3.103 \(\int \sqrt{5-6 x+9 x^2} \, dx\)

Optimal. Leaf size=38 \[ \frac{2}{3} \sinh ^{-1}\left (\frac{1}{2} (3 x-1)\right )-\frac{1}{6} (1-3 x) \sqrt{9 x^2-6 x+5} \]

[Out]

-((1 - 3*x)*Sqrt[5 - 6*x + 9*x^2])/6 + (2*ArcSinh[(-1 + 3*x)/2])/3

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Rubi [A]  time = 0.0273342, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2}{3} \sinh ^{-1}\left (\frac{1}{2} (3 x-1)\right )-\frac{1}{6} (1-3 x) \sqrt{9 x^2-6 x+5} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[5 - 6*x + 9*x^2],x]

[Out]

-((1 - 3*x)*Sqrt[5 - 6*x + 9*x^2])/6 + (2*ArcSinh[(-1 + 3*x)/2])/3

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Rubi in Sympy [A]  time = 2.02363, size = 44, normalized size = 1.16 \[ - \frac{\left (- 18 x + 6\right ) \sqrt{9 x^{2} - 6 x + 5}}{36} + \frac{2 \operatorname{atanh}{\left (\frac{18 x - 6}{6 \sqrt{9 x^{2} - 6 x + 5}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((9*x**2-6*x+5)**(1/2),x)

[Out]

-(-18*x + 6)*sqrt(9*x**2 - 6*x + 5)/36 + 2*atanh((18*x - 6)/(6*sqrt(9*x**2 - 6*x
+ 5)))/3

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Mathematica [A]  time = 0.0283147, size = 39, normalized size = 1.03 \[ \sqrt{9 x^2-6 x+5} \left (\frac{x}{2}-\frac{1}{6}\right )+\frac{2}{3} \sinh ^{-1}\left (\frac{1}{2} (3 x-1)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[5 - 6*x + 9*x^2],x]

[Out]

(-1/6 + x/2)*Sqrt[5 - 6*x + 9*x^2] + (2*ArcSinh[(-1 + 3*x)/2])/3

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Maple [A]  time = 0.005, size = 29, normalized size = 0.8 \[{\frac{18\,x-6}{36}\sqrt{9\,{x}^{2}-6\,x+5}}+{\frac{2}{3}{\it Arcsinh} \left ( -{\frac{1}{2}}+{\frac{3\,x}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((9*x^2-6*x+5)^(1/2),x)

[Out]

1/36*(18*x-6)*(9*x^2-6*x+5)^(1/2)+2/3*arcsinh(-1/2+3/2*x)

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Maxima [A]  time = 0.790748, size = 51, normalized size = 1.34 \[ \frac{1}{2} \, \sqrt{9 \, x^{2} - 6 \, x + 5} x - \frac{1}{6} \, \sqrt{9 \, x^{2} - 6 \, x + 5} + \frac{2}{3} \, \operatorname{arsinh}\left (\frac{3}{2} \, x - \frac{1}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(9*x^2 - 6*x + 5),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.215401, size = 177, normalized size = 4.66 \[ -\frac{324 \, x^{4} - 432 \, x^{3} + 351 \, x^{2} + 16 \,{\left (9 \, x^{2} - \sqrt{9 \, x^{2} - 6 \, x + 5}{\left (3 \, x - 1\right )} - 6 \, x + 3\right )} \log \left (-3 \, x + \sqrt{9 \, x^{2} - 6 \, x + 5} + 1\right ) -{\left (108 \, x^{3} - 108 \, x^{2} + 57 \, x - 11\right )} \sqrt{9 \, x^{2} - 6 \, x + 5} - 138 \, x + 17}{24 \,{\left (9 \, x^{2} - \sqrt{9 \, x^{2} - 6 \, x + 5}{\left (3 \, x - 1\right )} - 6 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(9*x^2 - 6*x + 5),x, algorithm="fricas")

[Out]

-1/24*(324*x^4 - 432*x^3 + 351*x^2 + 16*(9*x^2 - sqrt(9*x^2 - 6*x + 5)*(3*x - 1)
- 6*x + 3)*log(-3*x + sqrt(9*x^2 - 6*x + 5) + 1) - (108*x^3 - 108*x^2 + 57*x -
11)*sqrt(9*x^2 - 6*x + 5) - 138*x + 17)/(9*x^2 - sqrt(9*x^2 - 6*x + 5)*(3*x - 1)
- 6*x + 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{9 x^{2} - 6 x + 5}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((9*x**2-6*x+5)**(1/2),x)

[Out]

Integral(sqrt(9*x**2 - 6*x + 5), x)

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GIAC/XCAS [A]  time = 0.209735, size = 54, normalized size = 1.42 \[ \frac{1}{6} \, \sqrt{9 \, x^{2} - 6 \, x + 5}{\left (3 \, x - 1\right )} - \frac{2}{3} \,{\rm ln}\left (-3 \, x + \sqrt{9 \, x^{2} - 6 \, x + 5} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(9*x^2 - 6*x + 5),x, algorithm="giac")

[Out]

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