3.9 \(\int x^2 \log ^2(c x) \, dx\)

Optimal. Leaf size=32 \[ \frac{1}{3} x^3 \log ^2(c x)-\frac{2}{9} x^3 \log (c x)+\frac{2 x^3}{27} \]

[Out]

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

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Rubi [A]  time = 0.0197903, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2305, 2304} \[ \frac{1}{3} x^3 \log ^2(c x)-\frac{2}{9} x^3 \log (c x)+\frac{2 x^3}{27} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2305

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]

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

Mathematica [A]  time = 0.0012867, size = 32, normalized size = 1. \[ \frac{1}{3} x^3 \log ^2(c x)-\frac{2}{9} x^3 \log (c x)+\frac{2 x^3}{27} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.036, size = 27, normalized size = 0.8 \begin{align*}{\frac{2\,{x}^{3}}{27}}-{\frac{2\,{x}^{3}\ln \left ( cx \right ) }{9}}+{\frac{{x}^{3} \left ( \ln \left ( cx \right ) \right ) ^{2}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.991852, size = 28, normalized size = 0.88 \begin{align*} \frac{1}{27} \,{\left (9 \, \log \left (c x\right )^{2} - 6 \, \log \left (c x\right ) + 2\right )} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(9*log(c*x)^2 - 6*log(c*x) + 2)*x^3

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Fricas [A]  time = 0.755554, size = 68, normalized size = 2.12 \begin{align*} \frac{1}{3} \, x^{3} \log \left (c x\right )^{2} - \frac{2}{9} \, x^{3} \log \left (c x\right ) + \frac{2}{27} \, x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.108755, size = 29, normalized size = 0.91 \begin{align*} \frac{x^{3} \log{\left (c x \right )}^{2}}{3} - \frac{2 x^{3} \log{\left (c x \right )}}{9} + \frac{2 x^{3}}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19884, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{3} \, x^{3} \log \left (c x\right )^{2} - \frac{2}{9} \, x^{3} \log \left (c x\right ) + \frac{2}{27} \, x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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