Optimal. Leaf size=56 \[ \frac {149}{16} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-\frac {1}{8} \left (37-6 x^2\right ) \sqrt {x^4+5 x^2+3} \]
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Rubi [A] time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1251, 779, 621, 206} \[ \frac {149}{16} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-\frac {1}{8} \left (37-6 x^2\right ) \sqrt {x^4+5 x^2+3} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 1251
Rubi steps
\begin {align*} \int \frac {x^3 \left (2+3 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x (2+3 x)}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {1}{8} \left (37-6 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {149}{16} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {1}{8} \left (37-6 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {149}{8} \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 {1}{8} \left (37-6 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {149}{16} \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 = 56, normalized size = 1.00 \[ \frac {1}{16} \left (2 \sqrt {x^4+5 x^2+3} \left (6 x^2-37\right )+149 \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 46, normalized size = 0.82 \[ \frac {1}{8} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (6 \, x^{2} - 37\right )} - \frac {149}{16} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 46, normalized size = 0.82 \[ \frac {1}{8} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (6 \, x^{2} - 37\right )} - \frac {149}{16} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 53, normalized size = 0.95 \[ \frac {3 \sqrt {x^{4}+5 x^{2}+3}\, x^{2}}{4}+\frac {149 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{16}-\frac {37 \sqrt {x^{4}+5 x^{2}+3}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 56, normalized size = 1.00 \[ \frac {3}{4} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} - \frac {37}{8} \, \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {149}{16} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^3\,\left (3\,x^2+2\right )}{\sqrt {x^4+5\,x^2+3}} \,d x \]
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 \[ \int \frac {x^{3} \left (3 x^{2} + 2\right )}{\sqrt {x^{4} + 5 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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