Optimal. Leaf size=170 \[ \frac{4}{3} (2 x+1)^{3/2}-\frac{3^{3/4} \log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\frac{3^{3/4} \log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\sqrt{2} 3^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )-\sqrt{2} 3^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right ) \]
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Rubi [A] time = 0.308692, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{4}{3} (2 x+1)^{3/2}-\frac{3^{3/4} \log \left (2 x-\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\frac{3^{3/4} \log \left (2 x+\sqrt{2} \sqrt [4]{3} \sqrt{2 x+1}+\sqrt{3}+1\right )}{\sqrt{2}}+\sqrt{2} 3^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}\right )-\sqrt{2} 3^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{2 x+1}}{\sqrt [4]{3}}+1\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + 2*x)^(5/2)/(1 + x + x^2),x]
[Out]
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Rubi in Sympy [A] time = 34.5328, size = 163, normalized size = 0.96 \[ \frac{4 \left (2 x + 1\right )^{\frac{3}{2}}}{3} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x - \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x + \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )}}{2} - \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} - 1 \right )} - \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+2*x)**(5/2)/(x**2+x+1),x)
[Out]
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Mathematica [A] time = 0.0991131, size = 156, normalized size = 0.92 \[ \frac{4}{3} (2 x+1)^{3/2}-\frac{3^{3/4} \log \left (\sqrt{3} (2 x+1)-3^{3/4} \sqrt{4 x+2}+3\right )}{\sqrt{2}}+\frac{3^{3/4} \log \left (\sqrt{3} (2 x+1)+3^{3/4} \sqrt{4 x+2}+3\right )}{\sqrt{2}}+\sqrt{2} 3^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}\right )-\sqrt{2} 3^{3/4} \tan ^{-1}\left (\frac{\sqrt{4 x+2}}{\sqrt [4]{3}}+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + 2*x)^(5/2)/(1 + x + x^2),x]
[Out]
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Maple [A] time = 0.008, size = 120, normalized size = 0.7 \[{\frac{4}{3} \left ( 1+2\,x \right ) ^{{\frac{3}{2}}}}-{3}^{{\frac{3}{4}}}\arctan \left ( -1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) \sqrt{2}-{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{2}\ln \left ({1 \left ( 1+2\,x+\sqrt{3}-\sqrt [4]{3}\sqrt{2}\sqrt{1+2\,x} \right ) \left ( 1+2\,x+\sqrt{3}+\sqrt [4]{3}\sqrt{2}\sqrt{1+2\,x} \right ) ^{-1}} \right ) }-{3}^{{\frac{3}{4}}}\arctan \left ( 1+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{3}\sqrt{1+2\,x}} \right ) \sqrt{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+2*x)^(5/2)/(x^2+x+1),x)
[Out]
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Maxima [A] time = 0.768606, size = 190, normalized size = 1.12 \[ -3^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) - 3^{\frac{3}{4}} \sqrt{2} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) + \frac{1}{2} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{1}{2} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) + \frac{4}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 1)^(5/2)/(x^2 + x + 1),x, algorithm="maxima")
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Fricas [A] time = 0.227304, size = 278, normalized size = 1.64 \[ 2 \cdot 27^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{27^{\frac{3}{4}} \sqrt{2}}{27^{\frac{3}{4}} \sqrt{2} + 6 \, \sqrt{27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 18 \, x + 9 \, \sqrt{3} + 9} + 18 \, \sqrt{2 \, x + 1}}\right ) + 2 \cdot 27^{\frac{1}{4}} \sqrt{2} \arctan \left (-\frac{27^{\frac{3}{4}} \sqrt{2}}{27^{\frac{3}{4}} \sqrt{2} - 2 \, \sqrt{-9 \cdot 27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 162 \, x + 81 \, \sqrt{3} + 81} - 18 \, \sqrt{2 \, x + 1}}\right ) + \frac{1}{2} \cdot 27^{\frac{1}{4}} \sqrt{2} \log \left (9 \cdot 27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 162 \, x + 81 \, \sqrt{3} + 81\right ) - \frac{1}{2} \cdot 27^{\frac{1}{4}} \sqrt{2} \log \left (-9 \cdot 27^{\frac{3}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 162 \, x + 81 \, \sqrt{3} + 81\right ) + \frac{4}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 1)^(5/2)/(x^2 + x + 1),x, algorithm="fricas")
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Sympy [A] time = 26.627, size = 163, normalized size = 0.96 \[ \frac{4 \left (2 x + 1\right )^{\frac{3}{2}}}{3} - \sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x - \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )} + \sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (2 x + \sqrt{2} \sqrt [4]{3} \sqrt{2 x + 1} + 1 + \sqrt{3} \right )} - 2 \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} - 1 \right )} - 2 \sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{2 x + 1}}{3} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+2*x)**(5/2)/(x**2+x+1),x)
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GIAC/XCAS [A] time = 0.230681, size = 174, normalized size = 1.02 \[ \frac{4}{3} \,{\left (2 \, x + 1\right )}^{\frac{3}{2}} - 108^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} + 2 \, \sqrt{2 \, x + 1}\right )}\right ) - 108^{\frac{1}{4}} \arctan \left (-\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (3^{\frac{1}{4}} \sqrt{2} - 2 \, \sqrt{2 \, x + 1}\right )}\right ) + \frac{1}{2} \cdot 108^{\frac{1}{4}}{\rm ln}\left (3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) - \frac{1}{2} \cdot 108^{\frac{1}{4}}{\rm ln}\left (-3^{\frac{1}{4}} \sqrt{2} \sqrt{2 \, x + 1} + 2 \, x + \sqrt{3} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 1)^(5/2)/(x^2 + x + 1),x, algorithm="giac")
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