Optimal. Leaf size=74 \[ 3 \sqrt{\frac{3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac{9}{2} \sqrt{\frac{3}{2}} \sqrt{1-2 x} x \sqrt{2 x+1}+\frac{9}{4} \sqrt{\frac{3}{2}} \sin ^{-1}(2 x) \]
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Rubi [A] time = 0.0100821, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {38, 41, 216} \[ 3 \sqrt{\frac{3}{2}} (1-2 x)^{3/2} x (2 x+1)^{3/2}+\frac{9}{2} \sqrt{\frac{3}{2}} \sqrt{1-2 x} x \sqrt{2 x+1}+\frac{9}{4} \sqrt{\frac{3}{2}} \sin ^{-1}(2 x) \]
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 216
Rubi steps
\begin{align*} \int (3-6 x)^{3/2} (2+4 x)^{3/2} \, dx &=3 \sqrt{\frac{3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+\frac{9}{2} \int \sqrt{3-6 x} \sqrt{2+4 x} \, dx\\ &=\frac{9}{2} \sqrt{\frac{3}{2}} \sqrt{1-2 x} x \sqrt{1+2 x}+3 \sqrt{\frac{3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+\frac{27}{2} \int \frac{1}{\sqrt{3-6 x} \sqrt{2+4 x}} \, dx\\ &=\frac{9}{2} \sqrt{\frac{3}{2}} \sqrt{1-2 x} x \sqrt{1+2 x}+3 \sqrt{\frac{3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+\frac{27}{2} \int \frac{1}{\sqrt{6-24 x^2}} \, dx\\ &=\frac{9}{2} \sqrt{\frac{3}{2}} \sqrt{1-2 x} x \sqrt{1+2 x}+3 \sqrt{\frac{3}{2}} (1-2 x)^{3/2} x (1+2 x)^{3/2}+\frac{9}{4} \sqrt{\frac{3}{2}} \sin ^{-1}(2 x)\\ \end{align*}
Mathematica [A] time = 0.0335005, size = 39, normalized size = 0.53 \[ \frac{3}{4} \sqrt{\frac{3}{2}} \left (2 x \sqrt{1-4 x^2} \left (5-8 x^2\right )+3 \sin ^{-1}(2 x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 102, normalized size = 1.4 \begin{align*}{\frac{1}{16} \left ( 3-6\,x \right ) ^{{\frac{3}{2}}} \left ( 2+4\,x \right ) ^{{\frac{5}{2}}}}+{\frac{3}{16} \left ( 2+4\,x \right ) ^{{\frac{5}{2}}}\sqrt{3-6\,x}}-{\frac{3}{16} \left ( 2+4\,x \right ) ^{{\frac{3}{2}}}\sqrt{3-6\,x}}-{\frac{9}{8}\sqrt{3-6\,x}\sqrt{2+4\,x}}+{\frac{9\,\arcsin \left ( 2\,x \right ) \sqrt{6}}{8}\sqrt{ \left ( 2+4\,x \right ) \left ( 3-6\,x \right ) }{\frac{1}{\sqrt{3-6\,x}}}{\frac{1}{\sqrt{2+4\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50259, size = 46, normalized size = 0.62 \begin{align*} \frac{1}{4} \,{\left (-24 \, x^{2} + 6\right )}^{\frac{3}{2}} x + \frac{9}{4} \, \sqrt{-24 \, x^{2} + 6} x + \frac{9}{8} \, \sqrt{6} \arcsin \left (2 \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55817, size = 177, normalized size = 2.39 \begin{align*} -\frac{3}{4} \,{\left (8 \, x^{3} - 5 \, x\right )} \sqrt{4 \, x + 2} \sqrt{-6 \, x + 3} - \frac{9}{8} \, \sqrt{3} \sqrt{2} \arctan \left (\frac{\sqrt{3} \sqrt{2} \sqrt{4 \, x + 2} \sqrt{-6 \, x + 3}}{12 \, x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09449, size = 103, normalized size = 1.39 \begin{align*} -\frac{3}{8} \, \sqrt{3} \sqrt{2}{\left ({\left ({\left (4 \,{\left (2 \, x + 1\right )}{\left (x - 1\right )} + 5\right )}{\left (2 \, x + 1\right )} - 1\right )} \sqrt{2 \, x + 1} \sqrt{-2 \, x + 1} - 8 \, \sqrt{2 \, x + 1} x \sqrt{-2 \, x + 1} - 6 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{2 \, x + 1}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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