3.609 \(\int x^m \log ^2(x) \, dx\)

Optimal. Leaf size=42 \[ \frac{2 x^{m+1}}{(m+1)^3}+\frac{x^{m+1} \log ^2(x)}{m+1}-\frac{2 x^{m+1} \log (x)}{(m+1)^2} \]

[Out]

(2*x^(1 + m))/(1 + m)^3 - (2*x^(1 + m)*Log[x])/(1 + m)^2 + (x^(1 + m)*Log[x]^2)/(1 + m)

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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.

[In]

Int[x^m*Log[x]^2,x]

[Out]

(2*x^(1 + m))/(1 + m)^3 - (2*x^(1 + m)*Log[x])/(1 + m)^2 + (x^(1 + m)*Log[x]^2)/(1 + m)

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

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^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.

[In]

Integrate[x^m*Log[x]^2,x]

[Out]

(x^(1 + m)*(2 - 2*(1 + m)*Log[x] + (1 + m)^2*Log[x]^2))/(1 + m)^3

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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.

[In]

int(x^m*ln(x)^2,x)

[Out]

1/(1+m)*x*ln(x)^2*exp(m*ln(x))+2/(m^3+3*m^2+3*m+1)*x*exp(m*ln(x))-2/(m^2+2*m+1)*x*ln(x)*exp(m*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^m*log(x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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.

[In]

integrate(x^m*log(x)^2,x, algorithm="fricas")

[Out]

((m^2 + 2*m + 1)*x*log(x)^2 - 2*(m + 1)*x*log(x) + 2*x)*x^m/(m^3 + 3*m^2 + 3*m + 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**m*ln(x)**2,x)

[Out]

Exception raised: TypeError

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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.

[In]

integrate(x^m*log(x)^2,x, algorithm="giac")

[Out]

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