Optimal. Leaf size=82 \[ -\frac {9}{4} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right )+\sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-\frac {\left (4-9 x^2\right ) \sqrt {x^4+5}}{4 x^2}-\frac {\left (9 x^2+4\right ) \left (x^4+5\right )^{3/2}}{12 x^6} \]
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Rubi [A] time = 0.07, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1252, 811, 813, 844, 215, 266, 63, 207} \[ -\frac {\left (9 x^2+4\right ) \left (x^4+5\right )^{3/2}}{12 x^6}-\frac {\left (4-9 x^2\right ) \sqrt {x^4+5}}{4 x^2}+\sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-\frac {9}{4} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
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Rule 63
Rule 207
Rule 215
Rule 266
Rule 811
Rule 813
Rule 844
Rule 1252
Rubi steps
\begin {align*} \int \frac {\left (2+3 x^2\right ) \left (5+x^4\right )^{3/2}}{x^7} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(2+3 x) \left (5+x^2\right )^{3/2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac {\left (4+9 x^2\right ) \left (5+x^4\right )^{3/2}}{12 x^6}-\frac {1}{40} \operatorname {Subst}\left (\int \frac {(-40-90 x) \sqrt {5+x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (4-9 x^2\right ) \sqrt {5+x^4}}{4 x^2}-\frac {\left (4+9 x^2\right ) \left (5+x^4\right )^{3/2}}{12 x^6}+\frac {1}{80} \operatorname {Subst}\left (\int \frac {900+80 x}{x \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (4-9 x^2\right ) \sqrt {5+x^4}}{4 x^2}-\frac {\left (4+9 x^2\right ) \left (5+x^4\right )^{3/2}}{12 x^6}+\frac {45}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {5+x^2}} \, dx,x,x^2\right )+\operatorname {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\left (4-9 x^2\right ) \sqrt {5+x^4}}{4 x^2}-\frac {\left (4+9 x^2\right ) \left (5+x^4\right )^{3/2}}{12 x^6}+\sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+\frac {45}{8} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {5+x}} \, dx,x,x^4\right )\\ &=-\frac {\left (4-9 x^2\right ) \sqrt {5+x^4}}{4 x^2}-\frac {\left (4+9 x^2\right ) \left (5+x^4\right )^{3/2}}{12 x^6}+\sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+\frac {45}{4} \operatorname {Subst}\left (\int \frac {1}{-5+x^2} \, dx,x,\sqrt {5+x^4}\right )\\ &=-\frac {\left (4-9 x^2\right ) \sqrt {5+x^4}}{4 x^2}-\frac {\left (4+9 x^2\right ) \left (5+x^4\right )^{3/2}}{12 x^6}+\sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-\frac {9}{4} \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 60, normalized size = 0.73 \[ \frac {3}{250} \left (x^4+5\right )^{5/2} \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {x^4}{5}+1\right )-\frac {5 \sqrt {5} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {x^4}{5}\right )}{3 x^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 82, normalized size = 1.00 \[ \frac {27 \, \sqrt {5} x^{6} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{x^{2}}\right ) - 12 \, x^{6} \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) - 16 \, x^{6} + {\left (18 \, x^{6} - 16 \, x^{4} - 45 \, x^{2} - 20\right )} \sqrt {x^{4} + 5}}{12 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 158, normalized size = 1.93 \[ \frac {9}{4} \, \sqrt {5} \log \left (-\frac {x^{2} + \sqrt {5} - \sqrt {x^{4} + 5}}{x^{2} - \sqrt {5} - \sqrt {x^{4} + 5}}\right ) + \frac {3}{2} \, \sqrt {x^{4} + 5} + \frac {5 \, {\left (9 \, {\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{5} + 24 \, {\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{4} - 120 \, {\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{2} - 225 \, x^{2} + 225 \, \sqrt {x^{4} + 5} + 400\right )}}{6 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{2} - 5\right )}^{3}} - \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 73, normalized size = 0.89 \[ \arcsinh \left (\frac {\sqrt {5}\, x^{2}}{5}\right )-\frac {9 \sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )}{4}-\frac {4 \sqrt {x^{4}+5}}{3 x^{2}}-\frac {15 \sqrt {x^{4}+5}}{4 x^{4}}-\frac {5 \sqrt {x^{4}+5}}{3 x^{6}}+\frac {3 \sqrt {x^{4}+5}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 112, normalized size = 1.37 \[ \frac {9}{8} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{\sqrt {5} + \sqrt {x^{4} + 5}}\right ) + \frac {3}{2} \, \sqrt {x^{4} + 5} - \frac {\sqrt {x^{4} + 5}}{x^{2}} - \frac {15 \, \sqrt {x^{4} + 5}}{4 \, x^{4}} - \frac {{\left (x^{4} + 5\right )}^{\frac {3}{2}}}{3 \, x^{6}} + \frac {1}{2} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) - \frac {1}{2} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.95, size = 82, normalized size = 1.00 \[ \mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )+\frac {3\,\sqrt {x^4+5}}{2}+\sqrt {x^4+5}\,\left (\frac {2}{3\,x^2}-\frac {5}{3\,x^6}\right )-\frac {2\,\sqrt {x^4+5}}{x^2}-\frac {15\,\sqrt {x^4+5}}{4\,x^4}+\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {\sqrt {5}\,\sqrt {x^4+5}\,1{}\mathrm {i}}{5}\right )\,9{}\mathrm {i}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.56, size = 148, normalized size = 1.80 \[ - \frac {x^{2}}{\sqrt {x^{4} + 5}} - \frac {\sqrt {1 + \frac {5}{x^{4}}}}{3} + \frac {3 \sqrt {x^{4} + 5}}{2} + \frac {3 \sqrt {5} \log {\left (x^{4} \right )}}{4} - \frac {3 \sqrt {5} \log {\left (\sqrt {\frac {x^{4}}{5} + 1} + 1 \right )}}{2} - \frac {3 \sqrt {5} \operatorname {asinh}{\left (\frac {\sqrt {5}}{x^{2}} \right )}}{4} + \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )} - \frac {15 \sqrt {1 + \frac {5}{x^{4}}}}{4 x^{2}} - \frac {5}{x^{2} \sqrt {x^{4} + 5}} - \frac {5 \sqrt {1 + \frac {5}{x^{4}}}}{3 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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