Optimal. Leaf size=52 \[ -\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}-\frac{3}{128} (3-8 x) \sqrt{3 x-4 x^2}-\frac{27}{512} \sin ^{-1}\left (1-\frac{8 x}{3}\right ) \]
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Rubi [A] time = 0.0145858, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {640, 612, 619, 216} \[ -\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}-\frac{3}{128} (3-8 x) \sqrt{3 x-4 x^2}-\frac{27}{512} \sin ^{-1}\left (1-\frac{8 x}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 640
Rule 612
Rule 619
Rule 216
Rubi steps
\begin{align*} \int x \sqrt{3 x-4 x^2} \, dx &=-\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}+\frac{3}{8} \int \sqrt{3 x-4 x^2} \, dx\\ &=-\frac{3}{128} (3-8 x) \sqrt{3 x-4 x^2}-\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}+\frac{27}{256} \int \frac{1}{\sqrt{3 x-4 x^2}} \, dx\\ &=-\frac{3}{128} (3-8 x) \sqrt{3 x-4 x^2}-\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}-\frac{9}{512} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{9}}} \, dx,x,3-8 x\right )\\ &=-\frac{3}{128} (3-8 x) \sqrt{3 x-4 x^2}-\frac{1}{12} \left (3 x-4 x^2\right )^{3/2}-\frac{27}{512} \sin ^{-1}\left (1-\frac{8 x}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0434005, size = 63, normalized size = 1.21 \[ \frac{2 x \left (-512 x^3+480 x^2+36 x-81\right )-81 \sqrt{3-4 x} \sqrt{x} \sin ^{-1}\left (\sqrt{1-\frac{4 x}{3}}\right )}{768 \sqrt{-x (4 x-3)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 41, normalized size = 0.8 \begin{align*} -{\frac{1}{12} \left ( -4\,{x}^{2}+3\,x \right ) ^{{\frac{3}{2}}}}+{\frac{27}{512}\arcsin \left ( -1+{\frac{8\,x}{3}} \right ) }-{\frac{9-24\,x}{128}\sqrt{-4\,{x}^{2}+3\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.81818, size = 66, normalized size = 1.27 \begin{align*} -\frac{1}{12} \,{\left (-4 \, x^{2} + 3 \, x\right )}^{\frac{3}{2}} + \frac{3}{16} \, \sqrt{-4 \, x^{2} + 3 \, x} x - \frac{9}{128} \, \sqrt{-4 \, x^{2} + 3 \, x} - \frac{27}{512} \, \arcsin \left (-\frac{8}{3} \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96582, size = 122, normalized size = 2.35 \begin{align*} \frac{1}{384} \,{\left (128 \, x^{2} - 24 \, x - 27\right )} \sqrt{-4 \, x^{2} + 3 \, x} - \frac{27}{256} \, \arctan \left (\frac{\sqrt{-4 \, x^{2} + 3 \, x}}{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 x \sqrt{- x \left (4 x - 3\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21139, size = 43, normalized size = 0.83 \begin{align*} \frac{1}{384} \,{\left (8 \,{\left (16 \, x - 3\right )} x - 27\right )} \sqrt{-4 \, x^{2} + 3 \, x} + \frac{27}{512} \, \arcsin \left (\frac{8}{3} \, x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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