Optimal. Leaf size=38 \[ \frac {2}{3} \sinh ^{-1}\left (\frac {1}{2} (3 x-1)\right )-\frac {1}{6} (1-3 x) \sqrt {9 x^2-6 x+5} \]
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Rubi [A] time = 0.01, antiderivative size = 38, 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, 619, 215} \[ \frac {2}{3} \sinh ^{-1}\left (\frac {1}{2} (3 x-1)\right )-\frac {1}{6} (1-3 x) \sqrt {9 x^2-6 x+5} \]
Antiderivative was successfully verified.
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Rule 215
Rule 612
Rule 619
Rubi steps
\begin {align*} \int \sqrt {5-6 x+9 x^2} \, dx &=-\frac {1}{6} (1-3 x) \sqrt {5-6 x+9 x^2}+2 \int \frac {1}{\sqrt {5-6 x+9 x^2}} \, dx\\ &=-\frac {1}{6} (1-3 x) \sqrt {5-6 x+9 x^2}+\frac {1}{18} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{144}}} \, dx,x,-6+18 x\right )\\ &=-\frac {1}{6} (1-3 x) \sqrt {5-6 x+9 x^2}+\frac {2}{3} \sinh ^{-1}\left (\frac {1}{2} (-1+3 x)\right )\\ \end {align*}
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Mathematica [A] time = 0.02, 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.
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fricas [A] time = 0.91, size = 40, normalized size = 1.05 \[ \frac {1}{6} \, \sqrt {9 \, x^{2} - 6 \, x + 5} {\left (3 \, x - 1\right )} - \frac {2}{3} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} - 6 \, x + 5} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 40, normalized size = 1.05 \[ \frac {1}{6} \, \sqrt {9 \, x^{2} - 6 \, x + 5} {\left (3 \, x - 1\right )} - \frac {2}{3} \, \log \left (-3 \, x + \sqrt {9 \, x^{2} - 6 \, x + 5} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 29, normalized size = 0.76 \[ \frac {2 \arcsinh \left (\frac {3 x}{2}-\frac {1}{2}\right )}{3}+\frac {\left (18 x -6\right ) \sqrt {9 x^{2}-6 x +5}}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 38, normalized size = 1.00 \[ \frac {1}{2} \, \sqrt {9 \, x^{2} - 6 \, x + 5} x - \frac {1}{6} \, \sqrt {9 \, x^{2} - 6 \, x + 5} + \frac {2}{3} \, \operatorname {arsinh}\left (\frac {3}{2} \, x - \frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 39, normalized size = 1.03 \[ \frac {2\,\ln \left (3\,x+\sqrt {9\,x^2-6\,x+5}-1\right )}{3}+\left (\frac {x}{2}-\frac {1}{6}\right )\,\sqrt {9\,x^2-6\,x+5} \]
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 \[ \int \sqrt {9 x^{2} - 6 x + 5}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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