3.226 \(\int 16 x^3 \log ^2(x) \, dx\)

Optimal. Leaf size=24 \[ \frac {x^4}{2}+4 x^4 \log ^2(x)-2 x^4 \log (x) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {12, 2305, 2304} \[ \frac {x^4}{2}+4 x^4 \log ^2(x)-2 x^4 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[16*x^3*Log[x]^2,x]

[Out]

x^4/2 - 2*x^4*Log[x] + 4*x^4*Log[x]^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 16 x^3 \log ^2(x) \, dx &=16 \int x^3 \log ^2(x) \, dx\\ &=4 x^4 \log ^2(x)-8 \int x^3 \log (x) \, dx\\ &=\frac {x^4}{2}-2 x^4 \log (x)+4 x^4 \log ^2(x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.25 \[ 16 \left (\frac {x^4}{32}+\frac {1}{4} x^4 \log ^2(x)-\frac {1}{8} x^4 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[16*x^3*Log[x]^2,x]

[Out]

16*(x^4/32 - (x^4*Log[x])/8 + (x^4*Log[x]^2)/4)

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fricas [A]  time = 0.46, size = 22, normalized size = 0.92 \[ 4 \, x^{4} \log \relax (x)^{2} - 2 \, x^{4} \log \relax (x) + \frac {1}{2} \, x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*x^3*log(x)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 22, normalized size = 0.92 \[ 4 \, x^{4} \log \relax (x)^{2} - 2 \, x^{4} \log \relax (x) + \frac {1}{2} \, x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*x^3*log(x)^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 23, normalized size = 0.96 \[ 4 x^{4} \ln \relax (x )^{2}-2 x^{4} \ln \relax (x )+\frac {x^{4}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16*x^3*ln(x)^2,x)

[Out]

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

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maxima [A]  time = 0.44, size = 17, normalized size = 0.71 \[ \frac {1}{2} \, {\left (8 \, \log \relax (x)^{2} - 4 \, \log \relax (x) + 1\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*x^3*log(x)^2,x, algorithm="maxima")

[Out]

1/2*(8*log(x)^2 - 4*log(x) + 1)*x^4

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mupad [B]  time = 0.04, size = 17, normalized size = 0.71 \[ \frac {x^4\,\left (8\,{\ln \relax (x)}^2-4\,\ln \relax (x)+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16*x^3*log(x)^2,x)

[Out]

(x^4*(8*log(x)^2 - 4*log(x) + 1))/2

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sympy [A]  time = 0.11, size = 22, normalized size = 0.92 \[ 4 x^{4} \log {\relax (x )}^{2} - 2 x^{4} \log {\relax (x )} + \frac {x^{4}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*x**3*ln(x)**2,x)

[Out]

4*x**4*log(x)**2 - 2*x**4*log(x) + x**4/2

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