Optimal. Leaf size=81 \[ -\frac {1}{48} \left (59-18 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac {259}{128} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {3367}{256} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1251, 779, 612, 621, 206} \[ -\frac {1}{48} \left (59-18 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac {259}{128} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {3367}{256} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 1251
Rubi steps
\begin {align*} \int x^3 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (2+3 x) \sqrt {3+5 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {1}{48} \left (59-18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {259}{32} \operatorname {Subst}\left (\int \sqrt {3+5 x+x^2} \, dx,x,x^2\right )\\ &=\frac {259}{128} \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1}{48} \left (59-18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {3367}{256} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac {259}{128} \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1}{48} \left (59-18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {3367}{128} \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {5+2 x^2}{\sqrt {3+5 x^2+x^4}}\right )\\ &=\frac {259}{128} \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1}{48} \left (59-18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {3367}{256} \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 66, normalized size = 0.81 \[ \frac {1}{768} \left (2 \sqrt {x^4+5 x^2+3} \left (144 x^6+248 x^4-374 x^2+2469\right )-10101 \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 56, normalized size = 0.69 \[ \frac {1}{384} \, {\left (144 \, x^{6} + 248 \, x^{4} - 374 \, x^{2} + 2469\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {3367}{256} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 88, normalized size = 1.09 \[ \frac {1}{128} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + 5\right )} x^{2} - 89\right )} x^{2} + 1095\right )} + \frac {1}{24} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, x^{2} + 5\right )} x^{2} - 51\right )} + \frac {3367}{256} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 74, normalized size = 0.91 \[ \frac {3 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}} x^{2}}{8}-\frac {3367 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{256}-\frac {59 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{48}+\frac {259 \left (2 x^{2}+5\right ) \sqrt {x^{4}+5 x^{2}+3}}{128} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 87, normalized size = 1.07 \[ \frac {3}{8} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{2} + \frac {259}{64} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} - \frac {59}{48} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} + \frac {1295}{128} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {3367}{256} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 85, normalized size = 1.05 \[ \frac {3\,x^2\,{\left (x^4+5\,x^2+3\right )}^{3/2}}{8}-\frac {3367\,\ln \left (\sqrt {x^4+5\,x^2+3}+x^2+\frac {5}{2}\right )}{256}-\frac {9\,\left (\frac {x^2}{2}+\frac {5}{4}\right )\,\sqrt {x^4+5\,x^2+3}}{8}-\frac {59\,\sqrt {x^4+5\,x^2+3}\,\left (8\,x^4+10\,x^2-51\right )}{384} \]
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 \[ \int x^{3} \left (3 x^{2} + 2\right ) \sqrt {x^{4} + 5 x^{2} + 3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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