Optimal. Leaf size=56 \[ \frac {1}{6} \left (-4 x^2+16 x+9\right )^{3/2}-\frac {3}{2} (2-x) \sqrt {-4 x^2+16 x+9}-\frac {75}{4} \sin ^{-1}\left (\frac {2 (2-x)}{5}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {640, 612, 619, 216} \[ \frac {1}{6} \left (-4 x^2+16 x+9\right )^{3/2}-\frac {3}{2} (2-x) \sqrt {-4 x^2+16 x+9}-\frac {75}{4} \sin ^{-1}\left (\frac {2 (2-x)}{5}\right ) \]
Antiderivative was successfully verified.
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Rule 216
Rule 612
Rule 619
Rule 640
Rubi steps
\begin {align*} \int (7-2 x) \sqrt {9+16 x-4 x^2} \, dx &=\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} \operatorname {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*}
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Mathematica [A] time = 0.03, size = 43, normalized size = 0.77 \[ -\frac {1}{6} \sqrt {-4 x^2+16 x+9} \left (4 x^2-25 x+9\right )-\frac {75}{4} \sin ^{-1}\left (\frac {4}{5}-\frac {2 x}{5}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 47, normalized size = 0.84 \[ -\frac {1}{6} \, {\left (4 \, x^{2} - 25 \, x + 9\right )} \sqrt {-4 \, x^{2} + 16 \, x + 9} - \frac {75}{2} \, \arctan \left (\frac {\sqrt {-4 \, x^{2} + 16 \, x + 9} - 3}{2 \, x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 32, normalized size = 0.57 \[ -\frac {1}{6} \, {\left ({\left (4 \, x - 25\right )} x + 9\right )} \sqrt {-4 \, x^{2} + 16 \, x + 9} + \frac {75}{4} \, \arcsin \left (\frac {2}{5} \, x - \frac {4}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 43, normalized size = 0.77 \[ \frac {75 \arcsin \left (\frac {2 x}{5}-\frac {4}{5}\right )}{4}-\frac {3 \left (-8 x +16\right ) \sqrt {-4 x^{2}+16 x +9}}{16}+\frac {\left (-4 x^{2}+16 x +9\right )^{\frac {3}{2}}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.88, size = 52, normalized size = 0.93 \[ \frac {1}{6} \, {\left (-4 \, x^{2} + 16 \, x + 9\right )}^{\frac {3}{2}} + \frac {3}{2} \, \sqrt {-4 \, x^{2} + 16 \, x + 9} x - 3 \, \sqrt {-4 \, x^{2} + 16 \, x + 9} - \frac {75}{4} \, \arcsin \left (-\frac {2}{5} \, x + \frac {4}{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 74, normalized size = 1.32 \[ \frac {175\,\mathrm {asin}\left (\frac {2\,x}{5}-\frac {4}{5}\right )}{4}+7\,\left (\frac {x}{2}-1\right )\,\sqrt {-4\,x^2+16\,x+9}+\frac {\sqrt {-4\,x^2+16\,x+9}\,\left (-128\,x^2+128\,x+1056\right )}{192}+\ln \left (x-2-\frac {\sqrt {-4\,x^2+16\,x+9}\,1{}\mathrm {i}}{2}\right )\,25{}\mathrm {i} \]
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 \[ - \int 2 x \sqrt {- 4 x^{2} + 16 x + 9}\, dx - \int \left (- 7 \sqrt {- 4 x^{2} + 16 x + 9}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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