Optimal. Leaf size=42 \[ \frac{2 x^{m+1}}{(m+1)^3}+\frac{x^{m+1} \log ^2(x)}{m+1}-\frac{2 x^{m+1} \log (x)}{(m+1)^2} \]
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Rubi [A] time = 0.0238764, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2305, 2304} \[ \frac{2 x^{m+1}}{(m+1)^3}+\frac{x^{m+1} \log ^2(x)}{m+1}-\frac{2 x^{m+1} \log (x)}{(m+1)^2} \]
Antiderivative was successfully verified.
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Rule 2305
Rule 2304
Rubi steps
\begin{align*} \int x^m \log ^2(x) \, dx &=\frac{x^{1+m} \log ^2(x)}{1+m}-\frac{2 \int x^m \log (x) \, dx}{1+m}\\ &=\frac{2 x^{1+m}}{(1+m)^3}-\frac{2 x^{1+m} \log (x)}{(1+m)^2}+\frac{x^{1+m} \log ^2(x)}{1+m}\\ \end{align*}
Mathematica [A] time = 0.010345, size = 30, normalized size = 0.71 \[ \frac{x^{m+1} \left ((m+1)^2 \log ^2(x)-2 (m+1) \log (x)+2\right )}{(m+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 61, normalized size = 1.5 \begin{align*}{\frac{x \left ( \ln \left ( x \right ) \right ) ^{2}{{\rm e}^{m\ln \left ( x \right ) }}}{1+m}}+2\,{\frac{x{{\rm e}^{m\ln \left ( x \right ) }}}{{m}^{3}+3\,{m}^{2}+3\,m+1}}-2\,{\frac{x\ln \left ( x \right ){{\rm e}^{m\ln \left ( x \right ) }}}{{m}^{2}+2\,m+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 2.13438, size = 115, normalized size = 2.74 \begin{align*} \frac{{\left ({\left (m^{2} + 2 \, m + 1\right )} x \log \left (x\right )^{2} - 2 \,{\left (m + 1\right )} x \log \left (x\right ) + 2 \, x\right )} x^{m}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.09837, size = 113, normalized size = 2.69 \begin{align*} -\frac{2 \, m x x^{m} \log \left (x\right )}{{\left (m^{2} + 2 \, m + 1\right )}{\left (m + 1\right )}} + \frac{x^{m + 1} \log \left (x\right )^{2}}{m + 1} - \frac{2 \, x x^{m} \log \left (x\right )}{{\left (m^{2} + 2 \, m + 1\right )}{\left (m + 1\right )}} + \frac{2 \, x x^{m}}{{\left (m^{2} + 2 \, m + 1\right )}{\left (m + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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