Optimal. Leaf size=98 \[ -\frac {89}{48} \sqrt {x^4+5 x^2+3} x^4-\frac {1}{384} \left (24243-3802 x^2\right ) \sqrt {x^4+5 x^2+3}+\frac {32801}{256} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )+\frac {3}{8} \sqrt {x^4+5 x^2+3} x^6 \]
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Rubi [A] time = 0.09, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1251, 832, 779, 621, 206} \[ \frac {3}{8} \sqrt {x^4+5 x^2+3} x^6-\frac {89}{48} \sqrt {x^4+5 x^2+3} x^4-\frac {1}{384} \left (24243-3802 x^2\right ) \sqrt {x^4+5 x^2+3}+\frac {32801}{256} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 779
Rule 832
Rule 1251
Rubi steps
\begin {align*} \int \frac {x^7 \left (2+3 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^3 (2+3 x)}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac {3}{8} x^6 \sqrt {3+5 x^2+x^4}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {\left (-27-\frac {89 x}{2}\right ) x^2}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {89}{48} x^4 \sqrt {3+5 x^2+x^4}+\frac {3}{8} x^6 \sqrt {3+5 x^2+x^4}+\frac {1}{24} \operatorname {Subst}\left (\int \frac {x \left (267+\frac {1901 x}{4}\right )}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {89}{48} x^4 \sqrt {3+5 x^2+x^4}+\frac {3}{8} x^6 \sqrt {3+5 x^2+x^4}-\frac {1}{384} \left (24243-3802 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {32801}{256} \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {89}{48} x^4 \sqrt {3+5 x^2+x^4}+\frac {3}{8} x^6 \sqrt {3+5 x^2+x^4}-\frac {1}{384} \left (24243-3802 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {32801}{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 {89}{48} x^4 \sqrt {3+5 x^2+x^4}+\frac {3}{8} x^6 \sqrt {3+5 x^2+x^4}-\frac {1}{384} \left (24243-3802 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {32801}{256} \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 = 0.03, size = 66, normalized size = 0.67 \[ \frac {1}{768} \left (98403 \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )+2 \sqrt {x^4+5 x^2+3} \left (144 x^6-712 x^4+3802 x^2-24243\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 56, normalized size = 0.57 \[ \frac {1}{384} \, {\left (144 \, x^{6} - 712 \, x^{4} + 3802 \, x^{2} - 24243\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {32801}{256} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 60, normalized size = 0.61 \[ \frac {1}{384} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (18 \, x^{2} - 89\right )} x^{2} + 1901\right )} x^{2} - 24243\right )} - \frac {32801}{256} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 87, normalized size = 0.89 \[ \frac {3 \sqrt {x^{4}+5 x^{2}+3}\, x^{6}}{8}-\frac {89 \sqrt {x^{4}+5 x^{2}+3}\, x^{4}}{48}+\frac {1901 \sqrt {x^{4}+5 x^{2}+3}\, x^{2}}{192}+\frac {32801 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{256}-\frac {8081 \sqrt {x^{4}+5 x^{2}+3}}{128} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 90, normalized size = 0.92 \[ \frac {3}{8} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{6} - \frac {89}{48} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{4} + \frac {1901}{192} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} - \frac {8081}{128} \, \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {32801}{256} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
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 \[ \int \frac {x^7\,\left (3\,x^2+2\right )}{\sqrt {x^4+5\,x^2+3}} \,d x \]
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 \[ \int \frac {x^{7} \left (3 x^{2} + 2\right )}{\sqrt {x^{4} + 5 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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