Integrand size = 20, antiderivative size = 56 \[ \int (7-2 x) \sqrt {9+16 x-4 x^2} \, dx=-\frac {3}{2} (2-x) \sqrt {9+16 x-4 x^2}+\frac {1}{6} \left (9+16 x-4 x^2\right )^{3/2}-\frac {75}{4} \arcsin \left (\frac {2 (2-x)}{5}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {654, 626, 633, 222} \[ \int (7-2 x) \sqrt {9+16 x-4 x^2} \, dx=-\frac {75}{4} \arcsin \left (\frac {2 (2-x)}{5}\right )+\frac {1}{6} \left (-4 x^2+16 x+9\right )^{3/2}-\frac {3}{2} (2-x) \sqrt {-4 x^2+16 x+9} \]
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Rule 222
Rule 626
Rule 633
Rule 654
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \left (9+16 x-4 x^2\right )^{3/2}+3 \int \sqrt {9+16 x-4 x^2} \, dx \\ & = -\frac {3}{2} (2-x) \sqrt {9+16 x-4 x^2}+\frac {1}{6} \left (9+16 x-4 x^2\right )^{3/2}+\frac {75}{2} \int \frac {1}{\sqrt {9+16 x-4 x^2}} \, dx \\ & = -\frac {3}{2} (2-x) \sqrt {9+16 x-4 x^2}+\frac {1}{6} \left (9+16 x-4 x^2\right )^{3/2}-\frac {15}{16} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{400}}} \, dx,x,16-8 x\right ) \\ & = -\frac {3}{2} (2-x) \sqrt {9+16 x-4 x^2}+\frac {1}{6} \left (9+16 x-4 x^2\right )^{3/2}-\frac {75}{4} \sin ^{-1}\left (\frac {2 (2-x)}{5}\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int (7-2 x) \sqrt {9+16 x-4 x^2} \, dx=\frac {1}{6} \sqrt {9+16 x-4 x^2} \left (-9+25 x-4 x^2\right )-\frac {75}{2} \arctan \left (\frac {\sqrt {9+16 x-4 x^2}}{1+2 x}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.77
method | result | size |
default | \(-\frac {3 \left (-8 x +16\right ) \sqrt {-4 x^{2}+16 x +9}}{16}+\frac {75 \arcsin \left (-\frac {4}{5}+\frac {2 x}{5}\right )}{4}+\frac {\left (-4 x^{2}+16 x +9\right )^{\frac {3}{2}}}{6}\) | \(43\) |
risch | \(\frac {\left (4 x^{2}-25 x +9\right ) \left (4 x^{2}-16 x -9\right )}{6 \sqrt {-4 x^{2}+16 x +9}}+\frac {75 \arcsin \left (-\frac {4}{5}+\frac {2 x}{5}\right )}{4}\) | \(44\) |
trager | \(\left (-\frac {2}{3} x^{2}+\frac {25}{6} x -\frac {3}{2}\right ) \sqrt {-4 x^{2}+16 x +9}+\frac {75 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+\sqrt {-4 x^{2}+16 x +9}\right )}{4}\) | \(64\) |
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Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84 \[ \int (7-2 x) \sqrt {9+16 x-4 x^2} \, dx=-\frac {1}{6} \, {\left (4 \, x^{2} - 25 \, x + 9\right )} \sqrt {-4 \, x^{2} + 16 \, x + 9} - \frac {75}{2} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 16 \, x + 9} - 3}{2 \, x}\right ) \]
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Time = 0.45 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.73 \[ \int (7-2 x) \sqrt {9+16 x-4 x^2} \, dx=\sqrt {- 4 x^{2} + 16 x + 9} \left (- \frac {2 x^{2}}{3} + \frac {25 x}{6} - \frac {3}{2}\right ) + \frac {75 \operatorname {asin}{\left (\frac {2 x}{5} - \frac {4}{5} \right )}}{4} \]
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Time = 0.39 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.93 \[ \int (7-2 x) \sqrt {9+16 x-4 x^2} \, dx=\frac {1}{6} \, {\left (-4 \, x^{2} + 16 \, x + 9\right )}^{\frac {3}{2}} + \frac {3}{2} \, \sqrt {-4 \, x^{2} + 16 \, x + 9} x - 3 \, \sqrt {-4 \, x^{2} + 16 \, x + 9} - \frac {75}{4} \, \arcsin \left (-\frac {2}{5} \, x + \frac {4}{5}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.57 \[ \int (7-2 x) \sqrt {9+16 x-4 x^2} \, dx=-\frac {1}{6} \, {\left ({\left (4 \, x - 25\right )} x + 9\right )} \sqrt {-4 \, x^{2} + 16 \, x + 9} + \frac {75}{4} \, \arcsin \left (\frac {2}{5} \, x - \frac {4}{5}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.32 \[ \int (7-2 x) \sqrt {9+16 x-4 x^2} \, dx=\frac {175\,\mathrm {asin}\left (\frac {2\,x}{5}-\frac {4}{5}\right )}{4}+7\,\left (\frac {x}{2}-1\right )\,\sqrt {-4\,x^2+16\,x+9}+\frac {\sqrt {-4\,x^2+16\,x+9}\,\left (-128\,x^2+128\,x+1056\right )}{192}+\ln \left (x-2-\frac {\sqrt {-4\,x^2+16\,x+9}\,1{}\mathrm {i}}{2}\right )\,25{}\mathrm {i} \]
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