3.364 \(\int \frac{-3+x^3}{-7-6 x+x^2} \, dx\)

Optimal. Leaf size=29 \[ \frac{x^2}{2}+6 x+\frac{85}{2} \log (7-x)+\frac{1}{2} \log (x+1) \]

[Out]

6*x + x^2/2 + (85*Log[7 - x])/2 + Log[1 + x]/2

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Rubi [A]  time = 0.0359319, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{x^2}{2}+6 x+\frac{85}{2} \log (7-x)+\frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

[In]  Int[(-3 + x^3)/(-7 - 6*x + x^2),x]

[Out]

6*x + x^2/2 + (85*Log[7 - x])/2 + Log[1 + x]/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 6 x + \frac{85 \log{\left (- x + 7 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{2} + \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3-3)/(x**2-6*x-7),x)

[Out]

6*x + 85*log(-x + 7)/2 + log(x + 1)/2 + Integral(x, x)

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Mathematica [A]  time = 0.00906544, size = 29, normalized size = 1. \[ \frac{x^2}{2}+6 x+\frac{85}{2} \log (7-x)+\frac{1}{2} \log (x+1) \]

Antiderivative was successfully verified.

[In]  Integrate[(-3 + x^3)/(-7 - 6*x + x^2),x]

[Out]

6*x + x^2/2 + (85*Log[7 - x])/2 + Log[1 + x]/2

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Maple [A]  time = 0.008, size = 22, normalized size = 0.8 \[{\frac{{x}^{2}}{2}}+6\,x+{\frac{\ln \left ( 1+x \right ) }{2}}+{\frac{85\,\ln \left ( x-7 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3-3)/(x^2-6*x-7),x)

[Out]

1/2*x^2+6*x+1/2*ln(1+x)+85/2*ln(x-7)

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Maxima [A]  time = 0.825232, size = 28, normalized size = 0.97 \[ \frac{1}{2} \, x^{2} + 6 \, x + \frac{1}{2} \, \log \left (x + 1\right ) + \frac{85}{2} \, \log \left (x - 7\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 3)/(x^2 - 6*x - 7),x, algorithm="maxima")

[Out]

1/2*x^2 + 6*x + 1/2*log(x + 1) + 85/2*log(x - 7)

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Fricas [A]  time = 0.253094, size = 28, normalized size = 0.97 \[ \frac{1}{2} \, x^{2} + 6 \, x + \frac{1}{2} \, \log \left (x + 1\right ) + \frac{85}{2} \, \log \left (x - 7\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 3)/(x^2 - 6*x - 7),x, algorithm="fricas")

[Out]

1/2*x^2 + 6*x + 1/2*log(x + 1) + 85/2*log(x - 7)

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Sympy [A]  time = 0.212784, size = 22, normalized size = 0.76 \[ \frac{x^{2}}{2} + 6 x + \frac{85 \log{\left (x - 7 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3-3)/(x**2-6*x-7),x)

[Out]

x**2/2 + 6*x + 85*log(x - 7)/2 + log(x + 1)/2

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GIAC/XCAS [A]  time = 0.261119, size = 31, normalized size = 1.07 \[ \frac{1}{2} \, x^{2} + 6 \, x + \frac{1}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{85}{2} \,{\rm ln}\left ({\left | x - 7 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 - 3)/(x^2 - 6*x - 7),x, algorithm="giac")

[Out]

1/2*x^2 + 6*x + 1/2*ln(abs(x + 1)) + 85/2*ln(abs(x - 7))