Optimal. Leaf size=47 \[ \frac {3}{20} \sinh ^{-1}\left (x^{5/2}\right )+\frac {1}{10} \sqrt {x^5+1} x^{15/2}-\frac {3}{20} \sqrt {x^5+1} x^{5/2} \]
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Rubi [A] time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {321, 329, 275, 215} \[ \frac {1}{10} \sqrt {x^5+1} x^{15/2}-\frac {3}{20} \sqrt {x^5+1} x^{5/2}+\frac {3}{20} \sinh ^{-1}\left (x^{5/2}\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 275
Rule 321
Rule 329
Rubi steps
\begin {align*} \int \frac {x^{23/2}}{\sqrt {1+x^5}} \, dx &=\frac {1}{10} x^{15/2} \sqrt {1+x^5}-\frac {3}{4} \int \frac {x^{13/2}}{\sqrt {1+x^5}} \, dx\\ &=-\frac {3}{20} x^{5/2} \sqrt {1+x^5}+\frac {1}{10} x^{15/2} \sqrt {1+x^5}+\frac {3}{8} \int \frac {x^{3/2}}{\sqrt {1+x^5}} \, dx\\ &=-\frac {3}{20} x^{5/2} \sqrt {1+x^5}+\frac {1}{10} x^{15/2} \sqrt {1+x^5}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {3}{20} x^{5/2} \sqrt {1+x^5}+\frac {1}{10} x^{15/2} \sqrt {1+x^5}+\frac {3}{20} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^{5/2}\right )\\ &=-\frac {3}{20} x^{5/2} \sqrt {1+x^5}+\frac {1}{10} x^{15/2} \sqrt {1+x^5}+\frac {3}{20} \sinh ^{-1}\left (x^{5/2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 0.74 \[ \frac {1}{20} \left (3 \sinh ^{-1}\left (x^{5/2}\right )+\sqrt {x^5+1} \left (2 x^5-3\right ) x^{5/2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 46, normalized size = 0.98 \[ \frac {1}{20} \, {\left (2 \, x^{7} - 3 \, x^{2}\right )} \sqrt {x^{5} + 1} \sqrt {x} + \frac {3}{40} \, \log \left (2 \, x^{5} + 2 \, \sqrt {x^{5} + 1} x^{\frac {5}{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 36, normalized size = 0.77 \[ \frac {1}{20} \, {\left (2 \, x^{5} - 3\right )} \sqrt {x^{5} + 1} x^{\frac {5}{2}} - \frac {3}{20} \, \log \left (-x^{\frac {5}{2}} + \sqrt {x^{5} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 46, normalized size = 0.98 \[ \frac {\left (2 x^{5}-3\right ) \sqrt {x^{5}+1}\, x^{\frac {5}{2}}}{20}+\frac {3 \sqrt {\left (x^{5}+1\right ) x}\, \arcsinh \left (x^{\frac {5}{2}}\right )}{20 \sqrt {x^{5}+1}\, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.12, size = 86, normalized size = 1.83 \[ -\frac {\frac {5 \, \sqrt {x^{5} + 1}}{x^{\frac {5}{2}}} - \frac {3 \, {\left (x^{5} + 1\right )}^{\frac {3}{2}}}{x^{\frac {15}{2}}}}{20 \, {\left (\frac {2 \, {\left (x^{5} + 1\right )}}{x^{5}} - \frac {{\left (x^{5} + 1\right )}^{2}}{x^{10}} - 1\right )}} + \frac {3}{40} \, \log \left (\frac {\sqrt {x^{5} + 1}}{x^{\frac {5}{2}}} + 1\right ) - \frac {3}{40} \, \log \left (\frac {\sqrt {x^{5} + 1}}{x^{\frac {5}{2}}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^{23/2}}{\sqrt {x^5+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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