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 = 0.0171845, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 640
Rule 612
Rule 619
Rule 216
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*}
Mathematica [A] time = 0.0264351, 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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Maple [A] time = 0.046, size = 43, normalized size = 0.8 \begin{align*} -{\frac{-24\,x+48}{16}\sqrt{-4\,{x}^{2}+16\,x+9}}+{\frac{75}{4}\arcsin \left ( -{\frac{4}{5}}+{\frac{2\,x}{5}} \right ) }+{\frac{1}{6} \left ( -4\,{x}^{2}+16\,x+9 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49151, size = 70, normalized size = 1.25 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.39584, size = 135, normalized size = 2.41 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int 2 x \sqrt{- 4 x^{2} + 16 x + 9}\, dx - \int - 7 \sqrt{- 4 x^{2} + 16 x + 9}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08761, size = 43, normalized size = 0.77 \begin{align*} -\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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