Optimal. Leaf size=35 \[ -\frac {1}{36} \sqrt {5 x-9 x^2} (5-18 x)-\frac {25}{216} \sin ^{-1}\left (1-\frac {18 x}{5}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {612, 619, 216} \[ -\frac {1}{36} \sqrt {5 x-9 x^2} (5-18 x)-\frac {25}{216} \sin ^{-1}\left (1-\frac {18 x}{5}\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 612
Rule 619
Rubi steps
\begin {align*} \int \sqrt {5 x-9 x^2} \, dx &=-\frac {1}{36} (5-18 x) \sqrt {5 x-9 x^2}+\frac {25}{72} \int \frac {1}{\sqrt {5 x-9 x^2}} \, dx\\ &=-\frac {1}{36} (5-18 x) \sqrt {5 x-9 x^2}-\frac {5}{216} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{25}}} \, dx,x,5-18 x\right )\\ &=-\frac {1}{36} (5-18 x) \sqrt {5 x-9 x^2}-\frac {25}{216} \sin ^{-1}\left (1-\frac {18 x}{5}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 58, normalized size = 1.66 \[ \frac {-3 x \left (162 x^2-135 x+25\right )-25 \sqrt {5-9 x} \sqrt {x} \sin ^{-1}\left (\sqrt {1-\frac {9 x}{5}}\right )}{108 \sqrt {-x (9 x-5)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.31, size = 38, normalized size = 1.09 \[ \frac {1}{36} \, \sqrt {-9 \, x^{2} + 5 \, x} {\left (18 \, x - 5\right )} - \frac {25}{108} \, \arctan \left (\frac {\sqrt {-9 \, x^{2} + 5 \, x}}{3 \, x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 27, normalized size = 0.77 \[ \frac {1}{36} \, \sqrt {-9 \, x^{2} + 5 \, x} {\left (18 \, x - 5\right )} + \frac {25}{216} \, \arcsin \left (\frac {18}{5} \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 28, normalized size = 0.80 \[ \frac {25 \arcsin \left (\frac {18 x}{5}-1\right )}{216}-\frac {\left (-18 x +5\right ) \sqrt {-9 x^{2}+5 x}}{36} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.93, size = 36, normalized size = 1.03 \[ \frac {1}{2} \, \sqrt {-9 \, x^{2} + 5 \, x} x - \frac {5}{36} \, \sqrt {-9 \, x^{2} + 5 \, x} - \frac {25}{216} \, \arcsin \left (-\frac {18}{5} \, x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 26, normalized size = 0.74 \[ \frac {25\,\mathrm {asin}\left (\frac {18\,x}{5}-1\right )}{216}+\left (\frac {x}{2}-\frac {5}{36}\right )\,\sqrt {5\,x-9\,x^2} \]
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 \[ \int \sqrt {- 9 x^{2} + 5 x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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