Optimal. Leaf size=157 \[ \frac{\log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2} \sqrt [4]{3}}-\frac{\log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2} \sqrt [4]{3}}-\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}+\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt [4]{3}} \]
[Out]
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Rubi [A] time = 0.265317, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ \frac{\log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2} \sqrt [4]{3}}-\frac{\log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2} \sqrt [4]{3}}-\frac{\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )}{\sqrt [4]{3}}+\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right )}{\sqrt [4]{3}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 + 2*x]/(1 + x + x^2),x]
[Out]
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Rubi in Sympy [A] time = 30.6186, size = 155, normalized size = 0.99 \[ \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x - \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{6} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x + \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{6} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} - 1 \right )}}{3} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} + 1 \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+2*x)**(1/2)/(x**2+x+1),x)
[Out]
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Mathematica [A] time = 0.0584881, size = 114, normalized size = 0.73 \[ \frac{\log \left (\sqrt{3} (2 x+1)-3^{3/4} \sqrt{4 x+2}+3\right )-\log \left (\sqrt{3} (2 x+1)+3^{3/4} \sqrt{4 x+2}+3\right )-2 \tan ^{-1}\left (1-\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}+1\right )}{\sqrt{2} \sqrt [4]{3}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 + 2*x]/(1 + x + x^2),x]
[Out]
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Maple [A] time = 0.006, size = 111, normalized size = 0.7 \[{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\arctan \left ( -1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) }+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{6}\ln \left ({1 \left ( 1+2\,x+\sqrt{3}-\sqrt [4]{3}\sqrt{2}\sqrt{1+2\,x} \right ) \left ( 1+2\,x+\sqrt{3}+\sqrt [4]{3}\sqrt{2}\sqrt{1+2\,x} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\arctan \left ( 1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+2*x)^(1/2)/(x^2+x+1),x)
[Out]
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Maxima [A] time = 0.760656, size = 178, normalized size = 1.13 \[ \frac{1}{3} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) + \frac{1}{3} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) - \frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + \frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x + 1)/(x^2 + x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224639, size = 231, normalized size = 1.47 \[ -\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (4 \, \arctan \left (\frac{3}{3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + \sqrt{6 \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 6 \, \sqrt{3}{\left (2 \, x + 1\right )} + 18} + 3}\right ) + 4 \, \arctan \left (\frac{3}{3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + \sqrt{-6 \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 6 \, \sqrt{3}{\left (2 \, x + 1\right )} + 18} - 3}\right ) + \log \left (6 \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 6 \, \sqrt{3}{\left (2 \, x + 1\right )} + 18\right ) - \log \left (-6 \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 6 \, \sqrt{3}{\left (2 \, x + 1\right )} + 18\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x + 1)/(x^2 + x + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.27699, size = 158, normalized size = 1.01 \[ \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x - \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{3} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x + \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{3} + \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} - 1 \right )}}{3} + \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} + 1 \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+2*x)**(1/2)/(x**2+x+1),x)
[Out]
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GIAC/XCAS [A] time = 0.230614, size = 162, normalized size = 1.03 \[ \frac{1}{3} \cdot 108^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) + \frac{1}{3} \cdot 108^{\frac{1}{4}} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) - \frac{1}{6} \cdot 108^{\frac{1}{4}}{\rm ln}\left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + \frac{1}{6} \cdot 108^{\frac{1}{4}}{\rm ln}\left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(2*x + 1)/(x^2 + x + 1),x, algorithm="giac")
[Out]