Optimal. Leaf size=96 \[ \frac{2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{2}{9 \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right )}-\frac{2 \sqrt{x^3+1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x+1} \sqrt{x^2-x+1}} \]
[Out]
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Rubi [A] time = 0.112605, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}+\frac{2}{9 \sqrt{x+1} \sqrt{x^2-x+1} \left (x^3+1\right )}-\frac{2 \sqrt{x^3+1} \tanh ^{-1}\left (\sqrt{x^3+1}\right )}{3 \sqrt{x+1} \sqrt{x^2-x+1}} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(1 + x)^(5/2)*(1 - x + x^2)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 10.8541, size = 90, normalized size = 0.94 \[ \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{3 \left (x^{3} + 1\right )} + \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{9 \left (x^{3} + 1\right )^{2}} - \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1} \operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{3 \sqrt{x^{3} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(1+x)**(5/2)/(x**2-x+1)**(5/2),x)
[Out]
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Mathematica [C] time = 0.369333, size = 98, normalized size = 1.02 \[ \frac{2 \left (\frac{3 x^3+4}{\left (x^2-x+1\right )^{3/2}}-\frac{3 \sqrt{3} (x+1)^2 \Pi \left (1+\sqrt [3]{-1};\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{x+1}{1+\sqrt [3]{-1}}}}\right )}{9 (x+1)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(1 + x)^(5/2)*(1 - x + x^2)^(5/2)),x]
[Out]
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Maple [A] time = 0.071, size = 69, normalized size = 0.7 \[ -{\frac{2}{9\,{x}^{3}+9} \left ( 3\,{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) \sqrt{{x}^{3}+1}{x}^{3}-3\,{x}^{3}+3\,{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) \sqrt{{x}^{3}+1}-4 \right ){\frac{1}{\sqrt{1+x}}}{\frac{1}{\sqrt{{x}^{2}-x+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(1+x)^(5/2)/(x^2-x+1)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{5}{2}}{\left (x + 1\right )}^{\frac{5}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.277998, size = 153, normalized size = 1.59 \[ \frac{6 \, x^{3} - 3 \,{\left (x^{3} + 1\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1} \log \left (\sqrt{x^{2} - x + 1} \sqrt{x + 1} + 1\right ) + 3 \,{\left (x^{3} + 1\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1} \log \left (\sqrt{x^{2} - x + 1} \sqrt{x + 1} - 1\right ) + 8}{9 \,{\left (x^{3} + 1\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)*x),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(1/x/(1+x)**(5/2)/(x**2-x+1)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x^{2} - x + 1\right )}^{\frac{5}{2}}{\left (x + 1\right )}^{\frac{5}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((x^2 - x + 1)^(5/2)*(x + 1)^(5/2)*x),x, algorithm="giac")
[Out]