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

Optimal. Leaf size=79 \[ \frac{2 (1-x)}{3 x^2 \sqrt{x^2+x+1}}+\frac{37 \sqrt{x^2+x+1}}{12 x}-\frac{7 \sqrt{x^2+x+1}}{6 x^2}-\frac{3}{8} \tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right ) \]

[Out]

(2*(1 - x))/(3*x^2*Sqrt[1 + x + x^2]) - (7*Sqrt[1 + x + x^2])/(6*x^2) + (37*Sqrt
[1 + x + x^2])/(12*x) - (3*ArcTanh[(2 + x)/(2*Sqrt[1 + x + x^2])])/8

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Rubi [A]  time = 0.115994, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{2 (1-x)}{3 x^2 \sqrt{x^2+x+1}}+\frac{37 \sqrt{x^2+x+1}}{12 x}-\frac{7 \sqrt{x^2+x+1}}{6 x^2}-\frac{3}{8} \tanh ^{-1}\left (\frac{x+2}{2 \sqrt{x^2+x+1}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(1 + x + x^2)^(3/2)),x]

[Out]

(2*(1 - x))/(3*x^2*Sqrt[1 + x + x^2]) - (7*Sqrt[1 + x + x^2])/(6*x^2) + (37*Sqrt
[1 + x + x^2])/(12*x) - (3*ArcTanh[(2 + x)/(2*Sqrt[1 + x + x^2])])/8

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Rubi in Sympy [A]  time = 7.01911, size = 71, normalized size = 0.9 \[ - \frac{3 \operatorname{atanh}{\left (\frac{x + 2}{2 \sqrt{x^{2} + x + 1}} \right )}}{8} + \frac{37 \sqrt{x^{2} + x + 1}}{12 x} + \frac{2 \left (- x + 1\right )}{3 x^{2} \sqrt{x^{2} + x + 1}} - \frac{7 \sqrt{x^{2} + x + 1}}{6 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(x**2+x+1)**(3/2),x)

[Out]

-3*atanh((x + 2)/(2*sqrt(x**2 + x + 1)))/8 + 37*sqrt(x**2 + x + 1)/(12*x) + 2*(-
x + 1)/(3*x**2*sqrt(x**2 + x + 1)) - 7*sqrt(x**2 + x + 1)/(6*x**2)

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Mathematica [A]  time = 0.0453761, size = 57, normalized size = 0.72 \[ \frac{1}{24} \left (-9 \log \left (2 \sqrt{x^2+x+1}+x+2\right )+\frac{2 \left (37 x^3+23 x^2+15 x-6\right )}{x^2 \sqrt{x^2+x+1}}+9 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(1 + x + x^2)^(3/2)),x]

[Out]

((2*(-6 + 15*x + 23*x^2 + 37*x^3))/(x^2*Sqrt[1 + x + x^2]) + 9*Log[x] - 9*Log[2
+ x + 2*Sqrt[1 + x + x^2]])/24

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Maple [A]  time = 0.007, size = 69, normalized size = 0.9 \[ -{\frac{1}{2\,{x}^{2}}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}+{\frac{5}{4\,x}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}+{\frac{3}{8}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}+{\frac{37+74\,x}{24}{\frac{1}{\sqrt{{x}^{2}+x+1}}}}-{\frac{3}{8}{\it Artanh} \left ({\frac{2+x}{2}{\frac{1}{\sqrt{{x}^{2}+x+1}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^2+x+1)^(3/2),x)

[Out]

-1/2/x^2/(x^2+x+1)^(1/2)+5/4/x/(x^2+x+1)^(1/2)+3/8/(x^2+x+1)^(1/2)+37/24*(1+2*x)
/(x^2+x+1)^(1/2)-3/8*arctanh(1/2*(2+x)/(x^2+x+1)^(1/2))

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Maxima [A]  time = 1.59365, size = 96, normalized size = 1.22 \[ \frac{37 \, x}{12 \, \sqrt{x^{2} + x + 1}} + \frac{23}{12 \, \sqrt{x^{2} + x + 1}} + \frac{5}{4 \, \sqrt{x^{2} + x + 1} x} - \frac{1}{2 \, \sqrt{x^{2} + x + 1} x^{2}} - \frac{3}{8} \, \operatorname{arsinh}\left (\frac{\sqrt{3} x}{3 \,{\left | x \right |}} + \frac{2 \, \sqrt{3}}{3 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + x + 1)^(3/2)*x^3),x, algorithm="maxima")

