Optimal. Leaf size=132 \[ -\frac {161 \left (5 x^2+6\right ) \sqrt {x^4+5 x^2+3}}{5184 x^4}+\frac {2093 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{10368 \sqrt {3}}-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{15 x^{10}}-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{36 x^8}+\frac {173 \left (x^4+5 x^2+3\right )^{3/2}}{3240 x^6} \]
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Rubi [A] time = 0.11, antiderivative size = 132, 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, 834, 806, 720, 724, 206} \[ \frac {173 \left (x^4+5 x^2+3\right )^{3/2}}{3240 x^6}-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{36 x^8}-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{15 x^{10}}-\frac {161 \left (5 x^2+6\right ) \sqrt {x^4+5 x^2+3}}{5184 x^4}+\frac {2093 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{10368 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 806
Rule 834
Rule 1251
Rubi steps
\begin {align*} \int \frac {\left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4}}{x^{11}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(2+3 x) \sqrt {3+5 x+x^2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac {1}{30} \operatorname {Subst}\left (\int \frac {(-10+4 x) \sqrt {3+5 x+x^2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac {1}{360} \operatorname {Subst}\left (\int \frac {(-173-10 x) \sqrt {3+5 x+x^2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac {173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}+\frac {161}{432} \operatorname {Subst}\left (\int \frac {\sqrt {3+5 x+x^2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {161 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{5184 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac {173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}-\frac {2093 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )}{10368}\\ &=-\frac {161 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{5184 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac {173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}+\frac {2093 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {6+5 x^2}{\sqrt {3+5 x^2+x^4}}\right )}{5184}\\ &=-\frac {161 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{5184 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac {173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}+\frac {2093 \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{10368 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 84, normalized size = 0.64 \[ \frac {10465 \sqrt {3} \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )-\frac {6 \sqrt {x^4+5 x^2+3} \left (2641 x^8-1370 x^6+1176 x^4+10800 x^2+5184\right )}{x^{10}}}{155520} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 100, normalized size = 0.76 \[ \frac {10465 \, \sqrt {3} x^{10} \log \left (\frac {25 \, x^{2} + 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} + 6\right )} + 30}{x^{2}}\right ) - 15846 \, x^{10} - 6 \, {\left (2641 \, x^{8} - 1370 \, x^{6} + 1176 \, x^{4} + 10800 \, x^{2} + 5184\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{155520 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.56, size = 255, normalized size = 1.93 \[ -\frac {2093}{31104} \, \sqrt {3} \log \left (\frac {x^{2} + \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2} - \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}\right ) + \frac {10465 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{9} - 42830 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{7} + 1270080 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{6} + 7060800 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{5} + 15310080 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{4} + 16095870 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{3} + 7568640 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} + 1096335 \, x^{2} - 1096335 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 202176}{25920 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 152, normalized size = 1.15 \[ \frac {2093 \sqrt {3}\, \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right )}{31104}+\frac {805 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{15552 x^{2}}-\frac {161 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{2592 x^{4}}+\frac {173 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{3240 x^{6}}-\frac {\left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{36 x^{8}}-\frac {\left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{15 x^{10}}-\frac {2093 \sqrt {x^{4}+5 x^{2}+3}}{31104}-\frac {805 \left (2 x^{2}+5\right ) \sqrt {x^{4}+5 x^{2}+3}}{31104} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.73, size = 133, normalized size = 1.01 \[ \frac {2093}{31104} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) + \frac {161}{2592} \, \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {805 \, \sqrt {x^{4} + 5 \, x^{2} + 3}}{5184 \, x^{2}} - \frac {161 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{2592 \, x^{4}} + \frac {173 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{3240 \, x^{6}} - \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{36 \, x^{8}} - \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{15 \, x^{10}} \]
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 \[ \int \frac {\left (3\,x^2+2\right )\,\sqrt {x^4+5\,x^2+3}}{x^{11}} \,d x \]
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 \[ \int \frac {\left (3 x^{2} + 2\right ) \sqrt {x^{4} + 5 x^{2} + 3}}{x^{11}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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