Optimal. Leaf size=56 \[ -\frac{2 (x+2) x^2}{3 \sqrt{x^2+x+1}}+\frac{1}{3} (2 x+5) \sqrt{x^2+x+1}-\frac{3}{2} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]
[Out]
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Rubi [A] time = 0.068452, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{2 (x+2) x^2}{3 \sqrt{x^2+x+1}}+\frac{1}{3} (2 x+5) \sqrt{x^2+x+1}-\frac{3}{2} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^3/(1 + x + x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 4.17458, size = 58, normalized size = 1.04 \[ - \frac{2 x^{2} \left (x + 2\right )}{3 \sqrt{x^{2} + x + 1}} + \frac{\left (2 x + 5\right ) \sqrt{x^{2} + x + 1}}{3} - \frac{3 \operatorname{atanh}{\left (\frac{2 x + 1}{2 \sqrt{x^{2} + x + 1}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(x**2+x+1)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0327183, size = 41, normalized size = 0.73 \[ \frac{3 x^2+7 x+5}{3 \sqrt{x^2+x+1}}-\frac{3}{2} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(1 + x + x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.009, size = 61, normalized size = 1.1 \[{{x}^{2}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}+{\frac{3\,x}{2}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}+{\frac{5}{4}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}+{\frac{5+10\,x}{12}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}-{\frac{3}{2}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( x+{\frac{1}{2}} \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(x^2+x+1)^(3/2),x)
[Out]
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Maxima [A] time = 1.59714, size = 63, normalized size = 1.12 \[ \frac{x^{2}}{\sqrt{x^{2} + x + 1}} + \frac{7 \, x}{3 \, \sqrt{x^{2} + x + 1}} + \frac{5}{3 \, \sqrt{x^{2} + x + 1}} - \frac{3}{2} \, \operatorname{arsinh}\left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(x^2 + x + 1)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241266, size = 185, normalized size = 3.3 \[ -\frac{32 \, x^{4} + 56 \, x^{3} + 72 \, x^{2} - 6 \,{\left (8 \, x^{3} + 12 \, x^{2} -{\left (8 \, x^{2} + 8 \, x + 5\right )} \sqrt{x^{2} + x + 1} + 12 \, x + 4\right )} \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) -{\left (32 \, x^{3} + 40 \, x^{2} + 40 \, x - 5\right )} \sqrt{x^{2} + x + 1} + 24 \, x + 8}{4 \,{\left (8 \, x^{3} + 12 \, x^{2} -{\left (8 \, x^{2} + 8 \, x + 5\right )} \sqrt{x^{2} + x + 1} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(x^2 + x + 1)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\left (x^{2} + x + 1\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(x**2+x+1)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.205979, size = 51, normalized size = 0.91 \[ \frac{{\left (3 \, x + 7\right )} x + 5}{3 \, \sqrt{x^{2} + x + 1}} + \frac{3}{2} \,{\rm ln}\left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(x^2 + x + 1)^(3/2),x, algorithm="giac")
[Out]