Optimal. Leaf size=80 \[ \frac{1}{3} \left (2 x+\sqrt{2 x-1}\right )^{3/2}-\frac{1}{8} \left (2 \sqrt{2 x-1}+1\right ) \sqrt{2 x+\sqrt{2 x-1}}-\frac{3}{16} \sinh ^{-1}\left (\frac{2 \sqrt{2 x-1}+1}{\sqrt{3}}\right ) \]
[Out]
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Rubi [A] time = 0.0786265, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{1}{3} \left (2 x+\sqrt{2 x-1}\right )^{3/2}-\frac{1}{8} \left (2 \sqrt{2 x-1}+1\right ) \sqrt{2 x+\sqrt{2 x-1}}-\frac{3}{16} \sinh ^{-1}\left (\frac{2 \sqrt{2 x-1}+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2*x + Sqrt[-1 + 2*x]],x]
[Out]
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Rubi in Sympy [A] time = 2.52634, size = 78, normalized size = 0.98 \[ \frac{\left (2 x + \sqrt{2 x - 1}\right )^{\frac{3}{2}}}{3} - \frac{\sqrt{2 x + \sqrt{2 x - 1}} \left (2 \sqrt{2 x - 1} + 1\right )}{8} - \frac{3 \operatorname{atanh}{\left (\frac{2 \sqrt{2 x - 1} + 1}{2 \sqrt{2 x + \sqrt{2 x - 1}}} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*x+(-1+2*x)**(1/2))**(1/2),x)
[Out]
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Mathematica [A] time = 0.055038, size = 62, normalized size = 0.78 \[ \frac{1}{48} \left (2 \sqrt{2 x+\sqrt{2 x-1}} \left (16 x+2 \sqrt{2 x-1}-3\right )-9 \sinh ^{-1}\left (\frac{2 \sqrt{2 x-1}+1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[2*x + Sqrt[-1 + 2*x]],x]
[Out]
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Maple [A] time = 0.009, size = 60, normalized size = 0.8 \[{\frac{1}{3} \left ( 2\,x+\sqrt{2\,x-1} \right ) ^{{\frac{3}{2}}}}-{\frac{1}{8} \left ( 1+2\,\sqrt{2\,x-1} \right ) \sqrt{2\,x+\sqrt{2\,x-1}}}-{\frac{3}{16}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( \sqrt{2\,x-1}+{\frac{1}{2}} \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*x+(2*x-1)^(1/2))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{2 \, x + \sqrt{2 \, x - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x + sqrt(2*x - 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.574076, size = 99, normalized size = 1.24 \[ \frac{1}{24} \,{\left (16 \, x + 2 \, \sqrt{2 \, x - 1} - 3\right )} \sqrt{2 \, x + \sqrt{2 \, x - 1}} + \frac{3}{32} \, \log \left (-4 \, \sqrt{2 \, x + \sqrt{2 \, x - 1}}{\left (2 \, \sqrt{2 \, x - 1} + 1\right )} + 16 \, x + 8 \, \sqrt{2 \, x - 1} - 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x + sqrt(2*x - 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{2 x + \sqrt{2 x - 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x+(-1+2*x)**(1/2))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.287032, size = 92, normalized size = 1.15 \[ \frac{1}{24} \,{\left (2 \, \sqrt{2 \, x - 1}{\left (4 \, \sqrt{2 \, x - 1} + 1\right )} + 5\right )} \sqrt{2 \, x + \sqrt{2 \, x - 1}} + \frac{3}{16} \,{\rm ln}\left (2 \, \sqrt{2 \, x + \sqrt{2 \, x - 1}} - 2 \, \sqrt{2 \, x - 1} - 1\right ) - \frac{5}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x + sqrt(2*x - 1)),x, algorithm="giac")
[Out]