Optimal. Leaf size=25 \[ \frac {4}{693} \left (1+x^3\right )^{3/4} \left (32-24 x^3+21 x^6\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.60, 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 {4}{33} \left (x^3+1\right )^{11/4}-\frac {8}{21} \left (x^3+1\right )^{7/4}+\frac {4}{9} \left (x^3+1\right )^{3/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt [4]{1+x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{\sqrt [4]{1+x}}-2 (1+x)^{3/4}+(1+x)^{7/4}\right ) \, dx,x,x^3\right )\\ &=\frac {4}{9} \left (1+x^3\right )^{3/4}-\frac {8}{21} \left (1+x^3\right )^{7/4}+\frac {4}{33} \left (1+x^3\right )^{11/4}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 25, normalized size = 1.00 \begin {gather*} \frac {4}{693} \left (1+x^3\right )^{3/4} \left (32-24 x^3+21 x^6\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
2.
time = 0.28, size = 17, normalized size = 0.68
method | result | size |
meijerg | \(\frac {x^{9} \hypergeom \left (\left [\frac {1}{4}, 3\right ], \left [4\right ], -x^{3}\right )}{9}\) | \(17\) |
trager | \(\left (\frac {4}{33} x^{6}-\frac {32}{231} x^{3}+\frac {128}{693}\right ) \left (x^{3}+1\right )^{\frac {3}{4}}\) | \(21\) |
risch | \(\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}} \left (21 x^{6}-24 x^{3}+32\right )}{693}\) | \(22\) |
gosper | \(\frac {4 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (21 x^{6}-24 x^{3}+32\right )}{693 \left (x^{3}+1\right )^{\frac {1}{4}}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 28, normalized size = 1.12 \begin {gather*} \frac {4}{33} \, {\left (x^{3} + 1\right )}^{\frac {11}{4}} - \frac {8}{21} \, {\left (x^{3} + 1\right )}^{\frac {7}{4}} + \frac {4}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 21, normalized size = 0.84 \begin {gather*} \frac {4}{693} \, {\left (21 \, x^{6} - 24 \, x^{3} + 32\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.11, size = 41, normalized size = 1.64 \begin {gather*} \frac {4 x^{6} \left (x^{3} + 1\right )^{\frac {3}{4}}}{33} - \frac {32 x^{3} \left (x^{3} + 1\right )^{\frac {3}{4}}}{231} + \frac {128 \left (x^{3} + 1\right )^{\frac {3}{4}}}{693} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 28, normalized size = 1.12 \begin {gather*} \frac {4}{33} \, {\left (x^{3} + 1\right )}^{\frac {11}{4}} - \frac {8}{21} \, {\left (x^{3} + 1\right )}^{\frac {7}{4}} + \frac {4}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.23, size = 20, normalized size = 0.80 \begin {gather*} {\left (x^3+1\right )}^{3/4}\,\left (\frac {4\,x^6}{33}-\frac {32\,x^3}{231}+\frac {128}{693}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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