Optimal. Leaf size=16 \[ \frac {\left (x^8+1\right )^{3/2}}{6 x^6} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {449} \begin {gather*} \frac {\left (x^8+1\right )^{3/2}}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin {align*} \int \frac {\left (-1+x^8\right ) \sqrt {1+x^8}}{x^7} \, dx &=\frac {\left (1+x^8\right )^{3/2}}{6 x^6}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (x^8+1\right )^{3/2}}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.42, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^8\right )^{3/2}}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^{8} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.69, size = 25, normalized size = 1.56 \begin {gather*} \frac {1}{6} \, \sqrt {x^{8} + 1} x^{2} + \frac {\sqrt {\frac {1}{x^{8}} + 1}}{6 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 13, normalized size = 0.81
method | result | size |
gosper | \(\frac {\left (x^{8}+1\right )^{\frac {3}{2}}}{6 x^{6}}\) | \(13\) |
trager | \(\frac {\left (x^{8}+1\right )^{\frac {3}{2}}}{6 x^{6}}\) | \(13\) |
risch | \(\frac {x^{16}+2 x^{8}+1}{6 x^{6} \sqrt {x^{8}+1}}\) | \(23\) |
meijerg | \(\frac {\hypergeom \left (\left [-\frac {3}{4}, -\frac {1}{2}\right ], \left [\frac {1}{4}\right ], -x^{8}\right )}{6 x^{6}}+\frac {\hypergeom \left (\left [-\frac {1}{2}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], -x^{8}\right ) x^{2}}{2}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^{8} + 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 12, normalized size = 0.75 \begin {gather*} \frac {{\left (x^8+1\right )}^{3/2}}{6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.20, size = 66, normalized size = 4.12 \begin {gather*} \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 )} - \frac {\Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{2} \\ \frac {1}{4} \end {matrix}\middle | {x^{8} e^{i \pi }} \right )}}{8 x^{6} \Gamma \left (\frac {1}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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