Optimal. Leaf size=104 \[ \frac{1}{3} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}+\frac{5}{12} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{5}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{5}{8} \cosh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.149613, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{1}{3} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}+\frac{5}{12} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{5}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{5}{8} \cosh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]),x]
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Rubi in Sympy [A] time = 16.0636, size = 94, normalized size = 0.9 \[ \frac{x^{\frac{5}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{3} + \frac{5 x^{\frac{3}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{12} + \frac{5 \sqrt{x} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{8} + \frac{5 \operatorname{acosh}{\left (\sqrt{x} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0471159, size = 76, normalized size = 0.73 \[ \frac{1}{24} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x} \left (8 x^2+10 x+15\right )+\frac{5}{8} \log \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}+\sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]),x]
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Maple [A] time = 0.014, size = 65, normalized size = 0.6 \[{\frac{1}{24}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( 8\,{x}^{5/2}\sqrt{-1+x}+10\,{x}^{3/2}\sqrt{-1+x}+15\,\sqrt{x}\sqrt{-1+x}+15\,\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x)
[Out]
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Maxima [A] time = 1.42741, size = 63, normalized size = 0.61 \[ \frac{1}{3} \, \sqrt{x - 1} x^{\frac{5}{2}} + \frac{5}{12} \, \sqrt{x - 1} x^{\frac{3}{2}} + \frac{5}{8} \, \sqrt{x - 1} \sqrt{x} + \frac{5}{8} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213126, size = 261, normalized size = 2.51 \[ -\frac{2048 \, x^{6} - 1536 \, x^{5} + 1152 \, x^{4} - 3840 \, x^{3} - 2 \,{\left (1024 \, x^{5} - 256 \, x^{4} + 576 \, x^{3} - 1600 \, x^{2} + 448 \, x + 51\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} + 2304 \, x^{2} + 60 \,{\left (32 \, x^{3} - 2 \,{\left (16 \, x^{2} - 16 \, x + 3\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 48 \, x^{2} + 18 \, x - 1\right )} \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) - 54 \, x - 37}{192 \,{\left (32 \, x^{3} - 2 \,{\left (16 \, x^{2} - 16 \, x + 3\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 48 \, x^{2} + 18 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(sqrt(sqrt(x) + 1)*sqrt(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**(5/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="giac")
[Out]