3.63 \(\int x^2 \log ^2(x) \, dx\)

Optimal. Leaf size=28 \[ \frac{2 x^3}{27}+\frac{1}{3} x^3 \log ^2(x)-\frac{2}{9} x^3 \log (x) \]

[Out]

(2*x^3)/27 - (2*x^3*Log[x])/9 + (x^3*Log[x]^2)/3

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Rubi [A]  time = 0.0305475, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 x^3}{27}+\frac{1}{3} x^3 \log ^2(x)-\frac{2}{9} x^3 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[x^2*Log[x]^2,x]

[Out]

(2*x^3)/27 - (2*x^3*Log[x])/9 + (x^3*Log[x]^2)/3

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Rubi in Sympy [A]  time = 1.92764, size = 26, normalized size = 0.93 \[ \frac{x^{3} \log{\left (x \right )}^{2}}{3} - \frac{2 x^{3} \log{\left (x \right )}}{9} + \frac{2 x^{3}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*ln(x)**2,x)

[Out]

x**3*log(x)**2/3 - 2*x**3*log(x)/9 + 2*x**3/27

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Mathematica [A]  time = 0.00370572, size = 28, normalized size = 1. \[ \frac{2 x^3}{27}+\frac{1}{3} x^3 \log ^2(x)-\frac{2}{9} x^3 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Log[x]^2,x]

[Out]

(2*x^3)/27 - (2*x^3*Log[x])/9 + (x^3*Log[x]^2)/3

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Maple [A]  time = 0.001, size = 23, normalized size = 0.8 \[{\frac{2\,{x}^{3}}{27}}-{\frac{2\,{x}^{3}\ln \left ( x \right ) }{9}}+{\frac{{x}^{3} \left ( \ln \left ( x \right ) \right ) ^{2}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*ln(x)^2,x)

[Out]

2/27*x^3-2/9*x^3*ln(x)+1/3*x^3*ln(x)^2

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Maxima [A]  time = 1.41614, size = 23, normalized size = 0.82 \[ \frac{1}{27} \,{\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*log(x)^2,x, algorithm="maxima")

[Out]

1/27*(9*log(x)^2 - 6*log(x) + 2)*x^3

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Fricas [A]  time = 0.20615, size = 30, normalized size = 1.07 \[ \frac{1}{3} \, x^{3} \log \left (x\right )^{2} - \frac{2}{9} \, x^{3} \log \left (x\right ) + \frac{2}{27} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*log(x)^2,x, algorithm="fricas")

[Out]

1/3*x^3*log(x)^2 - 2/9*x^3*log(x) + 2/27*x^3

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Sympy [A]  time = 0.091174, size = 26, normalized size = 0.93 \[ \frac{x^{3} \log{\left (x \right )}^{2}}{3} - \frac{2 x^{3} \log{\left (x \right )}}{9} + \frac{2 x^{3}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*ln(x)**2,x)

[Out]

x**3*log(x)**2/3 - 2*x**3*log(x)/9 + 2*x**3/27

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GIAC/XCAS [A]  time = 0.23451, size = 30, normalized size = 1.07 \[ \frac{1}{3} \, x^{3}{\rm ln}\left (x\right )^{2} - \frac{2}{9} \, x^{3}{\rm ln}\left (x\right ) + \frac{2}{27} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*log(x)^2,x, algorithm="giac")

[Out]

1/3*x^3*ln(x)^2 - 2/9*x^3*ln(x) + 2/27*x^3