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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]