3.608 \(\int x^m \log (x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0210334, 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 \[ \frac{x^{m+1} \log (x)}{m+1}-\frac{x^{m+1}}{(m+1)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^m*Log[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.60283, size = 20, normalized size = 0.77 \[ \frac{x^{m + 1} \log{\left (x \right )}}{m + 1} - \frac{x^{m + 1}}{\left (m + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*ln(x),x)

[Out]

x**(m + 1)*log(x)/(m + 1) - x**(m + 1)/(m + 1)**2

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Mathematica [A]  time = 0.0138108, size = 19, normalized size = 0.73 \[ \frac{x^{m+1} ((m+1) \log (x)-1)}{(m+1)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*Log[x],x]

[Out]

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

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Maple [A]  time = 0.013, size = 34, normalized size = 1.3 \[{\frac{x\ln \left ( x \right ){{\rm e}^{m\ln \left ( x \right ) }}}{1+m}}-{\frac{x{{\rm e}^{m\ln \left ( x \right ) }}}{{m}^{2}+2\,m+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*ln(x),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m*log(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.214468, size = 34, normalized size = 1.31 \[ \frac{{\left ({\left (m + 1\right )} x \log \left (x\right ) - x\right )} x^{m}}{m^{2} + 2 \, m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m*log(x),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*ln(x),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{m} \log \left (x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m*log(x),x, algorithm="giac")

[Out]

integrate(x^m*log(x), x)