3.427 \(\int \frac{-x+x^3}{6+2 x} \, dx\)

Optimal. Leaf size=24 \[ \frac{x^3}{6}-\frac{3 x^2}{4}+4 x-12 \log (x+3) \]

[Out]

4*x - (3*x^2)/4 + x^3/6 - 12*Log[3 + x]

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Rubi [A]  time = 0.0411261, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^3}{6}-\frac{3 x^2}{4}+4 x-12 \log (x+3) \]

Antiderivative was successfully verified.

[In]  Int[(-x + x^3)/(6 + 2*x),x]

[Out]

4*x - (3*x^2)/4 + x^3/6 - 12*Log[3 + x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x^{3}}{6} + 4 x - 12 \log{\left (x + 3 \right )} - \frac{3 \int x\, dx}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3-x)/(6+2*x),x)

[Out]

x**3/6 + 4*x - 12*log(x + 3) - 3*Integral(x, x)/2

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Mathematica [A]  time = 0.00901904, size = 31, normalized size = 1.29 \[ \frac{1}{2} \left (\frac{x^3}{3}-\frac{3 x^2}{2}+8 x-24 \log (x+3)+\frac{93}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-x + x^3)/(6 + 2*x),x]

[Out]

(93/2 + 8*x - (3*x^2)/2 + x^3/3 - 24*Log[3 + x])/2

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Maple [A]  time = 0.004, size = 21, normalized size = 0.9 \[ 4\,x-{\frac{3\,{x}^{2}}{4}}+{\frac{{x}^{3}}{6}}-12\,\ln \left ( 3+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3-x)/(6+2*x),x)

[Out]

4*x-3/4*x^2+1/6*x^3-12*ln(3+x)

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Maxima [A]  time = 0.79585, size = 27, normalized size = 1.12 \[ \frac{1}{6} \, x^{3} - \frac{3}{4} \, x^{2} + 4 \, x - 12 \, \log \left (x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2*(x^3 - x)/(x + 3),x, algorithm="maxima")

[Out]

1/6*x^3 - 3/4*x^2 + 4*x - 12*log(x + 3)

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Fricas [A]  time = 0.272784, size = 27, normalized size = 1.12 \[ \frac{1}{6} \, x^{3} - \frac{3}{4} \, x^{2} + 4 \, x - 12 \, \log \left (x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2*(x^3 - x)/(x + 3),x, algorithm="fricas")

[Out]

1/6*x^3 - 3/4*x^2 + 4*x - 12*log(x + 3)

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Sympy [A]  time = 0.12358, size = 20, normalized size = 0.83 \[ \frac{x^{3}}{6} - \frac{3 x^{2}}{4} + 4 x - 12 \log{\left (x + 3 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3-x)/(6+2*x),x)

[Out]

x**3/6 - 3*x**2/4 + 4*x - 12*log(x + 3)

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GIAC/XCAS [A]  time = 0.260812, size = 28, normalized size = 1.17 \[ \frac{1}{6} \, x^{3} - \frac{3}{4} \, x^{2} + 4 \, x - 12 \,{\rm ln}\left ({\left | x + 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/2*(x^3 - x)/(x + 3),x, algorithm="giac")

[Out]

1/6*x^3 - 3/4*x^2 + 4*x - 12*ln(abs(x + 3))