Optimal. Leaf size=30 \[ \frac {2 \sqrt [4]{x^4-x^2} \left (4 x^4+x^2-5\right )}{45 x^5} \]
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Rubi [A] time = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.37, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2016, 2014} \begin {gather*} \frac {2 \left (x^4-x^2\right )^{5/4}}{9 x^7}+\frac {8 \left (x^4-x^2\right )^{5/4}}{45 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{-x^2+x^4}}{x^6} \, dx &=\frac {2 \left (-x^2+x^4\right )^{5/4}}{9 x^7}+\frac {4}{9} \int \frac {\sqrt [4]{-x^2+x^4}}{x^4} \, dx\\ &=\frac {2 \left (-x^2+x^4\right )^{5/4}}{9 x^7}+\frac {8 \left (-x^2+x^4\right )^{5/4}}{45 x^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.90 \begin {gather*} \frac {2 \left (x^2 \left (x^2-1\right )\right )^{5/4} \left (4 x^2+5\right )}{45 x^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 30, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{x^4-x^2} \left (4 x^4+x^2-5\right )}{45 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 26, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (4 \, x^{4} + x^{2} - 5\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}}{45 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 30, normalized size = 1.00 \begin {gather*} \frac {2}{9} \, {\left (\frac {1}{x^{2}} - 1\right )}^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - \frac {2}{5} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {2 \left (-1+x \right ) \left (1+x \right ) \left (4 x^{2}+5\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{45 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 27, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (4 \, x^{5} + x^{3} - 5 \, x\right )} {\left (x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}}}{45 \, x^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 26, normalized size = 0.87 \begin {gather*} \frac {2\,{\left (x^4-x^2\right )}^{1/4}\,\left (4\,x^4+x^2-5\right )}{45\,x^5} \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 {\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )}}{x^{6}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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