[Out]

37/12*x/sqrt(x^2 + x + 1) + 23/12/sqrt(x^2 + x + 1) + 5/4/(sqrt(x^2 + x + 1)*x)
- 1/2/(sqrt(x^2 + x + 1)*x^2) - 3/8*arcsinh(1/3*sqrt(3)*x/abs(x) + 2/3*sqrt(3)/a
bs(x))

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Fricas [A]  time = 0.234071, size = 354, normalized size = 4.48 \[ -\frac{96 \, x^{5} + 168 \, x^{4} + 182 \, x^{3} + 82 \, x^{2} + 3 \,{\left (32 \, x^{6} + 64 \, x^{5} + 78 \, x^{4} + 46 \, x^{3} + 14 \, x^{2} -{\left (32 \, x^{5} + 48 \, x^{4} + 42 \, x^{3} + 13 \, x^{2}\right )} \sqrt{x^{2} + x + 1}\right )} \log \left (-x + \sqrt{x^{2} + x + 1} + 1\right ) - 3 \,{\left (32 \, x^{6} + 64 \, x^{5} + 78 \, x^{4} + 46 \, x^{3} + 14 \, x^{2} -{\left (32 \, x^{5} + 48 \, x^{4} + 42 \, x^{3} + 13 \, x^{2}\right )} \sqrt{x^{2} + x + 1}\right )} \log \left (-x + \sqrt{x^{2} + x + 1} - 1\right ) - 2 \,{\left (48 \, x^{4} + 60 \, x^{3} + 43 \, x^{2} + 6 \, x - 28\right )} \sqrt{x^{2} + x + 1} - 38 \, x - 52}{8 \,{\left (32 \, x^{6} + 64 \, x^{5} + 78 \, x^{4} + 46 \, x^{3} + 14 \, x^{2} -{\left (32 \, x^{5} + 48 \, x^{4} + 42 \, x^{3} + 13 \, x^{2}\right )} \sqrt{x^{2} + x + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + x + 1)^(3/2)*x^3),x, algorithm="fricas")

[Out]

-1/8*(96*x^5 + 168*x^4 + 182*x^3 + 82*x^2 + 3*(32*x^6 + 64*x^5 + 78*x^4 + 46*x^3
+ 14*x^2 - (32*x^5 + 48*x^4 + 42*x^3 + 13*x^2)*sqrt(x^2 + x + 1))*log(-x + sqrt
(x^2 + x + 1) + 1) - 3*(32*x^6 + 64*x^5 + 78*x^4 + 46*x^3 + 14*x^2 - (32*x^5 + 4
8*x^4 + 42*x^3 + 13*x^2)*sqrt(x^2 + x + 1))*log(-x + sqrt(x^2 + x + 1) - 1) - 2*
(48*x^4 + 60*x^3 + 43*x^2 + 6*x - 28)*sqrt(x^2 + x + 1) - 38*x - 52)/(32*x^6 + 6
4*x^5 + 78*x^4 + 46*x^3 + 14*x^2 - (32*x^5 + 48*x^4 + 42*x^3 + 13*x^2)*sqrt(x^2
+ x + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \left (x^{2} + x + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(x**2+x+1)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.207246, size = 158, normalized size = 2. \[ \frac{2 \,{\left (2 \, x + 1\right )}}{3 \, \sqrt{x^{2} + x + 1}} - \frac{3 \,{\left (x - \sqrt{x^{2} + x + 1}\right )}^{3} + 8 \,{\left (x - \sqrt{x^{2} + x + 1}\right )}^{2} - 13 \, x + 13 \, \sqrt{x^{2} + x + 1} - 16}{4 \,{\left ({\left (x - \sqrt{x^{2} + x + 1}\right )}^{2} - 1\right )}^{2}} - \frac{3}{8} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x + 1} + 1 \right |}\right ) + \frac{3}{8} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + x + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 + x + 1)^(3/2)*x^3),x, algorithm="giac")

[Out]

2/3*(2*x + 1)/sqrt(x^2 + x + 1) - 1/4*(3*(x - sqrt(x^2 + x + 1))^3 + 8*(x - sqrt
(x^2 + x + 1))^2 - 13*x + 13*sqrt(x^2 + x + 1) - 16)/((x - sqrt(x^2 + x + 1))^2
- 1)^2 - 3/8*ln(abs(-x + sqrt(x^2 + x + 1) + 1)) + 3/8*ln(abs(-x + sqrt(x^2 + x
+ 1) - 1))