Optimal. Leaf size=49 \[ \frac {1}{6} (3 x+1) \sqrt {9 x^2+6 x-8}-\frac {3}{2} \tanh ^{-1}\left (\frac {3 x+1}{\sqrt {9 x^2+6 x-8}}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {612, 621, 206} \[ \frac {1}{6} (3 x+1) \sqrt {9 x^2+6 x-8}-\frac {3}{2} \tanh ^{-1}\left (\frac {3 x+1}{\sqrt {9 x^2+6 x-8}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rubi steps
\begin {align*} \int \sqrt {-8+6 x+9 x^2} \, dx &=\frac {1}{6} (1+3 x) \sqrt {-8+6 x+9 x^2}-\frac {9}{2} \int \frac {1}{\sqrt {-8+6 x+9 x^2}} \, dx\\ &=\frac {1}{6} (1+3 x) \sqrt {-8+6 x+9 x^2}-9 \operatorname {Subst}\left (\int \frac {1}{36-x^2} \, dx,x,\frac {6+18 x}{\sqrt {-8+6 x+9 x^2}}\right )\\ &=\frac {1}{6} (1+3 x) \sqrt {-8+6 x+9 x^2}-\frac {3}{2} \tanh ^{-1}\left (\frac {1+3 x}{\sqrt {-8+6 x+9 x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 1.00 \[ \left (\frac {x}{2}+\frac {1}{6}\right ) \sqrt {9 x^2+6 x-8}-\frac {3}{2} \log \left (\sqrt {9 x^2+6 x-8}+3 x+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 40, normalized size = 0.82 \[ \frac {1}{6} \, \sqrt {9 \, x^{2} + 6 \, x - 8} {\left (3 \, x + 1\right )} + \frac {3}{2} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} + 6 \, x - 8} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 41, normalized size = 0.84 \[ \frac {1}{6} \, \sqrt {9 \, x^{2} + 6 \, x - 8} {\left (3 \, x + 1\right )} + \frac {3}{2} \, \log \left ({\left | -3 \, x + \sqrt {9 \, x^{2} + 6 \, x - 8} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 50, normalized size = 1.02 \[ -\frac {\sqrt {9}\, \ln \left (\frac {\left (9 x +3\right ) \sqrt {9}}{9}+\sqrt {9 x^{2}+6 x -8}\right )}{2}+\frac {\left (18 x +6\right ) \sqrt {9 x^{2}+6 x -8}}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.87, size = 52, normalized size = 1.06 \[ \frac {1}{2} \, \sqrt {9 \, x^{2} + 6 \, x - 8} x + \frac {1}{6} \, \sqrt {9 \, x^{2} + 6 \, x - 8} - \frac {3}{2} \, \log \left (18 \, x + 6 \, \sqrt {9 \, x^{2} + 6 \, x - 8} + 6\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 39, normalized size = 0.80 \[ \left (\frac {x}{2}+\frac {1}{6}\right )\,\sqrt {9\,x^2+6\,x-8}-\frac {3\,\ln \left (3\,x+\sqrt {9\,x^2+6\,x-8}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {9 x^{2} + 6 x - 8}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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