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 = 0.0406657, 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 \[ \frac{2 x^{m+1}}{(m+1)^3}+\frac{x^{m+1} \log ^2(x)}{m+1}-\frac{2 x^{m+1} \log (x)}{(m+1)^2} \]

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)

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Rubi in Sympy [A]  time = 3.2409, size = 39, normalized size = 0.93 \[ \frac{x^{m + 1} \log{\left (x \right )}^{2}}{m + 1} - \frac{2 x^{m + 1} \log{\left (x \right )}}{\left (m + 1\right )^{2}} + \frac{2 x^{m + 1}}{\left (m + 1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*ln(x)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0188515, 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 = 0.014, size = 61, normalized size = 1.5 \[{\frac{x \left ( \ln \left ( x \right ) \right ) ^{2}{{\rm e}^{m\ln \left ( x \right ) }}}{1+m}}+2\,{\frac{x{{\rm e}^{m\ln \left ( x \right ) }}}{{m}^{3}+3\,{m}^{2}+3\,m+1}}-2\,{\frac{x\ln \left ( x \right ){{\rm e}^{m\ln \left ( x \right ) }}}{{m}^{2}+2\,m+1}} \]

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]  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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22421, size = 61, normalized size = 1.45 \[ \frac{{\left ({\left (m^{2} + 2 \, m + 1\right )} x \log \left (x\right )^{2} - 2 \,{\left (m + 1\right )} x \log \left (x\right ) + 2 \, x\right )} x^{m}}{m^{3} + 3 \, m^{2} + 3 \, m + 1} \]

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. \[ \text{Exception raised: TypeError} \]

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/XCAS [A]  time = 0.207145, size = 122, normalized size = 2.9 \[ -\frac{2 \, m x e^{\left (m{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right )}{{\left (m^{2} + 2 \, m + 1\right )}{\left (m + 1\right )}} + \frac{x^{m + 1}{\rm ln}\left (x\right )^{2}}{m + 1} - \frac{2 \, x e^{\left (m{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right )}{{\left (m^{2} + 2 \, m + 1\right )}{\left (m + 1\right )}} + \frac{2 \, x e^{\left (m{\rm ln}\left (x\right )\right )}}{{\left (m^{2} + 2 \, m + 1\right )}{\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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