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

Optimal. Leaf size=17 \[ \frac {1}{3} x^3 \log (x)-\frac {x^3}{9} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2304} \[ \frac {1}{3} x^3 \log (x)-\frac {x^3}{9} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Log[x],x]

[Out]

-x^3/9 + (x^3*Log[x])/3

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {align*} \int x^2 \log (x) \, dx &=-\frac {x^3}{9}+\frac {1}{3} x^3 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ \frac {1}{3} x^3 \log (x)-\frac {x^3}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Log[x],x]

[Out]

-1/9*x^3 + (x^3*Log[x])/3

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fricas [A]  time = 0.44, size = 13, normalized size = 0.76 \[ \frac {1}{3} \, x^{3} \log \relax (x) - \frac {1}{9} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*log(x) - 1/9*x^3

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giac [A]  time = 1.11, size = 13, normalized size = 0.76 \[ \frac {1}{3} \, x^{3} \log \relax (x) - \frac {1}{9} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*log(x) - 1/9*x^3

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maple [A]  time = 0.01, size = 14, normalized size = 0.82 \[ \frac {x^{3} \ln \relax (x )}{3}-\frac {x^{3}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*ln(x),x)

[Out]

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

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maxima [A]  time = 0.45, size = 13, normalized size = 0.76 \[ \frac {1}{3} \, x^{3} \log \relax (x) - \frac {1}{9} \, x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*log(x) - 1/9*x^3

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mupad [B]  time = 0.03, size = 9, normalized size = 0.53 \[ \frac {x^3\,\left (\ln \relax (x)-\frac {1}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*log(x),x)

[Out]

(x^3*(log(x) - 1/3))/3

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sympy [A]  time = 0.09, size = 12, normalized size = 0.71 \[ \frac {x^{3} \log {\relax (x )}}{3} - \frac {x^{3}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*ln(x),x)

[Out]

x**3*log(x)/3 - x**3/9

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