3.230 \(\int \frac{1}{\sqrt{9+3 x-5 x^2+x^3}} \, dx\)

Optimal. Leaf size=42 \[ \frac{(3-x) \sqrt{x+1} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{2}\right )}{\sqrt{x^3-5 x^2+3 x+9}} \]

[Out]

((3 - x)*Sqrt[1 + x]*ArcTanh[Sqrt[1 + x]/2])/Sqrt[9 + 3*x - 5*x^2 + x^3]

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Rubi [A]  time = 0.0797846, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{(3-x) \sqrt{x+1} \tanh ^{-1}\left (\frac{\sqrt{x+1}}{2}\right )}{\sqrt{x^3-5 x^2+3 x+9}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[9 + 3*x - 5*x^2 + x^3],x]

[Out]

((3 - x)*Sqrt[1 + x]*ArcTanh[Sqrt[1 + x]/2])/Sqrt[9 + 3*x - 5*x^2 + x^3]

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Rubi in Sympy [A]  time = 2.02922, size = 36, normalized size = 0.86 \[ \frac{\left (- x + 3\right ) \sqrt{x + 1} \operatorname{atanh}{\left (\frac{\sqrt{x + 1}}{2} \right )}}{\sqrt{x^{3} - 5 x^{2} + 3 x + 9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**3-5*x**2+3*x+9)**(1/2),x)

[Out]

(-x + 3)*sqrt(x + 1)*atanh(sqrt(x + 1)/2)/sqrt(x**3 - 5*x**2 + 3*x + 9)

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Mathematica [A]  time = 0.021747, size = 52, normalized size = 1.24 \[ \frac{(x-3) \sqrt{x+1} \left (\log \left (2-\sqrt{x+1}\right )-\log \left (\sqrt{x+1}+2\right )\right )}{2 \sqrt{(x-3)^2 (x+1)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[9 + 3*x - 5*x^2 + x^3],x]

[Out]

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

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Maple [A]  time = 0.013, size = 45, normalized size = 1.1 \[ -{\frac{-3+x}{2}\sqrt{1+x} \left ( \ln \left ( \sqrt{1+x}+2 \right ) -\ln \left ( \sqrt{1+x}-2 \right ) \right ){\frac{1}{\sqrt{{x}^{3}-5\,{x}^{2}+3\,x+9}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^3-5*x^2+3*x+9)^(1/2),x)

[Out]

-1/2*(-3+x)*(1+x)^(1/2)*(ln((1+x)^(1/2)+2)-ln((1+x)^(1/2)-2))/(x^3-5*x^2+3*x+9)^
(1/2)

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Maxima [A]  time = 1.54953, size = 28, normalized size = 0.67 \[ -\frac{1}{2} \, \log \left (\sqrt{x + 1} + 2\right ) + \frac{1}{2} \, \log \left (\sqrt{x + 1} - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^3 - 5*x^2 + 3*x + 9),x, algorithm="maxima")

[Out]

-1/2*log(sqrt(x + 1) + 2) + 1/2*log(sqrt(x + 1) - 2)

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Fricas [A]  time = 0.214519, size = 84, normalized size = 2. \[ -\frac{1}{2} \, \log \left (\frac{2 \, x + \sqrt{x^{3} - 5 \, x^{2} + 3 \, x + 9} - 6}{x - 3}\right ) + \frac{1}{2} \, \log \left (-\frac{2 \, x - \sqrt{x^{3} - 5 \, x^{2} + 3 \, x + 9} - 6}{x - 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^3 - 5*x^2 + 3*x + 9),x, algorithm="fricas")

[Out]

-1/2*log((2*x + sqrt(x^3 - 5*x^2 + 3*x + 9) - 6)/(x - 3)) + 1/2*log(-(2*x - sqrt
(x^3 - 5*x^2 + 3*x + 9) - 6)/(x - 3))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{3} - 5 x^{2} + 3 x + 9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**3-5*x**2+3*x+9)**(1/2),x)

[Out]

Integral(1/sqrt(x**3 - 5*x**2 + 3*x + 9), x)

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GIAC/XCAS [A]  time = 0.206842, size = 46, normalized size = 1.1 \[ -\frac{{\rm ln}\left (\sqrt{x + 1} + 2\right )}{2 \,{\rm sign}\left (x - 3\right )} + \frac{{\rm ln}\left ({\left | \sqrt{x + 1} - 2 \right |}\right )}{2 \,{\rm sign}\left (x - 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^3 - 5*x^2 + 3*x + 9),x, algorithm="giac")

[Out]

-1/2*ln(sqrt(x + 1) + 2)/sign(x - 3) + 1/2*ln(abs(sqrt(x + 1) - 2))/sign(x - 3)