Optimal. Leaf size=28 \[ -\frac {3 \left (x^3+x\right )^{2/3} \left (9 x^4-6 x^2+5\right )}{80 x^6} \]
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Rubi [A] time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2016, 2014} \begin {gather*} -\frac {3 \left (x^3+x\right )^{2/3}}{16 x^6}+\frac {9 \left (x^3+x\right )^{2/3}}{40 x^4}-\frac {27 \left (x^3+x\right )^{2/3}}{80 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx &=-\frac {3 \left (x+x^3\right )^{2/3}}{16 x^6}-\frac {3}{4} \int \frac {1}{x^4 \sqrt [3]{x+x^3}} \, dx\\ &=-\frac {3 \left (x+x^3\right )^{2/3}}{16 x^6}+\frac {9 \left (x+x^3\right )^{2/3}}{40 x^4}+\frac {9}{20} \int \frac {1}{x^2 \sqrt [3]{x+x^3}} \, dx\\ &=-\frac {3 \left (x+x^3\right )^{2/3}}{16 x^6}+\frac {9 \left (x+x^3\right )^{2/3}}{40 x^4}-\frac {27 \left (x+x^3\right )^{2/3}}{80 x^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x^3+x\right )^{2/3} \left (9 x^4-6 x^2+5\right )}{80 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.22, size = 28, normalized size = 1.00 \begin {gather*} -\frac {3 \left (x^3+x\right )^{2/3} \left (9 x^4-6 x^2+5\right )}{80 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 24, normalized size = 0.86 \begin {gather*} -\frac {3 \, {\left (9 \, x^{4} - 6 \, x^{2} + 5\right )} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{80 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 28, normalized size = 1.00 \begin {gather*} -\frac {3}{16} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {8}{3}} + \frac {3}{5} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{3}} - \frac {3}{4} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} \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 = 1.07 \begin {gather*} -\frac {3 \left (x^{2}+1\right ) \left (9 x^{4}-6 x^{2}+5\right )}{80 x^{5} \left (x^{3}+x \right )^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 31, normalized size = 1.11 \begin {gather*} -\frac {3 \, {\left (9 \, x^{7} + 3 \, x^{5} - x^{3} + 5 \, x\right )}}{80 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} x^{\frac {19}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 24, normalized size = 0.86 \begin {gather*} -\frac {3\,{\left (x^3+x\right )}^{2/3}\,\left (9\,x^4-6\,x^2+5\right )}{80\,x^6} \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 {1}{x^{6} \sqrt [3]{x \left (x^{2} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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