Optimal. Leaf size=16 \[ -\frac {2 \sqrt {x+x^4}}{3 x^2} \]
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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 {2 \sqrt {x^4+x}}{3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx &=-\frac {2 \sqrt {x+x^4}}{3 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 16, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {x+x^4}}{3 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.40, size = 13, normalized size = 0.81
method | result | size |
default | \(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}\) | \(13\) |
trager | \(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}\) | \(13\) |
meijerg | \(-\frac {2 \sqrt {x^{3}+1}}{3 x^{\frac {3}{2}}}\) | \(13\) |
elliptic | \(-\frac {2 \sqrt {x^{4}+x}}{3 x^{2}}\) | \(13\) |
risch | \(-\frac {2 \left (x^{3}+1\right )}{3 x \sqrt {x \left (x^{3}+1\right )}}\) | \(20\) |
gosper | \(-\frac {2 \left (1+x \right ) \left (x^{2}-x +1\right )}{3 x \sqrt {x^{4}+x}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs.
\(2 (12) = 24\).
time = 0.47, size = 25, normalized size = 1.56 \begin {gather*} -\frac {2 \, {\left (x^{4} + x\right )}}{3 \, \sqrt {x^{2} - x + 1} \sqrt {x + 1} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2 \, \sqrt {x^{4} + x}}{3 \, x^{2}} \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^{2} \sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 9, normalized size = 0.56 \begin {gather*} -\frac {2}{3} \, \sqrt {\frac {1}{x^{3}} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 12, normalized size = 0.75 \begin {gather*} -\frac {2\,\sqrt {x^4+x}}{3\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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