Optimal. Leaf size=18 \[ \frac {2 \left (x^6+x^2\right )^{3/4}}{3 x^3} \]
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Rubi [A] time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1590} \begin {gather*} \frac {2 \left (x^6+x^2\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin {align*} \int \frac {-1+x^4}{x^2 \sqrt [4]{x^2+x^6}} \, dx &=\frac {2 \left (x^2+x^6\right )^{3/4}}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^6+x^2\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^2+x^6\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 14, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{3 \, x^{3}} \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*} \int \frac {x^{4} - 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 15, normalized size = 0.83
method | result | size |
trager | \(\frac {2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(15\) |
gosper | \(\frac {\frac {2 x^{4}}{3}+\frac {2}{3}}{\left (x^{6}+x^{2}\right )^{\frac {1}{4}} x}\) | \(20\) |
risch | \(\frac {\frac {2 x^{4}}{3}+\frac {2}{3}}{x \left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\) | \(22\) |
meijerg | \(\frac {2 \hypergeom \left (\left [-\frac {3}{8}, \frac {1}{4}\right ], \left [\frac {5}{8}\right ], -x^{4}\right )}{3 x^{\frac {3}{2}}}+\frac {2 \hypergeom \left (\left [\frac {1}{4}, \frac {5}{8}\right ], \left [\frac {13}{8}\right ], -x^{4}\right ) x^{\frac {5}{2}}}{5}\) | \(34\) |
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*} \int \frac {x^{4} - 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 14, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (x^6+x^2\right )}^{3/4}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{2} \sqrt [4]{x^{2} \left (x^{4} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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