Integrand size = 13, antiderivative size = 30 \[ \int x^8 \sqrt [4]{1+x^3} \, dx=\frac {4 \sqrt [4]{1+x^3} \left (32-8 x^3+5 x^6+45 x^9\right )}{1755} \]
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Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int x^8 \sqrt [4]{1+x^3} \, dx=\frac {4}{39} \left (x^3+1\right )^{13/4}-\frac {8}{27} \left (x^3+1\right )^{9/4}+\frac {4}{15} \left (x^3+1\right )^{5/4} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int x^2 \sqrt [4]{1+x} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\sqrt [4]{1+x}-2 (1+x)^{5/4}+(1+x)^{9/4}\right ) \, dx,x,x^3\right ) \\ & = \frac {4}{15} \left (1+x^3\right )^{5/4}-\frac {8}{27} \left (1+x^3\right )^{9/4}+\frac {4}{39} \left (1+x^3\right )^{13/4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int x^8 \sqrt [4]{1+x^3} \, dx=\frac {4 \sqrt [4]{1+x^3} \left (32-8 x^3+5 x^6+45 x^9\right )}{1755} \]
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Result contains higher order function than in optimal. Order 5 vs. order 2.
Time = 0.82 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.57
method | result | size |
meijerg | \(\frac {x^{9} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, 3\right ], \left [4\right ], -x^{3}\right )}{9}\) | \(17\) |
pseudoelliptic | \(\frac {4 \left (x^{3}+1\right )^{\frac {5}{4}} \left (45 x^{6}-40 x^{3}+32\right )}{1755}\) | \(22\) |
trager | \(\left (\frac {4}{39} x^{9}+\frac {4}{351} x^{6}-\frac {32}{1755} x^{3}+\frac {128}{1755}\right ) \left (x^{3}+1\right )^{\frac {1}{4}}\) | \(26\) |
risch | \(\frac {4 \left (x^{3}+1\right )^{\frac {1}{4}} \left (45 x^{9}+5 x^{6}-8 x^{3}+32\right )}{1755}\) | \(27\) |
gosper | \(\frac {4 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (45 x^{6}-40 x^{3}+32\right ) \left (x^{3}+1\right )^{\frac {1}{4}}}{1755}\) | \(33\) |
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Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int x^8 \sqrt [4]{1+x^3} \, dx=\frac {4}{1755} \, {\left (45 \, x^{9} + 5 \, x^{6} - 8 \, x^{3} + 32\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (27) = 54\).
Time = 0.82 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87 \[ \int x^8 \sqrt [4]{1+x^3} \, dx=\frac {4 x^{9} \sqrt [4]{x^{3} + 1}}{39} + \frac {4 x^{6} \sqrt [4]{x^{3} + 1}}{351} - \frac {32 x^{3} \sqrt [4]{x^{3} + 1}}{1755} + \frac {128 \sqrt [4]{x^{3} + 1}}{1755} \]
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Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int x^8 \sqrt [4]{1+x^3} \, dx=\frac {4}{39} \, {\left (x^{3} + 1\right )}^{\frac {13}{4}} - \frac {8}{27} \, {\left (x^{3} + 1\right )}^{\frac {9}{4}} + \frac {4}{15} \, {\left (x^{3} + 1\right )}^{\frac {5}{4}} \]
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Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int x^8 \sqrt [4]{1+x^3} \, dx=\frac {4}{39} \, {\left (x^{3} + 1\right )}^{\frac {13}{4}} - \frac {8}{27} \, {\left (x^{3} + 1\right )}^{\frac {9}{4}} + \frac {4}{15} \, {\left (x^{3} + 1\right )}^{\frac {5}{4}} \]
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Time = 5.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int x^8 \sqrt [4]{1+x^3} \, dx={\left (x^3+1\right )}^{1/4}\,\left (\frac {4\,x^9}{39}+\frac {4\,x^6}{351}-\frac {32\,x^3}{1755}+\frac {128}{1755}\right ) \]
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