3.17 \(\int x \log ^3(c x) \, dx\)

Optimal. Leaf size=45 \[ \frac {1}{2} x^2 \log ^3(c x)-\frac {3}{4} x^2 \log ^2(c x)+\frac {3}{4} x^2 \log (c x)-\frac {3 x^2}{8} \]

[Out]

-3/8*x^2+3/4*x^2*ln(c*x)-3/4*x^2*ln(c*x)^2+1/2*x^2*ln(c*x)^3

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Rubi [A]  time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2305, 2304} \[ \frac {1}{2} x^2 \log ^3(c x)-\frac {3}{4} x^2 \log ^2(c x)+\frac {3}{4} x^2 \log (c x)-\frac {3 x^2}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x^2)/8 + (3*x^2*Log[c*x])/4 - (3*x^2*Log[c*x]^2)/4 + (x^2*Log[c*x]^3)/2

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
]

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]

Rubi steps

\begin {align*} \int x \log ^3(c x) \, dx &=\frac {1}{2} x^2 \log ^3(c x)-\frac {3}{2} \int x \log ^2(c x) \, dx\\ &=-\frac {3}{4} x^2 \log ^2(c x)+\frac {1}{2} x^2 \log ^3(c x)+\frac {3}{2} \int x \log (c x) \, dx\\ &=-\frac {3 x^2}{8}+\frac {3}{4} x^2 \log (c x)-\frac {3}{4} x^2 \log ^2(c x)+\frac {1}{2} x^2 \log ^3(c x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 45, normalized size = 1.00 \[ \frac {1}{2} x^2 \log ^3(c x)-\frac {3}{4} x^2 \log ^2(c x)+\frac {3}{4} x^2 \log (c x)-\frac {3 x^2}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x^2)/8 + (3*x^2*Log[c*x])/4 - (3*x^2*Log[c*x]^2)/4 + (x^2*Log[c*x]^3)/2

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fricas [A]  time = 0.41, size = 37, normalized size = 0.82 \[ \frac {1}{2} \, x^{2} \log \left (c x\right )^{3} - \frac {3}{4} \, x^{2} \log \left (c x\right )^{2} + \frac {3}{4} \, x^{2} \log \left (c x\right ) - \frac {3}{8} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(c*x)^3,x, algorithm="fricas")

[Out]

1/2*x^2*log(c*x)^3 - 3/4*x^2*log(c*x)^2 + 3/4*x^2*log(c*x) - 3/8*x^2

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giac [A]  time = 0.22, size = 37, normalized size = 0.82 \[ \frac {1}{2} \, x^{2} \log \left (c x\right )^{3} - \frac {3}{4} \, x^{2} \log \left (c x\right )^{2} + \frac {3}{4} \, x^{2} \log \left (c x\right ) - \frac {3}{8} \, x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(c*x)^3,x, algorithm="giac")

[Out]

1/2*x^2*log(c*x)^3 - 3/4*x^2*log(c*x)^2 + 3/4*x^2*log(c*x) - 3/8*x^2

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maple [A]  time = 0.03, size = 38, normalized size = 0.84 \[ \frac {x^{2} \ln \left (c x \right )^{3}}{2}-\frac {3 x^{2} \ln \left (c x \right )^{2}}{4}+\frac {3 x^{2} \ln \left (c x \right )}{4}-\frac {3 x^{2}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*x^2+3/4*x^2*ln(c*x)-3/4*x^2*ln(c*x)^2+1/2*x^2*ln(c*x)^3

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maxima [A]  time = 0.50, size = 29, normalized size = 0.64 \[ \frac {1}{8} \, {\left (4 \, \log \left (c x\right )^{3} - 6 \, \log \left (c x\right )^{2} + 6 \, \log \left (c x\right ) - 3\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*log(c*x)^3,x, algorithm="maxima")

[Out]

1/8*(4*log(c*x)^3 - 6*log(c*x)^2 + 6*log(c*x) - 3)*x^2

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mupad [B]  time = 3.48, size = 29, normalized size = 0.64 \[ \frac {x^2\,\left (4\,{\ln \left (c\,x\right )}^3-6\,{\ln \left (c\,x\right )}^2+6\,\ln \left (c\,x\right )-3\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*log(c*x)^3,x)

[Out]

(x^2*(6*log(c*x) - 6*log(c*x)^2 + 4*log(c*x)^3 - 3))/8

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sympy [A]  time = 0.13, size = 42, normalized size = 0.93 \[ \frac {x^{2} \log {\left (c x \right )}^{3}}{2} - \frac {3 x^{2} \log {\left (c x \right )}^{2}}{4} + \frac {3 x^{2} \log {\left (c x \right )}}{4} - \frac {3 x^{2}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*ln(c*x)**3,x)

[Out]

x**2*log(c*x)**3/2 - 3*x**2*log(c*x)**2/4 + 3*x**2*log(c*x)/4 - 3*x**2/8

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