Optimal. Leaf size=45 \[ \frac{1}{8} \sqrt{4 x^4+3 x^2}-\frac{3}{16} \tanh ^{-1}\left (\frac{2 x^2}{\sqrt{4 x^4+3 x^2}}\right ) \]
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Rubi [A] time = 0.0561752, antiderivative size = 45, 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, 620, 206} \[ \frac{1}{8} \sqrt{4 x^4+3 x^2}-\frac{3}{16} \tanh ^{-1}\left (\frac{2 x^2}{\sqrt{4 x^4+3 x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 2018
Rule 640
Rule 620
Rule 206
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{3}{8} \operatorname{Subst}\left (\int \frac{1}{1-4 x^2} \, dx,x,\frac{x^2}{\sqrt{3 x^2+4 x^4}}\right )\\ &=\frac{1}{8} \sqrt{3 x^2+4 x^4}-\frac{3}{16} \tanh ^{-1}\left (\frac{2 x^2}{\sqrt{3 x^2+4 x^4}}\right )\\ \end{align*}
Mathematica [A] time = 0.0097691, size = 51, normalized size = 1.13 \[ \frac{x \left (8 x^3-3 \sqrt{4 x^2+3} \sinh ^{-1}\left (\frac{2 x}{\sqrt{3}}\right )+6 x\right )}{16 \sqrt{x^2 \left (4 x^2+3\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 48, normalized size = 1.1 \begin{align*} -{\frac{x}{16}\sqrt{4\,{x}^{2}+3} \left ( -2\,x\sqrt{4\,{x}^{2}+3}+3\,{\it Arcsinh} \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 = 1.45947, size = 55, normalized size = 1.22 \begin{align*} \frac{1}{8} \, \sqrt{4 \, x^{4} + 3 \, x^{2}} - \frac{3}{32} \, \log \left (8 \, x^{2} + 4 \, \sqrt{4 \, x^{4} + 3 \, x^{2}} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23217, size = 95, normalized size = 2.11 \begin{align*} \frac{1}{8} \, \sqrt{4 \, x^{4} + 3 \, x^{2}} + \frac{3}{16} \, \log \left (-\frac{2 \, x^{2} - \sqrt{4 \, x^{4} + 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 \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 = 1.1693, size = 55, normalized size = 1.22 \begin{align*} \frac{1}{8} \, \sqrt{4 \, x^{4} + 3 \, x^{2}} + \frac{3}{32} \, \log \left (8 \, x^{2} - 4 \, \sqrt{4 \, x^{4} + 3 \, x^{2}} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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