3.271 \(\int x^2 \sqrt{1+x+x^2} \, dx\)

Optimal. Leaf size=65 \[ \frac{1}{4} x \left (x^2+x+1\right )^{3/2}-\frac{5}{24} \left (x^2+x+1\right )^{3/2}+\frac{1}{64} (2 x+1) \sqrt{x^2+x+1}+\frac{3}{128} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

[Out]

((1 + 2*x)*Sqrt[1 + x + x^2])/64 - (5*(1 + x + x^2)^(3/2))/24 + (x*(1 + x + x^2)
^(3/2))/4 + (3*ArcSinh[(1 + 2*x)/Sqrt[3]])/128

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Rubi [A]  time = 0.0562837, antiderivative size = 65, 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 \[ \frac{1}{4} x \left (x^2+x+1\right )^{3/2}-\frac{5}{24} \left (x^2+x+1\right )^{3/2}+\frac{1}{64} (2 x+1) \sqrt{x^2+x+1}+\frac{3}{128} \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[1 + x + x^2],x]

[Out]

((1 + 2*x)*Sqrt[1 + x + x^2])/64 - (5*(1 + x + x^2)^(3/2))/24 + (x*(1 + x + x^2)
^(3/2))/4 + (3*ArcSinh[(1 + 2*x)/Sqrt[3]])/128

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Rubi in Sympy [A]  time = 3.42105, size = 65, normalized size = 1. \[ \frac{x \left (x^{2} + x + 1\right )^{\frac{3}{2}}}{4} + \frac{\left (2 x + 1\right ) \sqrt{x^{2} + x + 1}}{64} - \frac{5 \left (x^{2} + x + 1\right )^{\frac{3}{2}}}{24} + \frac{3 \operatorname{atanh}{\left (\frac{2 x + 1}{2 \sqrt{x^{2} + x + 1}} \right )}}{128} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(x**2 + x + 1)**(3/2)/4 + (2*x + 1)*sqrt(x**2 + x + 1)/64 - 5*(x**2 + x + 1)**
(3/2)/24 + 3*atanh((2*x + 1)/(2*sqrt(x**2 + x + 1)))/128

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Mathematica [A]  time = 0.0343313, size = 46, normalized size = 0.71 \[ \frac{1}{384} \left (2 \sqrt{x^2+x+1} \left (48 x^3+8 x^2+14 x-37\right )+9 \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Sqrt[1 + x + x^2],x]

[Out]

(2*Sqrt[1 + x + x^2]*(-37 + 14*x + 8*x^2 + 48*x^3) + 9*ArcSinh[(1 + 2*x)/Sqrt[3]
])/384

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Maple [A]  time = 0.007, size = 49, normalized size = 0.8 \[{\frac{x}{4} \left ({x}^{2}+x+1 \right ) ^{{\frac{3}{2}}}}-{\frac{5}{24} \left ({x}^{2}+x+1 \right ) ^{{\frac{3}{2}}}}+{\frac{1+2\,x}{64}\sqrt{{x}^{2}+x+1}}+{\frac{3}{128}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( x+{\frac{1}{2}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(x^2+x+1)^(1/2),x)

[Out]

1/4*x*(x^2+x+1)^(3/2)-5/24*(x^2+x+1)^(3/2)+1/64*(1+2*x)*(x^2+x+1)^(1/2)+3/128*ar
csinh(2/3*3^(1/2)*(x+1/2))

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Maxima [A]  time = 1.62022, size = 76, normalized size = 1.17 \[ \frac{1}{4} \,{\left (x^{2} + x + 1\right )}^{\frac{3}{2}} x - \frac{5}{24} \,{\left (x^{2} + x + 1\right )}^{\frac{3}{2}} + \frac{1}{32} \, \sqrt{x^{2} + x + 1} x + \frac{1}{64} \, \sqrt{x^{2} + x + 1} + \frac{3}{128} \, \operatorname{arsinh}\left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(x^2 + x + 1)^(3/2)*x - 5/24*(x^2 + x + 1)^(3/2) + 1/32*sqrt(x^2 + x + 1)*x
+ 1/64*sqrt(x^2 + x + 1) + 3/128*arcsinh(1/3*sqrt(3)*(2*x + 1))

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Fricas [A]  time = 0.199672, size = 266, normalized size = 4.09 \[ -\frac{98304 \, x^{8} + 262144 \, x^{7} + 425984 \, x^{6} + 344064 \, x^{5} + 120960 \, x^{4} - 102144 \, x^{3} - 137952 \, x^{2} + 72 \,{\left (128 \, x^{4} + 256 \, x^{3} + 288 \, x^{2} - 8 \,{\left (16 \, x^{3} + 24 \, x^{2} + 18 \, x + 5\right )} \sqrt{x^{2} + x + 1} + 160 \, x + 41\right )} \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) - 8 \,{\left (12288 \, x^{7} + 26624 \, x^{6} + 35328 \, x^{5} + 17664 \, x^{4} - 2256 \, x^{3} - 12984 \, x^{2} - 8298 \, x - 2369\right )} \sqrt{x^{2} + x + 1} - 78688 \, x - 18227}{3072 \,{\left (128 \, x^{4} + 256 \, x^{3} + 288 \, x^{2} - 8 \,{\left (16 \, x^{3} + 24 \, x^{2} + 18 \, x + 5\right )} \sqrt{x^{2} + x + 1} + 160 \, x + 41\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3072*(98304*x^8 + 262144*x^7 + 425984*x^6 + 344064*x^5 + 120960*x^4 - 102144*
x^3 - 137952*x^2 + 72*(128*x^4 + 256*x^3 + 288*x^2 - 8*(16*x^3 + 24*x^2 + 18*x +
5)*sqrt(x^2 + x + 1) + 160*x + 41)*log(-2*x + 2*sqrt(x^2 + x + 1) - 1) - 8*(122
88*x^7 + 26624*x^6 + 35328*x^5 + 17664*x^4 - 2256*x^3 - 12984*x^2 - 8298*x - 236
9)*sqrt(x^2 + x + 1) - 78688*x - 18227)/(128*x^4 + 256*x^3 + 288*x^2 - 8*(16*x^3
+ 24*x^2 + 18*x + 5)*sqrt(x^2 + x + 1) + 160*x + 41)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**2*sqrt(x**2 + x + 1), x)

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GIAC/XCAS [A]  time = 0.206108, size = 59, normalized size = 0.91 \[ \frac{1}{192} \,{\left (2 \,{\left (4 \,{\left (6 \, x + 1\right )} x + 7\right )} x - 37\right )} \sqrt{x^{2} + x + 1} - \frac{3}{128} \,{\rm ln}\left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(2*(4*(6*x + 1)*x + 7)*x - 37)*sqrt(x^2 + x + 1) - 3/128*ln(-2*x + 2*sqrt(
x^2 + x + 1) - 1)