Optimal. Leaf size=67 \[ \frac {3}{10} \left (x^4+5\right )^{3/2} x^4-\frac {25}{8} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-\frac {5}{8} \sqrt {x^4+5} x^2-\frac {1}{4} \left (4-x^2\right ) \left (x^4+5\right )^{3/2} \]
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Rubi [A] time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1252, 833, 780, 195, 215} \[ \frac {3}{10} \left (x^4+5\right )^{3/2} x^4-\frac {5}{8} \sqrt {x^4+5} x^2-\frac {1}{4} \left (4-x^2\right ) \left (x^4+5\right )^{3/2}-\frac {25}{8} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 780
Rule 833
Rule 1252
Rubi steps
\begin {align*} \int x^5 \left (2+3 x^2\right ) \sqrt {5+x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 (2+3 x) \sqrt {5+x^2} \, dx,x,x^2\right )\\ &=\frac {3}{10} x^4 \left (5+x^4\right )^{3/2}+\frac {1}{10} \operatorname {Subst}\left (\int x (-30+10 x) \sqrt {5+x^2} \, dx,x,x^2\right )\\ &=\frac {3}{10} x^4 \left (5+x^4\right )^{3/2}-\frac {1}{4} \left (4-x^2\right ) \left (5+x^4\right )^{3/2}-\frac {5}{4} \operatorname {Subst}\left (\int \sqrt {5+x^2} \, dx,x,x^2\right )\\ &=-\frac {5}{8} x^2 \sqrt {5+x^4}+\frac {3}{10} x^4 \left (5+x^4\right )^{3/2}-\frac {1}{4} \left (4-x^2\right ) \left (5+x^4\right )^{3/2}-\frac {25}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {5}{8} x^2 \sqrt {5+x^4}+\frac {3}{10} x^4 \left (5+x^4\right )^{3/2}-\frac {1}{4} \left (4-x^2\right ) \left (5+x^4\right )^{3/2}-\frac {25}{8} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 50, normalized size = 0.75 \[ \frac {1}{40} \sqrt {x^4+5} \left (12 x^8+10 x^6+20 x^4+25 x^2-200\right )-\frac {25}{8} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 48, normalized size = 0.72 \[ \frac {1}{40} \, {\left (12 \, x^{8} + 10 \, x^{6} + 20 \, x^{4} + 25 \, x^{2} - 200\right )} \sqrt {x^{4} + 5} + \frac {25}{8} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 54, normalized size = 0.81 \[ \frac {1}{8} \, {\left (2 \, x^{4} + 5\right )} \sqrt {x^{4} + 5} x^{2} + \frac {3}{10} \, {\left (x^{4} + 5\right )}^{\frac {5}{2}} - \frac {5}{2} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} + \frac {25}{8} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 53, normalized size = 0.79 \[ \frac {\left (x^{4}+5\right )^{\frac {3}{2}} x^{2}}{4}-\frac {5 \sqrt {x^{4}+5}\, x^{2}}{8}-\frac {25 \arcsinh \left (\frac {\sqrt {5}\, x^{2}}{5}\right )}{8}+\frac {\left (x^{4}+5\right )^{\frac {3}{2}} \left (3 x^{4}-10\right )}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.01, size = 102, normalized size = 1.52 \[ \frac {3}{10} \, {\left (x^{4} + 5\right )}^{\frac {5}{2}} - \frac {5}{2} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} - \frac {25 \, {\left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + \frac {{\left (x^{4} + 5\right )}^{\frac {3}{2}}}{x^{6}}\right )}}{8 \, {\left (\frac {2 \, {\left (x^{4} + 5\right )}}{x^{4}} - \frac {{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} - \frac {25}{16} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) + \frac {25}{16} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 42, normalized size = 0.63 \[ \sqrt {x^4+5}\,\left (\frac {3\,x^8}{10}+\frac {x^6}{4}+\frac {x^4}{2}+\frac {5\,x^2}{8}-5\right )-\frac {25\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.01, size = 97, normalized size = 1.45 \[ \frac {x^{10}}{4 \sqrt {x^{4} + 5}} + \frac {3 x^{8} \sqrt {x^{4} + 5}}{10} + \frac {15 x^{6}}{8 \sqrt {x^{4} + 5}} + \frac {x^{4} \sqrt {x^{4} + 5}}{2} + \frac {25 x^{2}}{8 \sqrt {x^{4} + 5}} - 5 \sqrt {x^{4} + 5} - \frac {25 \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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