Optimal. Leaf size=23 \[ \frac {2 \left (-1+2 x^3\right ) \sqrt {x+x^4}}{9 x^5} \]
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Rubi [A]
time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2041, 2039}
\begin {gather*} \frac {4 \sqrt {x^4+x}}{9 x^2}-\frac {2 \sqrt {x^4+x}}{9 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt {x+x^4}} \, dx &=-\frac {2 \sqrt {x+x^4}}{9 x^5}-\frac {2}{3} \int \frac {1}{x^2 \sqrt {x+x^4}} \, dx\\ &=-\frac {2 \sqrt {x+x^4}}{9 x^5}+\frac {4 \sqrt {x+x^4}}{9 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 \left (-1+2 x^3\right ) \sqrt {x+x^4}}{9 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.55, size = 26, normalized size = 1.13
method | result | size |
trager | \(\frac {2 \left (2 x^{3}-1\right ) \sqrt {x^{4}+x}}{9 x^{5}}\) | \(20\) |
meijerg | \(-\frac {2 \left (-2 x^{3}+1\right ) \sqrt {x^{3}+1}}{9 x^{\frac {9}{2}}}\) | \(20\) |
risch | \(\frac {\frac {2}{9} x^{3}-\frac {2}{9}+\frac {4}{9} x^{6}}{x^{4} \sqrt {x \left (x^{3}+1\right )}}\) | \(25\) |
default | \(-\frac {2 \sqrt {x^{4}+x}}{9 x^{5}}+\frac {4 \sqrt {x^{4}+x}}{9 x^{2}}\) | \(26\) |
elliptic | \(-\frac {2 \sqrt {x^{4}+x}}{9 x^{5}}+\frac {4 \sqrt {x^{4}+x}}{9 x^{2}}\) | \(26\) |
gosper | \(\frac {2 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (2 x^{3}-1\right )}{9 x^{4} \sqrt {x^{4}+x}}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 32, normalized size = 1.39 \begin {gather*} \frac {2 \, {\left (2 \, x^{7} + x^{4} - x\right )}}{9 \, \sqrt {x^{2} - x + 1} \sqrt {x + 1} x^{\frac {11}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 \, \sqrt {x^{4} + x} {\left (2 \, x^{3} - 1\right )}}{9 \, 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 {1}{x^{5} \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 = 0.39, size = 19, normalized size = 0.83 \begin {gather*} -\frac {2}{9} \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {3}{2}} + \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 = 0.16, size = 19, normalized size = 0.83 \begin {gather*} \frac {2\,\left (2\,x^3-1\right )\,\sqrt {x^4+x}}{9\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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