3.1.56 \(\int x^p \log (x) \, dx\) [56]

Optimal. Leaf size=26 \[ -\frac {x^{1+p}}{(1+p)^2}+\frac {x^{1+p} \log (x)}{1+p} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 26, 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 {x^{p+1} \log (x)}{p+1}-\frac {x^{p+1}}{(p+1)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(x^(1 + p)/(1 + p)^2) + (x^(1 + p)*Log[x])/(1 + p)

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^p \log (x) \, dx &=-\frac {x^{1+p}}{(1+p)^2}+\frac {x^{1+p} \log (x)}{1+p}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 19, normalized size = 0.73 \begin {gather*} \frac {x^{1+p} (-1+(1+p) \log (x))}{(1+p)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + p)*(-1 + (1 + p)*Log[x]))/(1 + p)^2

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Maple [A]
time = 0.01, size = 19, normalized size = 0.73

method	result	size
risch	\(\frac {x \left (\ln \left (x \right ) p +\ln \left (x \right )-1\right ) x^{p}}{\left (1+p \right )^{2}}\)	\(19\)
norman	\(\frac {x \ln \left (x \right ) {\mathrm e}^{\ln \left (x \right ) p}}{1+p}-\frac {x \,{\mathrm e}^{\ln \left (x \right ) p}}{p^{2}+2 p +1}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 2.52, size = 26, normalized size = 1.00 \begin {gather*} \frac {x^{p + 1} \log \left (x\right )}{p + 1} - \frac {x^{p + 1}}{{\left (p + 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.78, size = 25, normalized size = 0.96 \begin {gather*} \frac {{\left ({\left (p + 1\right )} x \log \left (x\right ) - x\right )} x^{p}}{p^{2} + 2 \, p + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (20) = 40\).
time = 0.20, size = 56, normalized size = 2.15 \begin {gather*} \begin {cases} \frac {p x x^{p} \log {\left (x \right )}}{p^{2} + 2 p + 1} + \frac {x x^{p} \log {\left (x \right )}}{p^{2} + 2 p + 1} - \frac {x x^{p}}{p^{2} + 2 p + 1} & \text {for}\: p \neq -1 \\\frac {\log {\left (x \right )}^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**p*ln(x),x)

[Out]

Piecewise((p*x*x**p*log(x)/(p**2 + 2*p + 1) + x*x**p*log(x)/(p**2 + 2*p + 1) - x*x**p/(p**2 + 2*p + 1), Ne(p,
-1)), (log(x)**2/2, True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^p*log(x), x)

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Mupad [B]
time = 0.24, size = 32, normalized size = 1.23 \begin {gather*} \left \{\begin {array}{cl} \frac {{\ln \left (x\right )}^2}{2} & \text {\ if\ \ }p=-1\\ \frac {x^{p+1}\,\left (\ln \left (x\right )\,\left (p+1\right )-1\right )}{{\left (p+1\right )}^2} & \text {\ if\ \ }p\neq -1 \end {array}\right . \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

piecewise(p == -1, log(x)^2/2, p ~= -1, (x^(p + 1)*(log(x)*(p + 1) - 1))/(p + 1)^2)

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