Optimal. Leaf size=127 \[ \frac {3}{14} \left (x^4+5 x^2+3\right )^{5/2} x^4+\frac {\left (3313-1070 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}}{1680}-\frac {2183}{768} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac {28379 \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}}{2048}-\frac {368927 \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )}{4096} \]
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Rubi [A] time = 0.10, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1251, 832, 779, 612, 621, 206} \[ \frac {3}{14} \left (x^4+5 x^2+3\right )^{5/2} x^4+\frac {\left (3313-1070 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}}{1680}-\frac {2183}{768} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac {28379 \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}}{2048}-\frac {368927 \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )}{4096} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rule 1251
Rubi steps
\begin {align*} \int 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 x^2 (2+3 x) \left (3+5 x+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac {3}{14} x^4 \left (3+5 x^2+x^4\right )^{5/2}+\frac {1}{14} \operatorname {Subst}\left (\int \left (-18-\frac {107 x}{2}\right ) x \left (3+5 x+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac {3}{14} x^4 \left (3+5 x^2+x^4\right )^{5/2}+\frac {\left (3313-1070 x^2\right ) \left (3+5 x^2+x^4\right )^{5/2}}{1680}-\frac {2183}{96} \operatorname {Subst}\left (\int \left (3+5 x+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=-\frac {2183}{768} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {3}{14} x^4 \left (3+5 x^2+x^4\right )^{5/2}+\frac {\left (3313-1070 x^2\right ) \left (3+5 x^2+x^4\right )^{5/2}}{1680}+\frac {28379}{512} \operatorname {Subst}\left (\int \sqrt {3+5 x+x^2} \, dx,x,x^2\right )\\ &=\frac {28379 \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}}{2048}-\frac {2183}{768} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {3}{14} x^4 \left (3+5 x^2+x^4\right )^{5/2}+\frac {\left (3313-1070 x^2\right ) \left (3+5 x^2+x^4\right )^{5/2}}{1680}-\frac {368927 \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )}{4096}\\ &=\frac {28379 \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}}{2048}-\frac {2183}{768} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {3}{14} x^4 \left (3+5 x^2+x^4\right )^{5/2}+\frac {\left (3313-1070 x^2\right ) \left (3+5 x^2+x^4\right )^{5/2}}{1680}-\frac {368927 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {5+2 x^2}{\sqrt {3+5 x^2+x^4}}\right )}{2048}\\ &=\frac {28379 \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}}{2048}-\frac {2183}{768} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {3}{14} x^4 \left (3+5 x^2+x^4\right )^{5/2}+\frac {\left (3313-1070 x^2\right ) \left (3+5 x^2+x^4\right )^{5/2}}{1680}-\frac {368927 \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )}{4096}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 81, normalized size = 0.64 \[ \frac {2 \sqrt {x^4+5 x^2+3} \left (46080 x^{12}+323840 x^{10}+482944 x^8+154800 x^6+283304 x^4-1499570 x^2+9546951\right )-38737335 \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )}{430080} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 71, normalized size = 0.56 \[ \frac {1}{215040} \, {\left (46080 \, x^{12} + 323840 \, x^{10} + 482944 \, x^{8} + 154800 \, x^{6} + 283304 \, x^{4} - 1499570 \, x^{2} + 9546951\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {368927}{4096} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 207, normalized size = 1.63 \[ \frac {1}{71680} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, x^{2} + 5\right )} x^{2} - 203\right )} x^{2} + 7635\right )} x^{2} - 76083\right )} x^{2} + 1627215\right )} x^{2} - 20756241\right )} + \frac {17}{3072} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, x^{2} + 1\right )} x^{2} - 33\right )} x^{2} + 321\right )} x^{2} - 6837\right )} x^{2} + 87147\right )} + \frac {19}{3840} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{2} + 5\right )} x^{2} - 127\right )} x^{2} + 2635\right )} x^{2} - 33429\right )} + \frac {1}{64} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + 5\right )} x^{2} - 89\right )} x^{2} + 1095\right )} + \frac {368927}{4096} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 138, normalized size = 1.09 \[ \frac {3 \sqrt {x^{4}+5 x^{2}+3}\, x^{12}}{14}+\frac {253 \sqrt {x^{4}+5 x^{2}+3}\, x^{10}}{168}+\frac {539 \sqrt {x^{4}+5 x^{2}+3}\, x^{8}}{240}+\frac {645 \sqrt {x^{4}+5 x^{2}+3}\, x^{6}}{896}+\frac {5059 \sqrt {x^{4}+5 x^{2}+3}\, x^{4}}{3840}-\frac {149957 \sqrt {x^{4}+5 x^{2}+3}\, x^{2}}{21504}-\frac {368927 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4096}+\frac {3182317 \sqrt {x^{4}+5 x^{2}+3}}{71680} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 135, normalized size = 1.06 \[ \frac {3}{14} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}} x^{4} - \frac {107}{168} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}} x^{2} - \frac {2183}{384} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{2} + \frac {3313}{1680} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}} + \frac {28379}{1024} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} - \frac {10915}{768} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} + \frac {141895}{2048} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {368927}{4096} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
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 \[ \int x^5\,\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2} \,d x \]
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 \[ \int x^{5} \left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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