3.2222 \(\int \frac{7-3 x}{-5+2 x+x^2} \, dx\)

Optimal. Leaf size=47 \[ -\frac{1}{6} \left (9-5 \sqrt{6}\right ) \log \left (x-\sqrt{6}+1\right )-\frac{1}{6} \left (9+5 \sqrt{6}\right ) \log \left (x+\sqrt{6}+1\right ) \]

[Out]

-((9 - 5*Sqrt[6])*Log[1 - Sqrt[6] + x])/6 - ((9 + 5*Sqrt[6])*Log[1 + Sqrt[6] + x
])/6

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Rubi [A]  time = 0.0700014, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{1}{6} \left (9-5 \sqrt{6}\right ) \log \left (x-\sqrt{6}+1\right )-\frac{1}{6} \left (9+5 \sqrt{6}\right ) \log \left (x+\sqrt{6}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(7 - 3*x)/(-5 + 2*x + x^2),x]

[Out]

-((9 - 5*Sqrt[6])*Log[1 - Sqrt[6] + x])/6 - ((9 + 5*Sqrt[6])*Log[1 + Sqrt[6] + x
])/6

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Rubi in Sympy [A]  time = 4.69202, size = 49, normalized size = 1.04 \[ - \frac{\sqrt{6} \left (3 \sqrt{6} + 10\right ) \log{\left (x + 1 + \sqrt{6} \right )}}{12} + \frac{\sqrt{6} \left (- 3 \sqrt{6} + 10\right ) \log{\left (x - \sqrt{6} + 1 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((7-3*x)/(x**2+2*x-5),x)

[Out]

-sqrt(6)*(3*sqrt(6) + 10)*log(x + 1 + sqrt(6))/12 + sqrt(6)*(-3*sqrt(6) + 10)*lo
g(x - sqrt(6) + 1)/12

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Mathematica [A]  time = 0.0646039, size = 47, normalized size = 1. \[ \frac{1}{6} \left (5 \sqrt{6}-9\right ) \log \left (-x+\sqrt{6}-1\right )+\frac{1}{6} \left (-9-5 \sqrt{6}\right ) \log \left (x+\sqrt{6}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(7 - 3*x)/(-5 + 2*x + x^2),x]

[Out]

((-9 + 5*Sqrt[6])*Log[-1 + Sqrt[6] - x])/6 + ((-9 - 5*Sqrt[6])*Log[1 + Sqrt[6] +
x])/6

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Maple [A]  time = 0.005, size = 29, normalized size = 0.6 \[ -{\frac{3\,\ln \left ({x}^{2}+2\,x-5 \right ) }{2}}-{\frac{5\,\sqrt{6}}{3}{\it Artanh} \left ({\frac{ \left ( 2+2\,x \right ) \sqrt{6}}{12}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((7-3*x)/(x^2+2*x-5),x)

[Out]

-3/2*ln(x^2+2*x-5)-5/3*6^(1/2)*arctanh(1/12*(2+2*x)*6^(1/2))

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Maxima [A]  time = 0.912049, size = 47, normalized size = 1. \[ \frac{5}{6} \, \sqrt{6} \log \left (\frac{x - \sqrt{6} + 1}{x + \sqrt{6} + 1}\right ) - \frac{3}{2} \, \log \left (x^{2} + 2 \, x - 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x - 7)/(x^2 + 2*x - 5),x, algorithm="maxima")

[Out]

5/6*sqrt(6)*log((x - sqrt(6) + 1)/(x + sqrt(6) + 1)) - 3/2*log(x^2 + 2*x - 5)

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Fricas [A]  time = 0.220895, size = 78, normalized size = 1.66 \[ -\frac{1}{6} \, \sqrt{3}{\left (3 \, \sqrt{3} \log \left (x^{2} + 2 \, x - 5\right ) - 5 \, \sqrt{2} \log \left (\frac{\sqrt{3}{\left (x^{2} + 2 \, x + 7\right )} - 6 \, \sqrt{2}{\left (x + 1\right )}}{x^{2} + 2 \, x - 5}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x - 7)/(x^2 + 2*x - 5),x, algorithm="fricas")

[Out]

-1/6*sqrt(3)*(3*sqrt(3)*log(x^2 + 2*x - 5) - 5*sqrt(2)*log((sqrt(3)*(x^2 + 2*x +
7) - 6*sqrt(2)*(x + 1))/(x^2 + 2*x - 5)))

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Sympy [A]  time = 0.242424, size = 44, normalized size = 0.94 \[ - \left (\frac{3}{2} + \frac{5 \sqrt{6}}{6}\right ) \log{\left (x + 1 + \sqrt{6} \right )} - \left (- \frac{5 \sqrt{6}}{6} + \frac{3}{2}\right ) \log{\left (x - \sqrt{6} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((7-3*x)/(x**2+2*x-5),x)

[Out]

-(3/2 + 5*sqrt(6)/6)*log(x + 1 + sqrt(6)) - (-5*sqrt(6)/6 + 3/2)*log(x - sqrt(6)
+ 1)

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GIAC/XCAS [A]  time = 0.203772, size = 59, normalized size = 1.26 \[ \frac{5}{6} \, \sqrt{6}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{6} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt{6} + 2 \right |}}\right ) - \frac{3}{2} \,{\rm ln}\left ({\left | x^{2} + 2 \, x - 5 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x - 7)/(x^2 + 2*x - 5),x, algorithm="giac")

[Out]

5/6*sqrt(6)*ln(abs(2*x - 2*sqrt(6) + 2)/abs(2*x + 2*sqrt(6) + 2)) - 3/2*ln(abs(x
^2 + 2*x - 5))