3.16 \(\int x^2 \log ^3(c x) \, dx\)

Optimal. Leaf size=45 \[ \frac{1}{3} x^3 \log ^3(c x)-\frac{1}{3} x^3 \log ^2(c x)+\frac{2}{9} x^3 \log (c x)-\frac{2 x^3}{27} \]

[Out]

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

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Rubi [A]  time = 0.0314186, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2305, 2304} \[ \frac{1}{3} x^3 \log ^3(c x)-\frac{1}{3} x^3 \log ^2(c x)+\frac{2}{9} x^3 \log (c x)-\frac{2 x^3}{27} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Log[c*x]^3,x]

[Out]

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

Mathematica [A]  time = 0.0014576, size = 45, normalized size = 1. \[ \frac{1}{3} x^3 \log ^3(c x)-\frac{1}{3} x^3 \log ^2(c x)+\frac{2}{9} x^3 \log (c x)-\frac{2 x^3}{27} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Log[c*x]^3,x]

[Out]

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

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Maple [A]  time = 0.036, size = 38, 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}}+{\frac{{x}^{3} \left ( \ln \left ( cx \right ) \right ) ^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*ln(c*x)^3,x)

[Out]

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

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Maxima [A]  time = 1.07407, size = 39, normalized size = 0.87 \begin{align*} \frac{1}{27} \,{\left (9 \, \log \left (c x\right )^{3} - 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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.781229, size = 96, normalized size = 2.13 \begin{align*} \frac{1}{3} \, x^{3} \log \left (c x\right )^{3} - \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)^3,x, algorithm="fricas")

[Out]

1/3*x^3*log(c*x)^3 - 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 = 0.121807, size = 41, normalized size = 0.91 \begin{align*} \frac{x^{3} \log{\left (c x \right )}^{3}}{3} - \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)**3,x)

[Out]

x**3*log(c*x)**3/3 - 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 = 1.10201, size = 50, normalized size = 1.11 \begin{align*} \frac{1}{3} \, x^{3} \log \left (c x\right )^{3} - \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)^3,x, algorithm="giac")

[Out]

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