3.105 \(\int \sqrt{-8+6 x+9 x^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{1}{6} (3 x+1) \sqrt{9 x^2+6 x-8}-\frac{3}{2} \tanh ^{-1}\left (\frac{3 x+1}{\sqrt{9 x^2+6 x-8}}\right ) \]

[Out]

((1 + 3*x)*Sqrt[-8 + 6*x + 9*x^2])/6 - (3*ArcTanh[(1 + 3*x)/Sqrt[-8 + 6*x + 9*x^
2]])/2

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Rubi [A]  time = 0.0275822, antiderivative size = 49, 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 \[ \frac{1}{6} (3 x+1) \sqrt{9 x^2+6 x-8}-\frac{3}{2} \tanh ^{-1}\left (\frac{3 x+1}{\sqrt{9 x^2+6 x-8}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-8 + 6*x + 9*x^2],x]

[Out]

((1 + 3*x)*Sqrt[-8 + 6*x + 9*x^2])/6 - (3*ArcTanh[(1 + 3*x)/Sqrt[-8 + 6*x + 9*x^
2]])/2

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Rubi in Sympy [A]  time = 1.98259, size = 44, normalized size = 0.9 \[ \frac{\left (18 x + 6\right ) \sqrt{9 x^{2} + 6 x - 8}}{36} - \frac{3 \operatorname{atanh}{\left (\frac{18 x + 6}{6 \sqrt{9 x^{2} + 6 x - 8}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((9*x**2+6*x-8)**(1/2),x)

[Out]

(18*x + 6)*sqrt(9*x**2 + 6*x - 8)/36 - 3*atanh((18*x + 6)/(6*sqrt(9*x**2 + 6*x -
8)))/2

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Mathematica [A]  time = 0.0270837, size = 49, normalized size = 1. \[ \left (\frac{x}{2}+\frac{1}{6}\right ) \sqrt{9 x^2+6 x-8}-\frac{3}{2} \log \left (\sqrt{9 x^2+6 x-8}+3 x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-8 + 6*x + 9*x^2],x]

[Out]

(1/6 + x/2)*Sqrt[-8 + 6*x + 9*x^2] - (3*Log[1 + 3*x + Sqrt[-8 + 6*x + 9*x^2]])/2

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Maple [A]  time = 0.006, size = 50, normalized size = 1. \[{\frac{18\,x+6}{36}\sqrt{9\,{x}^{2}+6\,x-8}}-{\frac{\sqrt{9}}{2}\ln \left ({\frac{ \left ( 9\,x+3 \right ) \sqrt{9}}{9}}+\sqrt{9\,{x}^{2}+6\,x-8} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((9*x^2+6*x-8)^(1/2),x)

[Out]

1/36*(18*x+6)*(9*x^2+6*x-8)^(1/2)-1/2*ln(1/9*(9*x+3)*9^(1/2)+(9*x^2+6*x-8)^(1/2)
)*9^(1/2)

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Maxima [A]  time = 0.804273, size = 70, normalized size = 1.43 \[ \frac{1}{2} \, \sqrt{9 \, x^{2} + 6 \, x - 8} x + \frac{1}{6} \, \sqrt{9 \, x^{2} + 6 \, x - 8} - \frac{3}{2} \, \log \left (18 \, x + 6 \, \sqrt{9 \, x^{2} + 6 \, x - 8} + 6\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(9*x^2 + 6*x - 8),x, algorithm="maxima")

[Out]

1/2*sqrt(9*x^2 + 6*x - 8)*x + 1/6*sqrt(9*x^2 + 6*x - 8) - 3/2*log(18*x + 6*sqrt(
9*x^2 + 6*x - 8) + 6)

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Fricas [A]  time = 0.216237, size = 177, normalized size = 3.61 \[ -\frac{216 \, x^{4} + 288 \, x^{3} - 78 \, x^{2} - 12 \,{\left (18 \, x^{2} - 2 \, \sqrt{9 \, x^{2} + 6 \, x - 8}{\left (3 \, x + 1\right )} + 12 \, x - 7\right )} \log \left (-3 \, x + \sqrt{9 \, x^{2} + 6 \, x - 8} - 1\right ) - 2 \,{\left (36 \, x^{3} + 36 \, x^{2} - 7 \, x - 5\right )} \sqrt{9 \, x^{2} + 6 \, x - 8} - 116 \, x - 19}{8 \,{\left (18 \, x^{2} - 2 \, \sqrt{9 \, x^{2} + 6 \, x - 8}{\left (3 \, x + 1\right )} + 12 \, x - 7\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(9*x^2 + 6*x - 8),x, algorithm="fricas")

[Out]

-1/8*(216*x^4 + 288*x^3 - 78*x^2 - 12*(18*x^2 - 2*sqrt(9*x^2 + 6*x - 8)*(3*x + 1
) + 12*x - 7)*log(-3*x + sqrt(9*x^2 + 6*x - 8) - 1) - 2*(36*x^3 + 36*x^2 - 7*x -
5)*sqrt(9*x^2 + 6*x - 8) - 116*x - 19)/(18*x^2 - 2*sqrt(9*x^2 + 6*x - 8)*(3*x +
1) + 12*x - 7)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{9 x^{2} + 6 x - 8}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((9*x**2+6*x-8)**(1/2),x)

[Out]

Integral(sqrt(9*x**2 + 6*x - 8), x)

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GIAC/XCAS [A]  time = 0.20901, size = 55, normalized size = 1.12 \[ \frac{1}{6} \, \sqrt{9 \, x^{2} + 6 \, x - 8}{\left (3 \, x + 1\right )} + \frac{3}{2} \,{\rm ln}\left ({\left | -3 \, x + \sqrt{9 \, x^{2} + 6 \, x - 8} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(9*x^2 + 6*x - 8),x, algorithm="giac")

[Out]

1/6*sqrt(9*x^2 + 6*x - 8)*(3*x + 1) + 3/2*ln(abs(-3*x + sqrt(9*x^2 + 6*x - 8) -
1))