Optimal. Leaf size=111 \[ \frac {67 \left (5 x^2+6\right ) \sqrt {x^4+5 x^2+3}}{1728 x^4}-\frac {871 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{3456 \sqrt {3}}-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{12 x^8}-\frac {11 \left (x^4+5 x^2+3\right )^{3/2}}{216 x^6} \]
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Rubi [A] time = 0.09, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {11 \left (x^4+5 x^2+3\right )^{3/2}}{216 x^6}-\frac {\left (x^4+5 x^2+3\right )^{3/2}}{12 x^8}+\frac {67 \left (5 x^2+6\right ) \sqrt {x^4+5 x^2+3}}{1728 x^4}-\frac {871 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{3456 \sqrt {3}} \]
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^9} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(2+3 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}}{12 x^8}-\frac {1}{24} \operatorname {Subst}\left (\int \frac {(-11+2 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}}{12 x^8}-\frac {11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}-\frac {67}{144} \operatorname {Subst}\left (\int \frac {\sqrt {3+5 x+x^2}}{x^3} \, dx,x,x^2\right )\\ &=\frac {67 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{1728 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{12 x^8}-\frac {11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}+\frac {871 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {3+5 x+x^2}} \, dx,x,x^2\right )}{3456}\\ &=\frac {67 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{1728 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{12 x^8}-\frac {11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}-\frac {871 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {6+5 x^2}{\sqrt {3+5 x^2+x^4}}\right )}{1728}\\ &=\frac {67 \left (6+5 x^2\right ) \sqrt {3+5 x^2+x^4}}{1728 x^4}-\frac {\left (3+5 x^2+x^4\right )^{3/2}}{12 x^8}-\frac {11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}-\frac {871 \tanh ^{-1}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{3456 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 82, normalized size = 0.74 \[ \frac {6 \sqrt {x^4+5 x^2+3} \left (247 x^6-182 x^4-984 x^2-432\right )-871 \sqrt {3} x^8 \tanh ^{-1}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{10368 x^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 95, normalized size = 0.86 \[ \frac {871 \, \sqrt {3} x^{8} \log \left (\frac {25 \, x^{2} - 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} - 6\right )} + 30}{x^{2}}\right ) + 1482 \, x^{8} + 6 \, {\left (247 \, x^{6} - 182 \, x^{4} - 984 \, x^{2} - 432\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{10368 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 233, normalized size = 2.10 \[ \frac {871}{10368} \, \sqrt {3} \log \left (\frac {x^{2} + \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2} - \sqrt {3} - \sqrt {x^{4} + 5 \, x^{2} + 3}}\right ) - \frac {871 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{7} - 5184 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{6} - 57389 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{5} - 165888 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{4} - 204807 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{3} - 93312 \, {\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 2403 \, x^{2} + 2403 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5184}{1728 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5 \, x^{2} + 3}\right )}^{2} - 3\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 135, normalized size = 1.22 \[ -\frac {871 \sqrt {3}\, \arctanh \left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right )}{10368}-\frac {335 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{5184 x^{2}}+\frac {67 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{864 x^{4}}-\frac {11 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{216 x^{6}}-\frac {\left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{12 x^{8}}+\frac {871 \sqrt {x^{4}+5 x^{2}+3}}{10368}+\frac {335 \left (2 x^{2}+5\right ) \sqrt {x^{4}+5 x^{2}+3}}{10368} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.74, size = 116, normalized size = 1.05 \[ -\frac {871}{10368} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) - \frac {67}{864} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {335 \, \sqrt {x^{4} + 5 \, x^{2} + 3}}{1728 \, x^{2}} + \frac {67 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{864 \, x^{4}} - \frac {11 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{216 \, x^{6}} - \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}}}{12 \, x^{8}} \]
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 \[ \int \frac {\left (3\,x^2+2\right )\,\sqrt {x^4+5\,x^2+3}}{x^9} \,d x \]
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 \[ \int \frac {\left (3 x^{2} + 2\right ) \sqrt {x^{4} + 5 x^{2} + 3}}{x^{9}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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