Optimal. Leaf size=22 \[ \frac {1}{5} x^5 \, _2F_1\left (\frac {1}{2},\frac {5}{8};\frac {13}{8};-x^8\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {371}
\begin {gather*} \frac {1}{5} x^5 \, _2F_1\left (\frac {1}{2},\frac {5}{8};\frac {13}{8};-x^8\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rubi steps
\begin {align*} \int \frac {x^4}{\sqrt {1+x^8}} \, dx &=\frac {1}{5} x^5 \, _2F_1\left (\frac {1}{2},\frac {5}{8};\frac {13}{8};-x^8\right )\\ \end {align*}
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Mathematica [A]
time = 10.03, size = 22, normalized size = 1.00 \begin {gather*} \frac {1}{5} x^5 \, _2F_1\left (\frac {1}{2},\frac {5}{8};\frac {13}{8};-x^8\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 17, normalized size = 0.77
method | result | size |
meijerg | \(\frac {x^{5} \hypergeom \left (\left [\frac {1}{2}, \frac {5}{8}\right ], \left [\frac {13}{8}\right ], -x^{8}\right )}{5}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.35, size = 13, normalized size = 0.59 \begin {gather*} {\rm integral}\left (\frac {x^{4}}{\sqrt {x^{8} + 1}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.32, size = 29, normalized size = 1.32 \begin {gather*} \frac {x^{5} \Gamma \left (\frac {5}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{8} \\ \frac {13}{8} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac {13}{8}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {x^4}{\sqrt {x^8+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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