Optimal. Leaf size=65 \[ -\frac {3 \tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right )}{10 \sqrt {5}}+\frac {3 x^2+2}{10 x^2 \sqrt {x^4+5}}-\frac {2 \sqrt {x^4+5}}{25 x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1252, 823, 807, 266, 63, 207} \[ \frac {3 x^2+2}{10 x^2 \sqrt {x^4+5}}-\frac {2 \sqrt {x^4+5}}{25 x^2}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right )}{10 \sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rule 266
Rule 807
Rule 823
Rule 1252
Rubi steps
\begin {align*} \int \frac {2+3 x^2}{x^3 \left (5+x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {2+3 x}{x^2 \left (5+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {2+3 x^2}{10 x^2 \sqrt {5+x^4}}-\frac {1}{50} \operatorname {Subst}\left (\int \frac {-20-15 x}{x^2 \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {2+3 x^2}{10 x^2 \sqrt {5+x^4}}-\frac {2 \sqrt {5+x^4}}{25 x^2}+\frac {3}{10} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {2+3 x^2}{10 x^2 \sqrt {5+x^4}}-\frac {2 \sqrt {5+x^4}}{25 x^2}+\frac {3}{20} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {5+x}} \, dx,x,x^4\right )\\ &=\frac {2+3 x^2}{10 x^2 \sqrt {5+x^4}}-\frac {2 \sqrt {5+x^4}}{25 x^2}+\frac {3}{10} \operatorname {Subst}\left (\int \frac {1}{-5+x^2} \, dx,x,\sqrt {5+x^4}\right )\\ &=\frac {2+3 x^2}{10 x^2 \sqrt {5+x^4}}-\frac {2 \sqrt {5+x^4}}{25 x^2}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right )}{10 \sqrt {5}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 45, normalized size = 0.69 \[ \frac {15 x^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {x^4}{5}+1\right )-4 x^4-10}{50 x^2 \sqrt {x^4+5}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 77, normalized size = 1.18 \[ -\frac {4 \, x^{6} - 3 \, \sqrt {5} {\left (x^{6} + 5 \, x^{2}\right )} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{x^{2}}\right ) + 20 \, x^{2} + {\left (4 \, x^{4} - 15 \, x^{2} + 10\right )} \sqrt {x^{4} + 5}}{50 \, {\left (x^{6} + 5 \, x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 82, normalized size = 1.26 \[ \frac {3}{50} \, \sqrt {5} \log \left (-\frac {x^{2} + \sqrt {5} - \sqrt {x^{4} + 5}}{x^{2} - \sqrt {5} - \sqrt {x^{4} + 5}}\right ) - \frac {2 \, x^{2} - 15}{50 \, \sqrt {x^{4} + 5}} + \frac {2}{5 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{2} - 5\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.72 \[ -\frac {3 \sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )}{50}-\frac {2 x^{4}+5}{25 \sqrt {x^{4}+5}\, x^{2}}+\frac {3}{10 \sqrt {x^{4}+5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 68, normalized size = 1.05 \[ -\frac {x^{2}}{25 \, \sqrt {x^{4} + 5}} + \frac {3}{100} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{\sqrt {5} + \sqrt {x^{4} + 5}}\right ) + \frac {3}{10 \, \sqrt {x^{4} + 5}} - \frac {\sqrt {x^{4} + 5}}{25 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 47, normalized size = 0.72 \[ \frac {3}{10\,\sqrt {x^4+5}}-\frac {3\,\sqrt {5}\,\mathrm {atanh}\left (\frac {\sqrt {5}\,\sqrt {x^4+5}}{5}\right )}{50}-\frac {2\,x^4+5}{25\,x^2\,\sqrt {x^4+5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 12.98, size = 228, normalized size = 3.51 \[ \frac {3 x^{4} \log {\left (x^{4} \right )}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} - \frac {6 x^{4} \log {\left (\sqrt {\frac {x^{4}}{5} + 1} + 1 \right )}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} - \frac {3 x^{4} \log {\relax (5 )}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} + \frac {6 \sqrt {5} \sqrt {x^{4} + 5}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} + \frac {15 \log {\left (x^{4} \right )}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} - \frac {30 \log {\left (\sqrt {\frac {x^{4}}{5} + 1} + 1 \right )}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} - \frac {15 \log {\relax (5 )}}{20 \sqrt {5} x^{4} + 100 \sqrt {5}} - \frac {2}{25 \sqrt {1 + \frac {5}{x^{4}}}} - \frac {1}{5 x^{4} \sqrt {1 + \frac {5}{x^{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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