3.270 \(\int \frac{x^3}{\left (1+x+x^2\right )^{3/2}} \, dx\)

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]

(-2*x^2*(2 + x))/(3*Sqrt[1 + x + x^2]) + ((5 + 2*x)*Sqrt[1 + x + x^2])/3 - (3*Ar
cSinh[(1 + 2*x)/Sqrt[3]])/2

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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]

(-2*x^2*(2 + x))/(3*Sqrt[1 + x + x^2]) + ((5 + 2*x)*Sqrt[1 + x + x^2])/3 - (3*Ar
cSinh[(1 + 2*x)/Sqrt[3]])/2

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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]

-2*x**2*(x + 2)/(3*sqrt(x**2 + x + 1)) + (2*x + 5)*sqrt(x**2 + x + 1)/3 - 3*atan
h((2*x + 1)/(2*sqrt(x**2 + x + 1)))/2

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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]

(5 + 7*x + 3*x^2)/(3*Sqrt[1 + x + x^2]) - (3*ArcSinh[(1 + 2*x)/Sqrt[3]])/2

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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]

x^2/(x^2+x+1)^(1/2)+3/2*x/(x^2+x+1)^(1/2)+5/4/(x^2+x+1)^(1/2)+5/12*(1+2*x)/(x^2+
x+1)^(1/2)-3/2*arcsinh(2/3*3^(1/2)*(x+1/2))

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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]

x^2/sqrt(x^2 + x + 1) + 7/3*x/sqrt(x^2 + x + 1) + 5/3/sqrt(x^2 + x + 1) - 3/2*ar
csinh(1/3*sqrt(3)*(2*x + 1))

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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]

-1/4*(32*x^4 + 56*x^3 + 72*x^2 - 6*(8*x^3 + 12*x^2 - (8*x^2 + 8*x + 5)*sqrt(x^2
+ x + 1) + 12*x + 4)*log(-2*x + 2*sqrt(x^2 + x + 1) - 1) - (32*x^3 + 40*x^2 + 40
*x - 5)*sqrt(x^2 + x + 1) + 24*x + 8)/(8*x^3 + 12*x^2 - (8*x^2 + 8*x + 5)*sqrt(x
^2 + x + 1) + 12*x + 4)

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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]

Integral(x**3/(x**2 + x + 1)**(3/2), x)

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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]

1/3*((3*x + 7)*x + 5)/sqrt(x^2 + x + 1) + 3/2*ln(-2*x + 2*sqrt(x^2 + x + 1) - 1)