Optimal. Leaf size=78 \[ \frac {5}{16} \left (16+9 x^2\right ) \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {225}{16} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-5 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1266, 829, 858,
221, 272, 65, 213} \begin {gather*} -5 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right )+\frac {225}{16} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+\frac {1}{24} \left (9 x^2+8\right ) \left (x^4+5\right )^{3/2}+\frac {5}{16} \left (9 x^2+16\right ) \sqrt {x^4+5} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 213
Rule 221
Rule 272
Rule 829
Rule 858
Rule 1266
Rubi steps
\begin {align*} \int \frac {\left (2+3 x^2\right ) \left (5+x^4\right )^{3/2}}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(2+3 x) \left (5+x^2\right )^{3/2}}{x} \, dx,x,x^2\right )\\ &=\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {1}{8} \text {Subst}\left (\int \frac {(40+45 x) \sqrt {5+x^2}}{x} \, dx,x,x^2\right )\\ &=\frac {5}{16} \left (16+9 x^2\right ) \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {1}{16} \text {Subst}\left (\int \frac {400+225 x}{x \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {5}{16} \left (16+9 x^2\right ) \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {225}{16} \text {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )+25 \text {Subst}\left (\int \frac {1}{x \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {5}{16} \left (16+9 x^2\right ) \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {225}{16} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+\frac {25}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {5+x}} \, dx,x,x^4\right )\\ &=\frac {5}{16} \left (16+9 x^2\right ) \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {225}{16} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )+25 \text {Subst}\left (\int \frac {1}{-5+x^2} \, dx,x,\sqrt {5+x^4}\right )\\ &=\frac {5}{16} \left (16+9 x^2\right ) \sqrt {5+x^4}+\frac {1}{24} \left (8+9 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {225}{16} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-5 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 77, normalized size = 0.99 \begin {gather*} \frac {1}{48} \left (\sqrt {5+x^4} \left (320+225 x^2+16 x^4+18 x^6\right )+675 \tanh ^{-1}\left (\frac {x^2}{\sqrt {5+x^4}}\right )+480 \sqrt {5} \tanh ^{-1}\left (\frac {x^2-\sqrt {5+x^4}}{\sqrt {5}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 75, normalized size = 0.96
method | result | size |
trager | \(\left (\frac {3}{8} x^{6}+\frac {1}{3} x^{4}+\frac {75}{16} x^{2}+\frac {20}{3}\right ) \sqrt {x^{4}+5}-5 \RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-5\right )+\sqrt {x^{4}+5}}{x^{2}}\right )-\frac {225 \ln \left (x^{2}-\sqrt {x^{4}+5}\right )}{16}\) | \(70\) |
default | \(\frac {3 x^{6} \sqrt {x^{4}+5}}{8}+\frac {75 x^{2} \sqrt {x^{4}+5}}{16}+\frac {225 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{16}+\frac {x^{4} \sqrt {x^{4}+5}}{3}+\frac {20 \sqrt {x^{4}+5}}{3}-5 \sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )\) | \(75\) |
elliptic | \(\frac {3 x^{6} \sqrt {x^{4}+5}}{8}+\frac {75 x^{2} \sqrt {x^{4}+5}}{16}+\frac {225 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{16}+\frac {x^{4} \sqrt {x^{4}+5}}{3}+\frac {20 \sqrt {x^{4}+5}}{3}-5 \sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )\) | \(75\) |
meijerg | \(\frac {15 \sqrt {5}\, \left (-\frac {32 \sqrt {\pi }}{9}+\frac {2 \sqrt {\pi }\, \left (\frac {4 x^{4}}{5}+16\right ) \sqrt {1+\frac {x^{4}}{5}}}{9}-\frac {8 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{4}}{5}}}{2}\right )}{3}+\frac {4 \left (\frac {8}{3}-2 \ln \left (2\right )+4 \ln \left (x \right )-\ln \left (5\right )\right ) \sqrt {\pi }}{3}\right )}{8 \sqrt {\pi }}+\frac {\frac {15 \sqrt {\pi }\, x^{2} \sqrt {5}\, \left (\frac {x^{4}}{20}+\frac {5}{8}\right ) \sqrt {1+\frac {x^{4}}{5}}}{2}+\frac {225 \sqrt {\pi }\, \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{16}}{\sqrt {\pi }}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 138 vs.
\(2 (62) = 124\).
time = 0.49, size = 138, normalized size = 1.77 \begin {gather*} \frac {1}{3} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} + \frac {5}{2} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{\sqrt {5} + \sqrt {x^{4} + 5}}\right ) + 5 \, \sqrt {x^{4} + 5} + \frac {75 \, {\left (\frac {3 \, \sqrt {x^{4} + 5}}{x^{2}} - \frac {5 \, {\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 {225}{32} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) - \frac {225}{32} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 67, normalized size = 0.86 \begin {gather*} \frac {1}{48} \, {\left (18 \, x^{6} + 16 \, x^{4} + 225 \, x^{2} + 320\right )} \sqrt {x^{4} + 5} + 5 \, \sqrt {5} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{x^{2}}\right ) - \frac {225}{16} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 16.34, size = 114, normalized size = 1.46 \begin {gather*} \frac {3 x^{10}}{8 \sqrt {x^{4} + 5}} + \frac {105 x^{6}}{16 \sqrt {x^{4} + 5}} + \frac {375 x^{2}}{16 \sqrt {x^{4} + 5}} + \frac {\left (x^{4} + 5\right )^{\frac {3}{2}}}{3} + 5 \sqrt {x^{4} + 5} + \frac {5 \sqrt {5} \log {\left (x^{4} \right )}}{2} - 5 \sqrt {5} \log {\left (\sqrt {\frac {x^{4}}{5} + 1} + 1 \right )} + \frac {225 \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.03, size = 90, normalized size = 1.15 \begin {gather*} \frac {1}{48} \, \sqrt {x^{4} + 5} {\left ({\left (2 \, {\left (9 \, x^{2} + 8\right )} x^{2} + 225\right )} x^{2} + 320\right )} + 5 \, \sqrt {5} \log \left (-\frac {x^{2} + \sqrt {5} - \sqrt {x^{4} + 5}}{x^{2} - \sqrt {5} - \sqrt {x^{4} + 5}}\right ) - \frac {225}{16} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 55, normalized size = 0.71 \begin {gather*} \frac {225\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{16}-5\,\sqrt {5}\,\mathrm {atanh}\left (\frac {\sqrt {5}\,\sqrt {x^4+5}}{5}\right )+\sqrt {x^4+5}\,\left (\frac {3\,x^6}{8}+\frac {x^4}{3}+\frac {75\,x^2}{16}+\frac {20}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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