3.3.65 \(\int x^3 \log (x) \, dx\) [265]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = {2341} \begin {gather*} \frac {1}{4} x^4 \log (x)-\frac {x^4}{16} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*x^4 + (x^4*Log[x])/4

Rule 2341

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*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {x^4}{16}+\frac {1}{4} x^4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*x^4 + (x^4*Log[x])/4

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Mathics [A]
time = 1.69, size = 11, normalized size = 0.65 \begin {gather*} \frac {x^4 \left (-1+4 \text {Log}\left [x\right ]\right )}{16} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^3*Log[x],x]')

[Out]

x ^ 4 (-1 + 4 Log[x]) / 16

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Maple [A]
time = 0.00, size = 14, normalized size = 0.82

method	result	size
default	\(-\frac {x^{4}}{16}+\frac {x^{4} \ln \left (x \right )}{4}\)	\(14\)
norman	\(-\frac {x^{4}}{16}+\frac {x^{4} \ln \left (x \right )}{4}\)	\(14\)
risch	\(-\frac {x^{4}}{16}+\frac {x^{4} \ln \left (x \right )}{4}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*ln(x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, x^{4} \log \left (x\right ) - \frac {1}{16} \, x^{4} \end {gather*}

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 = 0.32, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, x^{4} \log \left (x\right ) - \frac {1}{16} \, x^{4} \end {gather*}

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 = 0.04, size = 12, normalized size = 0.71 \begin {gather*} \frac {x^{4} \log {\left (x \right )}}{4} - \frac {x^{4}}{16} \end {gather*}

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 = 0.00, size = 16, normalized size = 0.94 \begin {gather*} -\frac {x^{4}}{16}+\frac {1}{4} x^{4} \ln x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*log(x),x)

[Out]

1/4*x^4*log(x) - 1/16*x^4

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Mupad [B]
time = 0.19, size = 9, normalized size = 0.53 \begin {gather*} \frac {x^4\,\left (\ln \left (x\right )-\frac {1}{4}\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^4*(log(x) - 1/4))/4

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