Optimal. Leaf size=45 \[ \frac {1}{8} \sqrt {3 x^2+4 x^4}-\frac {3}{16} \tanh ^{-1}\left (\frac {2 x^2}{\sqrt {3 x^2+4 x^4}}\right ) \]
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Rubi [A]
time = 0.04, 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 = {2043, 654, 634,
212} \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 212
Rule 634
Rule 654
Rule 2043
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {3 x^2+4 x^4}} \, dx &=\frac {1}{2} \text {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} \text {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} \text {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 = 58, normalized size = 1.29 \begin {gather*} \frac {x \left (6 x+8 x^3+3 \sqrt {3+4 x^2} \log \left (-2 x+\sqrt {3+4 x^2}\right )\right )}{16 \sqrt {x^2 \left (3+4 x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 48, normalized size = 1.07
method | result | size |
meijerg | \(\frac {\frac {\sqrt {\pi }\, x \sqrt {3}\, \sqrt {\frac {4 x^{2}}{3}+1}}{8}-\frac {3 \sqrt {\pi }\, \arcsinh \left (\frac {2 x \sqrt {3}}{3}\right )}{16}}{\sqrt {\pi }}\) | \(37\) |
trager | \(\frac {\sqrt {4 x^{4}+3 x^{2}}}{8}+\frac {3 \ln \left (\frac {-2 x^{2}+\sqrt {4 x^{4}+3 x^{2}}}{x}\right )}{16}\) | \(43\) |
default | \(-\frac {x \sqrt {4 x^{2}+3}\, \left (-2 x \sqrt {4 x^{2}+3}+3 \arcsinh \left (\frac {2 x \sqrt {3}}{3}\right )\right )}{16 \sqrt {4 x^{4}+3 x^{2}}}\) | \(48\) |
risch | \(\frac {x^{2} \left (4 x^{2}+3\right )}{8 \sqrt {x^{2} \left (4 x^{2}+3\right )}}-\frac {3 \arcsinh \left (\frac {2 x \sqrt {3}}{3}\right ) x \sqrt {4 x^{2}+3}}{16 \sqrt {x^{2} \left (4 x^{2}+3\right )}}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, 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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Fricas [A]
time = 0.38, 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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\sqrt {x^{2} \cdot \left (4 x^{2} + 3\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.91, size = 43, normalized size = 0.96 \begin {gather*} -\frac {3}{32} \, \log \left (3\right ) \mathrm {sgn}\left (x\right ) + \frac {\sqrt {4 \, x^{2} + 3} x}{8 \, \mathrm {sgn}\left (x\right )} + \frac {3 \, \log \left (-2 \, x + \sqrt {4 \, x^{2} + 3}\right )}{16 \, \mathrm {sgn}\left (x\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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