Optimal. Leaf size=16 \[ -\frac {3 \left (x+x^3\right )^{4/3}}{8 x^4} \]
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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2039}
\begin {gather*} -\frac {3 \left (x^3+x\right )^{4/3}}{8 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{x+x^3}}{x^4} \, dx &=-\frac {3 \left (x+x^3\right )^{4/3}}{8 x^4}\\ \end {align*}
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Mathematica [A]
time = 0.46, size = 21, normalized size = 1.31 \begin {gather*} -\frac {3 \left (1+x^2\right ) \sqrt [3]{x+x^3}}{8 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 13, normalized size = 0.81
method | result | size |
meijerg | \(-\frac {3 \left (x^{2}+1\right )^{\frac {4}{3}}}{8 x^{\frac {8}{3}}}\) | \(13\) |
gosper | \(-\frac {3 \left (x^{2}+1\right ) \left (x^{3}+x \right )^{\frac {1}{3}}}{8 x^{3}}\) | \(18\) |
trager | \(-\frac {3 \left (x^{2}+1\right ) \left (x^{3}+x \right )^{\frac {1}{3}}}{8 x^{3}}\) | \(18\) |
risch | \(-\frac {3 \left (\left (x^{2}+1\right ) x \right )^{\frac {1}{3}} \left (x^{4}+2 x^{2}+1\right )}{8 x^{3} \left (x^{2}+1\right )}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 17, normalized size = 1.06 \begin {gather*} -\frac {3 \, {\left (x^{3} + x\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}{8 \, x^{\frac {11}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 17, normalized size = 1.06 \begin {gather*} -\frac {3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{8 \, 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 {\sqrt [3]{x \left (x^{2} + 1\right )}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 9, normalized size = 0.56 \begin {gather*} -\frac {3}{8} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 27, normalized size = 1.69 \begin {gather*} -\frac {3\,{\left (x^3+x\right )}^{1/3}+3\,x^2\,{\left (x^3+x\right )}^{1/3}}{8\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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