3.285 \(\int x^2 \log (1+x) \, dx\)

Optimal. Leaf size=39 \[ -\frac{x^3}{9}+\frac{1}{3} x^3 \log (x+1)+\frac{x^2}{6}-\frac{x}{3}+\frac{1}{3} \log (x+1) \]

[Out]

-x/3 + x^2/6 - x^3/9 + Log[1 + x]/3 + (x^3*Log[1 + x])/3

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Rubi [A]  time = 0.0310959, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{x^3}{9}+\frac{1}{3} x^3 \log (x+1)+\frac{x^2}{6}-\frac{x}{3}+\frac{1}{3} \log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[x^2*Log[1 + x],x]

[Out]

-x/3 + x^2/6 - x^3/9 + Log[1 + x]/3 + (x^3*Log[1 + x])/3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3} \log{\left (x + 1 \right )}}{3} - \frac{x^{3}}{9} - \frac{x}{3} + \frac{\log{\left (x + 1 \right )}}{3} + \frac{\int x\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*ln(1+x),x)

[Out]

x**3*log(x + 1)/3 - x**3/9 - x/3 + log(x + 1)/3 + Integral(x, x)/3

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Mathematica [A]  time = 0.00314863, size = 39, normalized size = 1. \[ -\frac{x^3}{9}+\frac{1}{3} x^3 \log (x+1)+\frac{x^2}{6}-\frac{x}{3}+\frac{1}{3} \log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*Log[1 + x],x]

[Out]

-x/3 + x^2/6 - x^3/9 + Log[1 + x]/3 + (x^3*Log[1 + x])/3

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Maple [A]  time = 0.004, size = 46, normalized size = 1.2 \[{\frac{ \left ( 1+x \right ) ^{3}\ln \left ( 1+x \right ) }{3}}-{\frac{{x}^{3}}{9}}+{\frac{{x}^{2}}{6}}-{\frac{x}{3}}-{\frac{11}{18}}- \left ( 1+x \right ) ^{2}\ln \left ( 1+x \right ) +\ln \left ( 1+x \right ) \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*ln(1+x),x)

[Out]

1/3*(1+x)^3*ln(1+x)-1/9*x^3+1/6*x^2-1/3*x-11/18-(1+x)^2*ln(1+x)+ln(1+x)*(1+x)

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Maxima [A]  time = 1.34187, size = 39, normalized size = 1. \[ \frac{1}{3} \, x^{3} \log \left (x + 1\right ) - \frac{1}{9} \, x^{3} + \frac{1}{6} \, x^{2} - \frac{1}{3} \, x + \frac{1}{3} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*log(x + 1),x, algorithm="maxima")

[Out]

1/3*x^3*log(x + 1) - 1/9*x^3 + 1/6*x^2 - 1/3*x + 1/3*log(x + 1)

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Fricas [A]  time = 0.205935, size = 34, normalized size = 0.87 \[ -\frac{1}{9} \, x^{3} + \frac{1}{6} \, x^{2} + \frac{1}{3} \,{\left (x^{3} + 1\right )} \log \left (x + 1\right ) - \frac{1}{3} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*log(x + 1),x, algorithm="fricas")

[Out]

-1/9*x^3 + 1/6*x^2 + 1/3*(x^3 + 1)*log(x + 1) - 1/3*x

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Sympy [A]  time = 0.109592, size = 29, normalized size = 0.74 \[ \frac{x^{3} \log{\left (x + 1 \right )}}{3} - \frac{x^{3}}{9} + \frac{x^{2}}{6} - \frac{x}{3} + \frac{\log{\left (x + 1 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*ln(1+x),x)

[Out]

x**3*log(x + 1)/3 - x**3/9 + x**2/6 - x/3 + log(x + 1)/3

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GIAC/XCAS [A]  time = 0.22657, size = 66, normalized size = 1.69 \[ \frac{1}{3} \,{\left (x + 1\right )}^{3}{\rm ln}\left (x + 1\right ) - \frac{1}{9} \,{\left (x + 1\right )}^{3} -{\left (x + 1\right )}^{2}{\rm ln}\left (x + 1\right ) + \frac{1}{2} \,{\left (x + 1\right )}^{2} +{\left (x + 1\right )}{\rm ln}\left (x + 1\right ) - x - 1 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*log(x + 1),x, algorithm="giac")

[Out]

1/3*(x + 1)^3*ln(x + 1) - 1/9*(x + 1)^3 - (x + 1)^2*ln(x + 1) + 1/2*(x + 1)^2 +
(x + 1)*ln(x + 1) - x - 1