Optimal. Leaf size=30 \[ \frac {4 \sqrt [4]{x^3+1} \left (45 x^9+5 x^6-8 x^3+32\right )}{1755} \]
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Rubi [A] time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.33, 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 = {266, 43} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^8 \sqrt [4]{1+x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x^2 \sqrt [4]{1+x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {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*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.83 \begin {gather*} \frac {4 \left (x^3+1\right )^{5/4} \left (45 x^6-40 x^3+32\right )}{1755} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 30, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{1+x^3} \left (32-8 x^3+5 x^6+45 x^9\right )}{1755} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 26, normalized size = 0.87 \begin {gather*} \frac {4}{1755} \, {\left (45 \, x^{9} + 5 \, x^{6} - 8 \, x^{3} + 32\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 28, normalized size = 0.93 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 17, normalized size = 0.57
method | result | size |
meijerg | \(\frac {\hypergeom \left (\left [-\frac {1}{4}, 3\right ], \relax [4], -x^{3}\right ) x^{9}}{9}\) | \(17\) |
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 | \(\frac {4 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (45 x^{6}-40 x^{3}+32\right ) \left (x^{3}+1\right )^{\frac {1}{4}}}{1755}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 28, normalized size = 0.93 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 25, normalized size = 0.83 \begin {gather*} {\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.41, size = 56, normalized size = 1.87 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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