3.56 \(\int x^p \log (x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0098902, antiderivative size = 26, 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} \[ \frac{x^{p+1} \log (x)}{p+1}-\frac{x^{p+1}}{(p+1)^2} \]

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

Mathematica [A]  time = 0.0074999, size = 19, normalized size = 0.73 \[ \frac{x^{p+1} ((p+1) \log (x)-1)}{(p+1)^2} \]

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 = 0.007, size = 34, normalized size = 1.3 \begin{align*}{\frac{x\ln \left ( x \right ){{\rm e}^{\ln \left ( x \right ) p}}}{1+p}}-{\frac{x{{\rm e}^{\ln \left ( x \right ) p}}}{{p}^{2}+2\,p+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.91599, size = 59, normalized size = 2.27 \begin{align*} \frac{{\left ({\left (p + 1\right )} x \log \left (x\right ) - x\right )} x^{p}}{p^{2} + 2 \, p + 1} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{p} \log \left (x\right )\,{d x} \end{align*}

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)