Optimal. Leaf size=99 \[ \frac {3}{10} \left (x^4+5 x^2+3\right )^{5/2}-\frac {11}{32} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac {429}{256} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {5577}{512} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1247, 640, 612, 621, 206} \[ \frac {3}{10} \left (x^4+5 x^2+3\right )^{5/2}-\frac {11}{32} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac {429}{256} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {5577}{512} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 640
Rule 1247
Rubi steps
\begin {align*} \int x \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (2+3 x) \left (3+5 x+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac {3}{10} \left (3+5 x^2+x^4\right )^{5/2}-\frac {11}{4} \operatorname {Subst}\left (\int \left (3+5 x+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=-\frac {11}{32} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {3}{10} \left (3+5 x^2+x^4\right )^{5/2}+\frac {429}{64} \operatorname {Subst}\left (\int \sqrt {3+5 x+x^2} \, dx,x,x^2\right )\\ &=\frac {429}{256} \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {11}{32} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {3}{10} \left (3+5 x^2+x^4\right )^{5/2}-\frac {5577}{512} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac {429}{256} \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {11}{32} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {3}{10} \left (3+5 x^2+x^4\right )^{5/2}-\frac {5577}{256} \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 {429}{256} \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {11}{32} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {3}{10} \left (3+5 x^2+x^4\right )^{5/2}-\frac {5577}{512} \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 = 71, normalized size = 0.72 \[ \frac {2 \sqrt {x^4+5 x^2+3} \left (384 x^8+2960 x^6+5304 x^4+2170 x^2+7581\right )-27885 \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )}{2560} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 61, normalized size = 0.62 \[ \frac {1}{1280} \, {\left (384 \, x^{8} + 2960 \, x^{6} + 5304 \, x^{4} + 2170 \, x^{2} + 7581\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {5577}{512} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 151, normalized size = 1.53 \[ \frac {1}{1280} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{2} + 5\right )} x^{2} - 127\right )} x^{2} + 2635\right )} x^{2} - 33429\right )} + \frac {17}{384} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + 5\right )} x^{2} - 89\right )} x^{2} + 1095\right )} + \frac {19}{48} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, x^{2} + 5\right )} x^{2} - 51\right )} + \frac {3}{4} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, x^{2} + 5\right )} + \frac {5577}{512} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 104, normalized size = 1.05 \[ \frac {3 \sqrt {x^{4}+5 x^{2}+3}\, x^{8}}{10}+\frac {37 \sqrt {x^{4}+5 x^{2}+3}\, x^{6}}{16}+\frac {663 \sqrt {x^{4}+5 x^{2}+3}\, x^{4}}{160}+\frac {217 \sqrt {x^{4}+5 x^{2}+3}\, x^{2}}{128}-\frac {5577 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{512}+\frac {7581 \sqrt {x^{4}+5 x^{2}+3}}{1280} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 101, normalized size = 1.02 \[ -\frac {11}{16} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{2} + \frac {3}{10} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}} + \frac {429}{128} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} - \frac {55}{32} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} + \frac {2145}{256} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {5577}{512} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.53, size = 127, normalized size = 1.28 \[ \frac {\left (x^2+\frac {5}{2}\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2}}{4}-\frac {15\,x^2\,{\left (x^4+5\,x^2+3\right )}^{3/2}}{16}-\frac {5577\,\ln \left (\sqrt {x^4+5\,x^2+3}+x^2+\frac {5}{2}\right )}{512}+\frac {585\,\left (2\,x^2+5\right )\,\sqrt {x^4+5\,x^2+3}}{256}-\frac {39\,\left (\frac {x^2}{2}+\frac {5}{4}\right )\,\sqrt {x^4+5\,x^2+3}}{16}-\frac {75\,{\left (x^4+5\,x^2+3\right )}^{3/2}}{32}+\frac {3\,{\left (x^4+5\,x^2+3\right )}^{5/2}}{10} \]
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 \[ \int x \left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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