3.62 \(\int x^n \log (a x) \, dx\)

Optimal. Leaf size=28 \[ \frac{x^{n+1} \log (a x)}{n+1}-\frac{x^{n+1}}{(n+1)^2} \]

[Out]

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

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Rubi [A]  time = 0.0098877, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2304} \[ \frac{x^{n+1} \log (a x)}{n+1}-\frac{x^{n+1}}{(n+1)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^n*Log[a*x],x]

[Out]

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

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

Mathematica [A]  time = 0.0052956, size = 21, normalized size = 0.75 \[ \frac{x^{n+1} ((n+1) \log (a x)-1)}{(n+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^n*Log[a*x],x]

[Out]

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

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Maple [A]  time = 0.013, size = 36, normalized size = 1.3 \begin{align*}{\frac{x\ln \left ( ax \right ){{\rm e}^{n\ln \left ( x \right ) }}}{1+n}}-{\frac{x{{\rm e}^{n\ln \left ( x \right ) }}}{{n}^{2}+2\,n+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^n*ln(a*x),x)

[Out]

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

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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^n*log(a*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.519222, size = 85, normalized size = 3.04 \begin{align*} \frac{{\left ({\left (n + 1\right )} x \log \left (a\right ) +{\left (n + 1\right )} x \log \left (x\right ) - x\right )} x^{n}}{n^{2} + 2 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((n + 1)*x*log(a) + (n + 1)*x*log(x) - x)*x^n/(n^2 + 2*n + 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**n*ln(a*x),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^n*log(a*x), x)