Optimal. Leaf size=62 \[ \frac {1}{6} \sqrt {x^8+1} x^2+\frac {\left (x^4+1\right ) \sqrt {\frac {x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{6 \sqrt {x^8+1}} \]
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Rubi [A] time = 0.02, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {275, 195, 220} \[ \frac {1}{6} \sqrt {x^8+1} x^2+\frac {\left (x^4+1\right ) \sqrt {\frac {x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{6 \sqrt {x^8+1}} \]
Antiderivative was successfully verified.
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Rule 195
Rule 220
Rule 275
Rubi steps
\begin {align*} \int x \sqrt {1+x^8} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \sqrt {1+x^4} \, dx,x,x^2\right )\\ &=\frac {1}{6} x^2 \sqrt {1+x^8}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,x^2\right )\\ &=\frac {1}{6} x^2 \sqrt {1+x^8}+\frac {\left (1+x^4\right ) \sqrt {\frac {1+x^8}{\left (1+x^4\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac {1}{2}\right )}{6 \sqrt {1+x^8}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 22, normalized size = 0.35 \[ \frac {1}{2} x^2 \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-x^8\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {x^{8} + 1} x, x\right ) \]
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 \[ \int \sqrt {x^{8} + 1} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.15, size = 30, normalized size = 0.48 \[ \frac {x^{2} \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], -x^{8}\right )}{3}+\frac {\sqrt {x^{8}+1}\, x^{2}}{6} \]
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 \[ \int \sqrt {x^{8} + 1} x\,{d x} \]
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 \[ \int x\,\sqrt {x^8+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.76, size = 31, normalized size = 0.50 \[ \frac {x^{2} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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