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 = 0.06, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2018, 640, 620, 206} \begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 2018
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*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 1.13 \begin {gather*} \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 )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 49, normalized size = 1.09 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 45, normalized size = 1.00 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 41, normalized size = 0.91 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 48, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {4 x^{2}+3}\, \left (-2 \sqrt {4 x^{2}+3}\, x +3 \arcsinh \left (\frac {2 \sqrt {3}\, x}{3}\right )\right ) x}{16 \sqrt {4 x^{4}+3 x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.05, size = 41, normalized size = 0.91 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.40, size = 40, normalized size = 0.89 \begin {gather*} \frac {\sqrt {4\,x^4+3\,x^2}}{8}-\frac {3\,\ln \left (\frac {\sqrt {4\,x^2+3}\,\sqrt {x^2}}{2}+x^2+\frac {3}{8}\right )}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\sqrt {x^{2} \left (4 x^{2} + 3\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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