Optimal. Leaf size=34 \[ -\frac{1}{8} \sqrt{-4 x^4-3 x^2}-\frac{3}{32} \sin ^{-1}\left (\frac{8 x^2}{3}+1\right ) \]
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Rubi [A] time = 0.0572119, 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} \[ -\frac{1}{8} \sqrt{-4 x^4-3 x^2}-\frac{3}{32} \sin ^{-1}\left (\frac{8 x^2}{3}+1\right ) \]
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 = 0.0169786, size = 52, normalized size = 1.53 \[ \frac{x \left (8 x^3-3 \sqrt{4 x^2+3} \sinh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )+6 x\right )}{16 \sqrt{-x^2 \left (4 x^2+3\right )}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 54, normalized size = 1.6 \begin{align*} -{\frac{x}{16}\sqrt{-4\,{x}^{2}-3} \left ( 2\,x\sqrt{-4\,{x}^{2}-3}+3\,\arctan \left ( 2\,{\frac{x}{\sqrt{-4\,{x}^{2}-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 = 1.47273, 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 [C] time = 1.1557, size = 158, normalized size = 4.65 \begin{align*} -\frac{1}{8} \, \sqrt{-4 \, x^{2} - 3} x - \frac{3}{32} i \, \log \left (-\frac{8 \, x + 4 i \, \sqrt{-4 \, x^{2} - 3}}{x}\right ) + \frac{3}{32} i \, \log \left (-\frac{8 \, x - 4 i \, \sqrt{-4 \, x^{2} - 3}}{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 \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 = 1.1691, size = 36, normalized size = 1.06 \begin{align*} -\frac{1}{8} \, \sqrt{4 \, x^{4} + 3 \, x^{2}} i - \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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