Optimal. Leaf size=60 \[ \frac {3}{10} \left (x^4+5\right )^{5/2}+\frac {75}{8} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+\frac {1}{4} x^2 \left (x^4+5\right )^{3/2}+\frac {15}{8} x^2 \sqrt {x^4+5} \]
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Rubi [A] time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1248, 641, 195, 215} \[ \frac {3}{10} \left (x^4+5\right )^{5/2}+\frac {1}{4} x^2 \left (x^4+5\right )^{3/2}+\frac {15}{8} x^2 \sqrt {x^4+5}+\frac {75}{8} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 641
Rule 1248
Rubi steps
\begin {align*} \int x \left (2+3 x^2\right ) \left (5+x^4\right )^{3/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (2+3 x) \left (5+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac {3}{10} \left (5+x^4\right )^{5/2}+\operatorname {Subst}\left (\int \left (5+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac {1}{4} x^2 \left (5+x^4\right )^{3/2}+\frac {3}{10} \left (5+x^4\right )^{5/2}+\frac {15}{4} \operatorname {Subst}\left (\int \sqrt {5+x^2} \, dx,x,x^2\right )\\ &=\frac {15}{8} x^2 \sqrt {5+x^4}+\frac {1}{4} x^2 \left (5+x^4\right )^{3/2}+\frac {3}{10} \left (5+x^4\right )^{5/2}+\frac {75}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {15}{8} x^2 \sqrt {5+x^4}+\frac {1}{4} x^2 \left (5+x^4\right )^{3/2}+\frac {3}{10} \left (5+x^4\right )^{5/2}+\frac {75}{8} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 0.93 \[ \frac {75}{8} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+\frac {1}{2} \sqrt {x^4+5} \left (\frac {3 x^8}{5}+\frac {x^6}{2}+6 x^4+\frac {25 x^2}{4}+15\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 48, normalized size = 0.80 \[ \frac {1}{40} \, {\left (12 \, x^{8} + 10 \, x^{6} + 120 \, x^{4} + 125 \, x^{2} + 300\right )} \sqrt {x^{4} + 5} - \frac {75}{8} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 57, normalized size = 0.95 \[ \frac {1}{8} \, {\left (2 \, x^{4} + 5\right )} \sqrt {x^{4} + 5} x^{2} + \frac {3}{10} \, {\left (x^{4} + 5\right )}^{\frac {5}{2}} + \frac {5}{2} \, \sqrt {x^{4} + 5} x^{2} - \frac {75}{8} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 0.77 \[ \frac {\sqrt {x^{4}+5}\, x^{6}}{4}+\frac {25 \sqrt {x^{4}+5}\, x^{2}}{8}+\frac {75 \arcsinh \left (\frac {\sqrt {5}\, x^{2}}{5}\right )}{8}+\frac {3 \left (x^{4}+5\right )^{\frac {5}{2}}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.60, size = 95, normalized size = 1.58 \[ \frac {3}{10} \, {\left (x^{4} + 5\right )}^{\frac {5}{2}} + \frac {25 \, {\left (\frac {3 \, \sqrt {x^{4} + 5}}{x^{2}} - \frac {5 \, {\left (x^{4} + 5\right )}^{\frac {3}{2}}}{x^{6}}\right )}}{8 \, {\left (\frac {2 \, {\left (x^{4} + 5\right )}}{x^{4}} - \frac {{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} + \frac {75}{16} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) - \frac {75}{16} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 42, normalized size = 0.70 \[ \frac {75\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{8}+\sqrt {x^4+5}\,\left (\frac {3\,x^8}{10}+\frac {x^6}{4}+3\,x^4+\frac {25\,x^2}{8}+\frac {15}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 8.19, size = 109, normalized size = 1.82 \[ \frac {x^{10}}{4 \sqrt {x^{4} + 5}} + \frac {3 x^{8} \sqrt {x^{4} + 5}}{10} + \frac {35 x^{6}}{8 \sqrt {x^{4} + 5}} + \frac {x^{4} \sqrt {x^{4} + 5}}{2} + \frac {125 x^{2}}{8 \sqrt {x^{4} + 5}} + \frac {5 \left (x^{4} + 5\right )^{\frac {3}{2}}}{2} - 5 \sqrt {x^{4} + 5} + \frac {75 \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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