3.267 \(\int \frac{x^3}{\sqrt{1+x+x^2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{1}{3} \sqrt{x^2+x+1} x^2-\frac{1}{24} (10 x+1) \sqrt{x^2+x+1}+\frac{7}{16} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

[Out]

(x^2*Sqrt[1 + x + x^2])/3 - ((1 + 10*x)*Sqrt[1 + x + x^2])/24 + (7*ArcSinh[(1 +
2*x)/Sqrt[3]])/16

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Rubi [A]  time = 0.0695073, antiderivative size = 53, 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 \[ \frac{1}{3} \sqrt{x^2+x+1} x^2-\frac{1}{24} (10 x+1) \sqrt{x^2+x+1}+\frac{7}{16} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[1 + x + x^2],x]

[Out]

(x^2*Sqrt[1 + x + x^2])/3 - ((1 + 10*x)*Sqrt[1 + x + x^2])/24 + (7*ArcSinh[(1 +
2*x)/Sqrt[3]])/16

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Rubi in Sympy [A]  time = 3.93157, size = 56, normalized size = 1.06 \[ \frac{x^{2} \sqrt{x^{2} + x + 1}}{3} - \frac{\left (\frac{5 x}{2} + \frac{1}{4}\right ) \sqrt{x^{2} + x + 1}}{6} + \frac{7 \operatorname{atanh}{\left (\frac{2 x + 1}{2 \sqrt{x^{2} + x + 1}} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(x**2 + x + 1)/3 - (5*x/2 + 1/4)*sqrt(x**2 + x + 1)/6 + 7*atanh((2*x +
1)/(2*sqrt(x**2 + x + 1)))/16

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Mathematica [A]  time = 0.0264348, size = 41, normalized size = 0.77 \[ \frac{1}{24} \sqrt{x^2+x+1} \left (8 x^2-10 x-1\right )+\frac{7}{16} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/Sqrt[1 + x + x^2],x]

[Out]

(Sqrt[1 + x + x^2]*(-1 - 10*x + 8*x^2))/24 + (7*ArcSinh[(1 + 2*x)/Sqrt[3]])/16

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Maple [A]  time = 0.008, size = 47, normalized size = 0.9 \[{\frac{{x}^{2}}{3}\sqrt{{x}^{2}+x+1}}-{\frac{5\,x}{12}\sqrt{{x}^{2}+x+1}}-{\frac{1}{24}\sqrt{{x}^{2}+x+1}}+{\frac{7}{16}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( x+{\frac{1}{2}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(x^2+x+1)^(1/2),x)

[Out]

1/3*x^2*(x^2+x+1)^(1/2)-5/12*x*(x^2+x+1)^(1/2)-1/24*(x^2+x+1)^(1/2)+7/16*arcsinh
(2/3*3^(1/2)*(x+1/2))

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Maxima [A]  time = 1.64278, size = 65, normalized size = 1.23 \[ \frac{1}{3} \, \sqrt{x^{2} + x + 1} x^{2} - \frac{5}{12} \, \sqrt{x^{2} + x + 1} x - \frac{1}{24} \, \sqrt{x^{2} + x + 1} + \frac{7}{16} \, \operatorname{arsinh}\left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(x^2 + x + 1)*x^2 - 5/12*sqrt(x^2 + x + 1)*x - 1/24*sqrt(x^2 + x + 1) +
7/16*arcsinh(1/3*sqrt(3)*(2*x + 1))

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Fricas [A]  time = 0.20298, size = 212, normalized size = 4. \[ -\frac{2048 \, x^{6} + 1536 \, x^{5} - 384 \, x^{4} - 3840 \, x^{3} - 3456 \, x^{2} + 84 \,{\left (32 \, x^{3} + 48 \, x^{2} - 2 \,{\left (16 \, x^{2} + 16 \, x + 7\right )} \sqrt{x^{2} + x + 1} + 42 \, x + 13\right )} \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) - 2 \,{\left (1024 \, x^{5} + 256 \, x^{4} - 704 \, x^{3} - 1472 \, x^{2} - 704 \, x - 59\right )} \sqrt{x^{2} + x + 1} - 1530 \, x - 125}{192 \,{\left (32 \, x^{3} + 48 \, x^{2} - 2 \,{\left (16 \, x^{2} + 16 \, x + 7\right )} \sqrt{x^{2} + x + 1} + 42 \, x + 13\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/192*(2048*x^6 + 1536*x^5 - 384*x^4 - 3840*x^3 - 3456*x^2 + 84*(32*x^3 + 48*x^
2 - 2*(16*x^2 + 16*x + 7)*sqrt(x^2 + x + 1) + 42*x + 13)*log(-2*x + 2*sqrt(x^2 +
x + 1) - 1) - 2*(1024*x^5 + 256*x^4 - 704*x^3 - 1472*x^2 - 704*x - 59)*sqrt(x^2
+ x + 1) - 1530*x - 125)/(32*x^3 + 48*x^2 - 2*(16*x^2 + 16*x + 7)*sqrt(x^2 + x
+ 1) + 42*x + 13)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**3/sqrt(x**2 + x + 1), x)

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GIAC/XCAS [A]  time = 0.203827, size = 53, normalized size = 1. \[ \frac{1}{24} \,{\left (2 \,{\left (4 \, x - 5\right )} x - 1\right )} \sqrt{x^{2} + x + 1} - \frac{7}{16} \,{\rm ln}\left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(2*(4*x - 5)*x - 1)*sqrt(x^2 + x + 1) - 7/16*ln(-2*x + 2*sqrt(x^2 + x + 1)
- 1)