Optimal. Leaf size=73 \[ \frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{3}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{3}{4} \cosh ^{-1}\left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.112956, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{1}{2} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{3}{4} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{3}{4} \cosh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]),x]
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Rubi in Sympy [A] time = 13.133, size = 65, normalized size = 0.89 \[ \frac{x^{\frac{3}{2}} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{2} + \frac{3 \sqrt{x} \sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}{4} + \frac{3 \operatorname{acosh}{\left (\sqrt{x} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)
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Mathematica [A] time = 0.0461284, size = 70, normalized size = 0.96 \[ \frac{1}{4} \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x} (2 x+3)+3 \log \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}+\sqrt{x}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)/(Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]),x]
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Maple [A] time = 0.014, size = 55, normalized size = 0.8 \[{\frac{1}{4}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( 2\,{x}^{3/2}\sqrt{-1+x}+3\,\sqrt{x}\sqrt{-1+x}+3\,\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)/(-1+x^(1/2))^(1/2)/(1+x^(1/2))^(1/2),x)
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Maxima [A] time = 1.37467, size = 50, normalized size = 0.68 \[ \frac{1}{2} \, \sqrt{x - 1} x^{\frac{3}{2}} + \frac{3}{4} \, \sqrt{x - 1} \sqrt{x} + \frac{3}{4} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="maxima")
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Fricas [A] time = 0.214932, size = 200, normalized size = 2.74 \[ \frac{128 \, x^{4} - 4 \,{\left (32 \, x^{3} + 16 \, x^{2} - 30 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 168 \, x^{2} - 12 \,{\left (4 \,{\left (2 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 8 \, x^{2} + 8 \, x - 1\right )} \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) + 40 \, x + 7}{32 \,{\left (4 \,{\left (2 \, x - 1\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 8 \, x^{2} + 8 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2}}}{\sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)/(-1+x**(1/2))**(1/2)/(1+x**(1/2))**(1/2),x)
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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^(3/2)/(sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1)),x, algorithm="giac")
[Out]