Optimal. Leaf size=14 \[ \frac {3}{2} x \sqrt [3]{x^3+x} \]
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Rubi [A] time = 0.08, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2056, 449} \begin {gather*} \frac {3}{2} x \sqrt [3]{x^3+x} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 2056
Rubi steps
\begin {align*} \int \frac {\left (2+3 x^2\right ) \sqrt [3]{x+x^3}}{1+x^2} \, dx &=\frac {\sqrt [3]{x+x^3} \int \frac {\sqrt [3]{x} \left (2+3 x^2\right )}{\left (1+x^2\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {3}{2} x \sqrt [3]{x+x^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 14, normalized size = 1.00 \begin {gather*} \frac {3}{2} x \sqrt [3]{x^3+x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 14, normalized size = 1.00 \begin {gather*} \frac {3}{2} x \sqrt [3]{x+x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 10, normalized size = 0.71 \begin {gather*} \frac {3}{2} \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 12, normalized size = 0.86 \begin {gather*} \frac {3}{2} \, x^{2} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 11, normalized size = 0.79
method | result | size |
gosper | \(\frac {3 x \left (x^{3}+x \right )^{\frac {1}{3}}}{2}\) | \(11\) |
trager | \(\frac {3 x \left (x^{3}+x \right )^{\frac {1}{3}}}{2}\) | \(11\) |
risch | \(\frac {3 \left (\left (x^{2}+1\right ) x \right )^{\frac {1}{3}} x}{2}\) | \(13\) |
meijerg | \(\frac {3 \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{2}\right ) x^{\frac {4}{3}}}{2}+\frac {9 \hypergeom \left (\left [\frac {2}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], -x^{2}\right ) x^{\frac {10}{3}}}{10}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + x\right )}^{\frac {1}{3}} {\left (3 \, x^{2} + 2\right )}}{x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 10, normalized size = 0.71 \begin {gather*} \frac {3\,x\,{\left (x^3+x\right )}^{1/3}}{2} \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 {\sqrt [3]{x \left (x^{2} + 1\right )} \left (3 x^{2} + 2\right )}{x^{2} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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