### 3.265 $$\int x^3 \log (x) \, dx$$

Optimal. Leaf size=17 $\frac{1}{4} x^4 \log (x)-\frac{x^4}{16}$

[Out]

-x^4/16 + (x^4*Log[x])/4

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Rubi [A]  time = 0.0066092, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.167, Rules used = {2304} $\frac{1}{4} x^4 \log (x)-\frac{x^4}{16}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-x^4/16 + (x^4*Log[x])/4

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^3 \log (x) \, dx &=-\frac{x^4}{16}+\frac{1}{4} x^4 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0008132, size = 17, normalized size = 1. $\frac{1}{4} x^4 \log (x)-\frac{x^4}{16}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-x^4/16 + (x^4*Log[x])/4

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Maple [A]  time = 0.001, size = 14, normalized size = 0.8 \begin{align*} -{\frac{{x}^{4}}{16}}+{\frac{{x}^{4}\ln \left ( x \right ) }{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*x^4+1/4*x^4*ln(x)

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Maxima [A]  time = 0.973134, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{4} \, x^{4} \log \left (x\right ) - \frac{1}{16} \, x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*log(x) - 1/16*x^4

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Fricas [A]  time = 1.88875, size = 36, normalized size = 2.12 \begin{align*} \frac{1}{4} \, x^{4} \log \left (x\right ) - \frac{1}{16} \, x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*log(x) - 1/16*x^4

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Sympy [A]  time = 0.08583, size = 12, normalized size = 0.71 \begin{align*} \frac{x^{4} \log{\left (x \right )}}{4} - \frac{x^{4}}{16} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4*log(x)/4 - x**4/16

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Giac [A]  time = 1.05809, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{4} \, x^{4} \log \left (x\right ) - \frac{1}{16} \, x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*log(x) - 1/16*x^4