3.1.63 \(\int x^2 \log ^2(x) \, dx\) [63]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2342, 2341} \begin {gather*} \frac {2 x^3}{27}+\frac {1}{3} x^3 \log ^2(x)-\frac {2}{9} x^3 \log (x) \end {gather*}

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

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
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps

\begin {align*} \int x^2 \log ^2(x) \, dx &=\frac {1}{3} x^3 \log ^2(x)-\frac {2}{3} \int x^2 \log (x) \, dx\\ &=\frac {2 x^3}{27}-\frac {2}{9} x^3 \log (x)+\frac {1}{3} x^3 \log ^2(x)\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {2 x^3}{27}-\frac {2}{9} x^3 \log (x)+\frac {1}{3} x^3 \log ^2(x) \end {gather*}

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 = 0.00, size = 23, normalized size = 0.82

method	result	size
default	\(\frac {2 x^{3}}{27}-\frac {2 x^{3} \ln \left (x \right )}{9}+\frac {x^{3} \ln \left (x \right )^{2}}{3}\)	\(23\)
norman	\(\frac {2 x^{3}}{27}-\frac {2 x^{3} \ln \left (x \right )}{9}+\frac {x^{3} \ln \left (x \right )^{2}}{3}\)	\(23\)
risch	\(\frac {2 x^{3}}{27}-\frac {2 x^{3} \ln \left (x \right )}{9}+\frac {x^{3} \ln \left (x \right )^{2}}{3}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 4.00, size = 17, normalized size = 0.61 \begin {gather*} \frac {1}{27} \, {\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} \end {gather*}

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 = 0.98, size = 22, normalized size = 0.79 \begin {gather*} \frac {1}{3} \, x^{3} \log \left (x\right )^{2} - \frac {2}{9} \, x^{3} \log \left (x\right ) + \frac {2}{27} \, x^{3} \end {gather*}

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 = 0.04, size = 26, normalized size = 0.93 \begin {gather*} \frac {x^{3} \log {\left (x \right )}^{2}}{3} - \frac {2 x^{3} \log {\left (x \right )}}{9} + \frac {2 x^{3}}{27} \end {gather*}

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 [A]
time = 0.49, size = 22, normalized size = 0.79 \begin {gather*} \frac {1}{3} \, x^{3} \log \left (x\right )^{2} - \frac {2}{9} \, x^{3} \log \left (x\right ) + \frac {2}{27} \, x^{3} \end {gather*}

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*log(x)^2 - 2/9*x^3*log(x) + 2/27*x^3

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Mupad [B]
time = 0.03, size = 17, normalized size = 0.61 \begin {gather*} \frac {2\,x^3\,\left (\frac {9\,{\ln \left (x\right )}^2}{2}-3\,\ln \left (x\right )+1\right )}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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