3.1.55 \(\int x^2 \log (x) \, dx\) [55]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, 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 = {2341} \begin {gather*} \frac {1}{3} x^3 \log (x)-\frac {x^3}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*x^3 + (x^3*Log[x])/3

Rule 2341

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*}

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*x^3 + (x^3*Log[x])/3

________________________________________________________________________________________

Maple [A]
time = 0.00, size = 14, normalized size = 0.82

method	result	size
default	\(-\frac {x^{3}}{9}+\frac {x^{3} \ln \left (x \right )}{3}\)	\(14\)
norman	\(-\frac {x^{3}}{9}+\frac {x^{3} \ln \left (x \right )}{3}\)	\(14\)
risch	\(-\frac {x^{3}}{9}+\frac {x^{3} \ln \left (x \right )}{3}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*ln(x),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 2.03, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, x^{3} \log \left (x\right ) - \frac {1}{9} \, x^{3} \end {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.71, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, x^{3} \log \left (x\right ) - \frac {1}{9} \, x^{3} \end {gather*}

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

________________________________________________________________________________________

Sympy [A]
time = 0.03, size = 12, normalized size = 0.71 \begin {gather*} \frac {x^{3} \log {\left (x \right )}}{3} - \frac {x^{3}}{9} \end {gather*}

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

________________________________________________________________________________________

Giac [A]
time = 1.07, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, x^{3} \log \left (x\right ) - \frac {1}{9} \, x^{3} \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 9, normalized size = 0.53 \begin {gather*} \frac {x^3\,\left (\ln \left (x\right )-\frac {1}{3}\right )}{3} \end {gather*}

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

________________________________________________________________________________________