Optimal. Leaf size=51 \[ -\frac {75}{16} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-\frac {15}{16} \sqrt {x^4+5} x^2+\frac {1}{24} \left (9 x^2+8\right ) \left (x^4+5\right )^{3/2} \]
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Rubi [A] time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1252, 780, 195, 215} \[ -\frac {15}{16} \sqrt {x^4+5} x^2+\frac {1}{24} \left (9 x^2+8\right ) \left (x^4+5\right )^{3/2}-\frac {75}{16} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 780
Rule 1252
Rubi steps
\begin {align*} \int x^3 \left (2+3 x^2\right ) \sqrt {5+x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (2+3 x) \sqrt {5+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}-\frac {15}{8} \operatorname {Subst}\left (\int \sqrt {5+x^2} \, dx,x,x^2\right )\\ &=-\frac {15}{16} x^2 \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}-\frac {75}{16} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {15}{16} x^2 \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}-\frac {75}{16} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 44, normalized size = 0.86 \[ \frac {1}{48} \left (\sqrt {x^4+5} \left (18 x^6+16 x^4+45 x^2+80\right )-225 \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 43, normalized size = 0.84 \[ \frac {1}{48} \, {\left (18 \, x^{6} + 16 \, x^{4} + 45 \, x^{2} + 80\right )} \sqrt {x^{4} + 5} + \frac {75}{16} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 45, normalized size = 0.88 \[ \frac {3}{16} \, {\left (2 \, x^{4} + 5\right )} \sqrt {x^{4} + 5} x^{2} + \frac {1}{3} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} + \frac {75}{16} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 0.90 \[ \frac {3 \left (x^{4}+5\right )^{\frac {3}{2}} x^{2}}{8}-\frac {15 \sqrt {x^{4}+5}\, x^{2}}{16}-\frac {75 \arcsinh \left (\frac {\sqrt {5}\, x^{2}}{5}\right )}{16}+\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.02, size = 93, normalized size = 1.82 \[ \frac {1}{3} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} - \frac {75 \, {\left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + \frac {{\left (x^{4} + 5\right )}^{\frac {3}{2}}}{x^{6}}\right )}}{16 \, {\left (\frac {2 \, {\left (x^{4} + 5\right )}}{x^{4}} - \frac {{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} - \frac {75}{32} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) + \frac {75}{32} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 37, normalized size = 0.73 \[ \sqrt {x^4+5}\,\left (\frac {3\,x^6}{8}+\frac {x^4}{3}+\frac {15\,x^2}{16}+\frac {5}{3}\right )-\frac {75\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.31, size = 70, normalized size = 1.37 \[ \frac {3 x^{10}}{8 \sqrt {x^{4} + 5}} + \frac {45 x^{6}}{16 \sqrt {x^{4} + 5}} + \frac {75 x^{2}}{16 \sqrt {x^{4} + 5}} + \frac {\left (x^{4} + 5\right )^{\frac {3}{2}}}{3} - \frac {75 \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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