Optimal. Leaf size=27 \[ -\frac {2}{3} \sqrt {1+x^3}+\frac {2}{9} \left (1+x^3\right )^{3/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {272, 45}
\begin {gather*} \frac {2}{9} \left (x^3+1\right )^{3/2}-\frac {2 \sqrt {x^3+1}}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {1+x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {1}{\sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,x^3\right )\\ &=-\frac {2}{3} \sqrt {1+x^3}+\frac {2}{9} \left (1+x^3\right )^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} \frac {2}{9} \left (-2+x^3\right ) \sqrt {1+x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 23, normalized size = 0.85
method | result | size |
risch | \(\frac {2 \left (x^{3}-2\right ) \sqrt {x^{3}+1}}{9}\) | \(15\) |
trager | \(\left (\frac {2 x^{3}}{9}-\frac {4}{9}\right ) \sqrt {x^{3}+1}\) | \(16\) |
default | \(\frac {2 x^{3} \sqrt {x^{3}+1}}{9}-\frac {4 \sqrt {x^{3}+1}}{9}\) | \(23\) |
elliptic | \(\frac {2 x^{3} \sqrt {x^{3}+1}}{9}-\frac {4 \sqrt {x^{3}+1}}{9}\) | \(23\) |
gosper | \(\frac {2 \left (x +1\right ) \left (x^{2}-x +1\right ) \left (x^{3}-2\right )}{9 \sqrt {x^{3}+1}}\) | \(26\) |
meijerg | \(\frac {\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-4 x^{3}+8\right ) \sqrt {x^{3}+1}}{6}}{3 \sqrt {\pi }}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 19, normalized size = 0.70 \begin {gather*} \frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {x^{3} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 14, normalized size = 0.52 \begin {gather*} \frac {2}{9} \, \sqrt {x^{3} + 1} {\left (x^{3} - 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 26, normalized size = 0.96 \begin {gather*} \frac {2 x^{3} \sqrt {x^{3} + 1}}{9} - \frac {4 \sqrt {x^{3} + 1}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.55, size = 19, normalized size = 0.70 \begin {gather*} \frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {x^{3} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 14, normalized size = 0.52 \begin {gather*} \frac {2\,\sqrt {x^3+1}\,\left (x^3-2\right )}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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