Optimal. Leaf size=119 \[ \frac {1}{48} \left (18 x^2+61\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac {1}{128} \left (199-74 x^2\right ) \sqrt {x^4+5 x^2+3}+\frac {2401}{256} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-3 \sqrt {3} \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1251, 814, 843, 621, 206, 724} \[ \frac {1}{48} \left (18 x^2+61\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac {1}{128} \left (199-74 x^2\right ) \sqrt {x^4+5 x^2+3}+\frac {2401}{256} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-3 \sqrt {3} \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 814
Rule 843
Rule 1251
Rubi steps
\begin {align*} \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(2+3 x) \left (3+5 x+x^2\right )^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{48} \left (61+18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {1}{16} \operatorname {Subst}\left (\int \frac {\left (-48+\frac {37 x}{2}\right ) \sqrt {3+5 x+x^2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{128} \left (199-74 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{48} \left (61+18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {1}{64} \operatorname {Subst}\left (\int \frac {576+\frac {2401 x}{4}}{x \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{128} \left (199-74 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{48} \left (61+18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+9 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )+\frac {2401}{256} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{128} \left (199-74 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{48} \left (61+18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-18 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {6+5 x^2}{\sqrt {3+5 x^2+x^4}}\right )+\frac {2401}{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 {1}{128} \left (199-74 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {1}{48} \left (61+18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {2401}{256} \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )-3 \sqrt {3} \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 104, normalized size = 0.87 \[ \frac {2401}{256} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )-3 \sqrt {3} \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )+\frac {1}{384} \sqrt {x^4+5 x^2+3} \left (144 x^6+1208 x^4+2650 x^2+2061\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 106, normalized size = 0.89 \[ \frac {1}{384} \, {\left (144 \, x^{6} + 1208 \, x^{4} + 2650 \, x^{2} + 2061\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} + 3 \, \sqrt {3} \log \left (\frac {25 \, x^{2} - 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} - 6\right )} + 30}{x^{2}}\right ) - \frac {2401}{256} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 113, normalized size = 0.95 \[ \frac {1}{384} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (18 \, x^{2} + 151\right )} x^{2} + 1325\right )} x^{2} + 2061\right )} + 3 \, \sqrt {3} \log \left (\frac {x^{2} + \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2} - \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}\right ) - \frac {2401}{256} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 117, normalized size = 0.98 \[ \frac {3 \sqrt {x^{4}+5 x^{2}+3}\, x^{6}}{8}+\frac {151 \sqrt {x^{4}+5 x^{2}+3}\, x^{4}}{48}+\frac {1325 \sqrt {x^{4}+5 x^{2}+3}\, x^{2}}{192}-3 \sqrt {3}\, \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right )+\frac {2401 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{256}+\frac {687 \sqrt {x^{4}+5 x^{2}+3}}{128} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 120, normalized size = 1.01 \[ \frac {3}{8} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{2} - \frac {37}{64} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} + \frac {61}{48} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} - 3 \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) + \frac {199}{128} \, \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {2401}{256} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2}}{x} \,d x \]
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 \[ \int \frac {\left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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