Optimal. Leaf size=34 \[ -\frac{1}{8} \sqrt{3 x^2-4 x^4}-\frac{3}{32} \sin ^{-1}\left (1-\frac{8 x^2}{3}\right ) \]
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Rubi [A] time = 0.0582664, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2018, 640, 619, 216} \[ -\frac{1}{8} \sqrt{3 x^2-4 x^4}-\frac{3}{32} \sin ^{-1}\left (1-\frac{8 x^2}{3}\right ) \]
Antiderivative was successfully verified.
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Rule 2018
Rule 640
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{3 x^2-4 x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{3 x-4 x^2}} \, dx,x,x^2\right )\\ &=-\frac{1}{8} \sqrt{3 x^2-4 x^4}+\frac{3}{16} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3 x-4 x^2}} \, dx,x,x^2\right )\\ &=-\frac{1}{8} \sqrt{3 x^2-4 x^4}-\frac{1}{32} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{9}}} \, dx,x,3-8 x^2\right )\\ &=-\frac{1}{8} \sqrt{3 x^2-4 x^4}-\frac{3}{32} \sin ^{-1}\left (1-\frac{8 x^2}{3}\right )\\ \end{align*}
Mathematica [A] time = 0.0188481, size = 57, normalized size = 1.68 \[ \frac{x \left (8 x^3+3 \sqrt{4 x^2-3} \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-3}}\right )-6 x\right )}{16 \sqrt{3 x^2-4 x^4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 48, normalized size = 1.4 \begin{align*}{\frac{x}{16}\sqrt{-4\,{x}^{2}+3} \left ( -2\,x\sqrt{-4\,{x}^{2}+3}+3\,\arcsin \left ( 2/3\,x\sqrt{3} \right ) \right ){\frac{1}{\sqrt{-4\,{x}^{4}+3\,{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44845, size = 35, normalized size = 1.03 \begin{align*} -\frac{1}{8} \, \sqrt{-4 \, x^{4} + 3 \, x^{2}} - \frac{3}{32} \, \arcsin \left (-\frac{8}{3} \, x^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25666, size = 96, normalized size = 2.82 \begin{align*} -\frac{1}{8} \, \sqrt{-4 \, x^{4} + 3 \, x^{2}} - \frac{3}{16} \, \arctan \left (\frac{\sqrt{-4 \, x^{4} + 3 \, x^{2}}}{2 \, x^{2}}\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 \frac{x^{3}}{\sqrt{- x^{2} \left (4 x^{2} - 3\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15154, size = 35, normalized size = 1.03 \begin{align*} -\frac{1}{8} \, \sqrt{-4 \, x^{4} + 3 \, x^{2}} + \frac{3}{32} \, \arcsin \left (\frac{8}{3} \, x^{2} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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