Optimal. Leaf size=77 \[ -\frac {\left (47 x^2+33\right ) x^2}{13 \sqrt {x^4+5 x^2+3}}+\frac {133}{26} \sqrt {x^4+5 x^2+3}-\frac {41}{4} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 77, 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, 818, 640, 621, 206} \[ -\frac {\left (47 x^2+33\right ) x^2}{13 \sqrt {x^4+5 x^2+3}}+\frac {133}{26} \sqrt {x^4+5 x^2+3}-\frac {41}{4} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 640
Rule 818
Rule 1251
Rubi steps
\begin {align*} \int \frac {x^5 \left (2+3 x^2\right )}{\left (3+5 x^2+x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2 (2+3 x)}{\left (3+5 x+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {x^2 \left (33+47 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {1}{13} \operatorname {Subst}\left (\int \frac {33+\frac {133 x}{2}}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {x^2 \left (33+47 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {133}{26} \sqrt {3+5 x^2+x^4}-\frac {41}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {x^2 \left (33+47 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {133}{26} \sqrt {3+5 x^2+x^4}-\frac {41}{2} \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 {x^2 \left (33+47 x^2\right )}{13 \sqrt {3+5 x^2+x^4}}+\frac {133}{26} \sqrt {3+5 x^2+x^4}-\frac {41}{4} \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 = 72, normalized size = 0.94 \[ \frac {78 x^4+1198 x^2-533 \sqrt {x^4+5 x^2+3} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )+798}{52 \sqrt {x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 86, normalized size = 1.12 \[ \frac {1811 \, x^{4} + 9055 \, x^{2} + 1066 \, {\left (x^{4} + 5 \, x^{2} + 3\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) + 4 \, {\left (39 \, x^{4} + 599 \, x^{2} + 399\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} + 5433}{104 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 52, normalized size = 0.68 \[ \frac {{\left (39 \, x^{2} + 599\right )} x^{2} + 399}{26 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} + \frac {41}{4} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 91, normalized size = 1.18 \[ \frac {3 x^{4}}{2 \sqrt {x^{4}+5 x^{2}+3}}+\frac {41 x^{2}}{4 \sqrt {x^{4}+5 x^{2}+3}}-\frac {41 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4}-\frac {133}{8 \sqrt {x^{4}+5 x^{2}+3}}+\frac {\frac {665 x^{2}}{52}+\frac {3325}{104}}{\sqrt {x^{4}+5 x^{2}+3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 73, normalized size = 0.95 \[ \frac {3 \, x^{4}}{2 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} + \frac {599 \, x^{2}}{26 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} + \frac {399}{26 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} - \frac {41}{4} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
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 \[ \int \frac {x^5\,\left (3\,x^2+2\right )}{{\left (x^4+5\,x^2+3\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{5} \left (3 x^{2} + 2\right )}{\left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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