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

Optimal. Leaf size=23 \[ \frac{x^2}{2}+2 x+9 \log (3-x)+\log (x+1) \]

[Out]

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

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Rubi [A]  time = 0.0402839, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{x^2}{2}+2 x+9 \log (3-x)+\log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 x + 9 \log{\left (- x + 3 \right )} + \log{\left (x + 1 \right )} + \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0075788, size = 23, normalized size = 1. \[ \frac{x^2}{2}+2 x+9 \log (3-x)+\log (x+1) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 20, normalized size = 0.9 \[{\frac{{x}^{2}}{2}}+2\,x+9\,\ln \left ( -3+x \right ) +\ln \left ( 1+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+3*x)/(x^2-2*x-3),x)

[Out]

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

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Maxima [A]  time = 1.42222, size = 26, normalized size = 1.13 \[ \frac{1}{2} \, x^{2} + 2 \, x + \log \left (x + 1\right ) + 9 \, \log \left (x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.195934, size = 26, normalized size = 1.13 \[ \frac{1}{2} \, x^{2} + 2 \, x + \log \left (x + 1\right ) + 9 \, \log \left (x - 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.096035, size = 19, normalized size = 0.83 \[ \frac{x^{2}}{2} + 2 x + 9 \log{\left (x - 3 \right )} + \log{\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220625, size = 28, normalized size = 1.22 \[ \frac{1}{2} \, x^{2} + 2 \, x +{\rm ln}\left ({\left | x + 1 \right |}\right ) + 9 \,{\rm ln}\left ({\left | x - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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