### 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]

x^4/2 - 2*x^4*Log[x] + 4*x^4*Log[x]^2

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Rubi [A]  time = 0.0228337, antiderivative size = 24, normalized size of antiderivative = 1., 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 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 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*}

Mathematica [A]  time = 0.0012288, 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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Maple [A]  time = 0.003, size = 23, normalized size = 1. \begin{align*}{\frac{{x}^{4}}{2}}-2\,{x}^{4}\ln \left ( x \right ) +4\,{x}^{4} \left ( \ln \left ( x \right ) \right ) ^{2} \end{align*}

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 = 1.04629, size = 23, normalized size = 0.96 \begin{align*} \frac{1}{2} \,{\left (8 \, \log \left (x\right )^{2} - 4 \, \log \left (x\right ) + 1\right )} x^{4} \end{align*}

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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Fricas [A]  time = 1.78746, size = 55, normalized size = 2.29 \begin{align*} 4 \, x^{4} \log \left (x\right )^{2} - 2 \, x^{4} \log \left (x\right ) + \frac{1}{2} \, x^{4} \end{align*}

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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Sympy [A]  time = 0.105006, size = 22, normalized size = 0.92 \begin{align*} 4 x^{4} \log{\left (x \right )}^{2} - 2 x^{4} \log{\left (x \right )} + \frac{x^{4}}{2} \end{align*}

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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Giac [A]  time = 1.34682, size = 30, normalized size = 1.25 \begin{align*} 4 \, x^{4} \log \left (x\right )^{2} - 2 \, x^{4} \log \left (x\right ) + \frac{1}{2} \, x^{4} \end{align*}

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