Optimal. Leaf size=58 \[ -\frac {3 \sqrt {x^4+5}}{4 x^4}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right )}{4 \sqrt {5}}-\frac {\left (x^4+5\right )^{3/2}}{15 x^6} \]
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Rubi [A] time = 0.05, antiderivative size = 58, 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, 807, 266, 47, 63, 207} \[ -\frac {\left (x^4+5\right )^{3/2}}{15 x^6}-\frac {3 \sqrt {x^4+5}}{4 x^4}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right )}{4 \sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 207
Rule 266
Rule 807
Rule 1252
Rubi steps
\begin {align*} \int \frac {\left (2+3 x^2\right ) \sqrt {5+x^4}}{x^7} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(2+3 x) \sqrt {5+x^2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {\left (5+x^4\right )^{3/2}}{15 x^6}+\frac {3}{2} \operatorname {Subst}\left (\int \frac {\sqrt {5+x^2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\left (5+x^4\right )^{3/2}}{15 x^6}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {\sqrt {5+x}}{x^2} \, dx,x,x^4\right )\\ &=-\frac {3 \sqrt {5+x^4}}{4 x^4}-\frac {\left (5+x^4\right )^{3/2}}{15 x^6}+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {5+x}} \, dx,x,x^4\right )\\ &=-\frac {3 \sqrt {5+x^4}}{4 x^4}-\frac {\left (5+x^4\right )^{3/2}}{15 x^6}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{-5+x^2} \, dx,x,\sqrt {5+x^4}\right )\\ &=-\frac {3 \sqrt {5+x^4}}{4 x^4}-\frac {\left (5+x^4\right )^{3/2}}{15 x^6}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right )}{4 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 72, normalized size = 1.24 \[ -\frac {3 \left (5 x^4+\sqrt {5} \sqrt {x^4+5} x^4 \tanh ^{-1}\left (\sqrt {\frac {x^4}{5}+1}\right )+25\right )}{20 x^4 \sqrt {x^4+5}}-\frac {\left (x^4+5\right )^{3/2}}{15 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 59, normalized size = 1.02 \[ \frac {9 \, \sqrt {5} x^{6} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{x^{2}}\right ) - 4 \, x^{6} - {\left (4 \, x^{4} + 45 \, x^{2} + 20\right )} \sqrt {x^{4} + 5}}{60 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 116, normalized size = 2.00 \[ \frac {3}{20} \, \sqrt {5} \log \left (-\frac {x^{2} + \sqrt {5} - \sqrt {x^{4} + 5}}{x^{2} - \sqrt {5} - \sqrt {x^{4} + 5}}\right ) + \frac {9 \, {\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{5} + 12 \, {\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{4} - 225 \, x^{2} + 225 \, \sqrt {x^{4} + 5} + 100}{6 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{2} - 5\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.90 \[ -\frac {3 \sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )}{20}-\frac {3 \left (x^{4}+5\right )^{\frac {3}{2}}}{20 x^{4}}-\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{15 x^{6}}+\frac {3 \sqrt {x^{4}+5}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.61, size = 59, normalized size = 1.02 \[ \frac {3}{40} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{\sqrt {5} + \sqrt {x^{4} + 5}}\right ) - \frac {3 \, \sqrt {x^{4} + 5}}{4 \, x^{4}} - \frac {{\left (x^{4} + 5\right )}^{\frac {3}{2}}}{15 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 43, normalized size = 0.74 \[ -\frac {3\,\sqrt {5}\,\mathrm {atanh}\left (\frac {\sqrt {5}\,\sqrt {x^4+5}}{5}\right )}{20}-\frac {3\,\sqrt {x^4+5}}{4\,x^4}-\frac {{\left (x^4+5\right )}^{3/2}}{15\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.96, size = 63, normalized size = 1.09 \[ - \frac {\sqrt {1 + \frac {5}{x^{4}}}}{15} - \frac {3 \sqrt {5} \operatorname {asinh}{\left (\frac {\sqrt {5}}{x^{2}} \right )}}{20} - \frac {3 \sqrt {1 + \frac {5}{x^{4}}}}{4 x^{2}} - \frac {\sqrt {1 + \frac {5}{x^{4}}}}{3 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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