \frac{11 \operatorname{asin}{\left (x \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(1+x)**(3/2)/(1-x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0598163, size = 59, normalized size = 0.5 \[ \frac{11}{8} \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-\frac{1}{240} \sqrt{1-x^2} \left (40 x^5+96 x^4+110 x^3+128 x^2+165 x+256\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(1 + x)^(3/2))/Sqrt[1 - x],x]
[Out]
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Maple [A] time = 0.021, size = 108, normalized size = 0.9 \[{\frac{1}{240}\sqrt{1-x}\sqrt{1+x} \left ( -40\,{x}^{5}\sqrt{-{x}^{2}+1}-96\,{x}^{4}\sqrt{-{x}^{2}+1}-110\,{x}^{3}\sqrt{-{x}^{2}+1}-128\,{x}^{2}\sqrt{-{x}^{2}+1}-165\,x\sqrt{-{x}^{2}+1}+165\,\arcsin \left ( x \right ) -256\,\sqrt{-{x}^{2}+1} \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(1+x)^(3/2)/(1-x)^(1/2),x)
[Out]
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Maxima [A] time = 1.49661, size = 113, normalized size = 0.96 \[ -\frac{1}{6} \, \sqrt{-x^{2} + 1} x^{5} - \frac{2}{5} \, \sqrt{-x^{2} + 1} x^{4} - \frac{11}{24} \, \sqrt{-x^{2} + 1} x^{3} - \frac{8}{15} \, \sqrt{-x^{2} + 1} x^{2} - \frac{11}{16} \, \sqrt{-x^{2} + 1} x - \frac{16}{15} \, \sqrt{-x^{2} + 1} + \frac{11}{16} \, \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(3/2)*x^4/sqrt(-x + 1),x, algorithm="maxima")
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Fricas [A] time = 0.213631, size = 284, normalized size = 2.41 \[ \frac{240 \, x^{11} + 576 \, x^{10} - 860 \, x^{9} - 2880 \, x^{8} - 630 \, x^{7} + 2560 \, x^{6} - 510 \, x^{5} + 7040 \, x^{3} -{\left (40 \, x^{11} + 96 \, x^{10} - 610 \, x^{9} - 1600 \, x^{8} + 105 \, x^{7} + 2560 \, x^{6} + 1030 \, x^{5} + 4400 \, x^{3} - 5280 \, x\right )} \sqrt{x + 1} \sqrt{-x + 1} - 330 \,{\left (x^{6} - 18 \, x^{4} + 48 \, x^{2} + 2 \,{\left (3 \, x^{4} - 16 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - 32\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) - 5280 \, x}{240 \,{\left (x^{6} - 18 \, x^{4} + 48 \, x^{2} + 2 \,{\left (3 \, x^{4} - 16 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} - 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(3/2)*x^4/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(x**4*(1+x)**(3/2)/(1-x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.235303, size = 80, normalized size = 0.68 \[ -\frac{1}{240} \,{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \, x - 8\right )}{\left (x + 1\right )} + 63\right )}{\left (x + 1\right )} - 13\right )}{\left (x + 1\right )} + 55\right )}{\left (x + 1\right )} + 165\right )} \sqrt{x + 1} \sqrt{-x + 1} + \frac{11}{8} \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(3/2)*x^4/sqrt(-x + 1),x, algorithm="giac")
[Out]