Optimal. Leaf size=136 \[ \frac {3}{8} \sqrt [3]{x^4+x^2}-\frac {\log \left (2^{2/3} \sqrt [3]{x^4+x^2}+1\right )}{8\ 2^{2/3}}+\frac {\log \left (-2 \sqrt [3]{2} \left (x^4+x^2\right )^{2/3}+2^{2/3} \sqrt [3]{x^4+x^2}-1\right )}{16\ 2^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2\ 2^{2/3} \sqrt [3]{x^4+x^2}}{\sqrt {3}}\right )}{8\ 2^{2/3}} \]
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Rubi [A] time = 0.21, antiderivative size = 109, normalized size of antiderivative = 0.80, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2034, 694, 266, 50, 58, 617, 204, 31} \begin {gather*} \frac {3 \sqrt [3]{\left (2 x^2+1\right )^2-1}}{8\ 2^{2/3}}+\frac {\log \left (2 x^2+1\right )}{8\ 2^{2/3}}-\frac {3 \log \left (\sqrt [3]{\left (2 x^2+1\right )^2-1}+1\right )}{16\ 2^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-2 \sqrt [3]{\left (2 x^2+1\right )^2-1}}{\sqrt {3}}\right )}{8\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 58
Rule 204
Rule 266
Rule 617
Rule 694
Rule 2034
Rubi steps
\begin {align*} \int \frac {x \sqrt [3]{x^2+x^4}}{1+2 x^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt [3]{x+x^2}}{1+2 x} \, dx,x,x^2\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1}{4}+\frac {x^2}{4}}}{x} \, dx,x,1+2 x^2\right )\\ &=\frac {1}{8} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-\frac {1}{4}+\frac {x}{4}}}{x} \, dx,x,\left (1+2 x^2\right )^2\right )\\ &=\frac {3 \sqrt [3]{-1+\left (1+2 x^2\right )^2}}{8\ 2^{2/3}}-\frac {1}{32} \operatorname {Subst}\left (\int \frac {1}{\left (-\frac {1}{4}+\frac {x}{4}\right )^{2/3} x} \, dx,x,\left (1+2 x^2\right )^2\right )\\ &=\frac {3 \sqrt [3]{-1+\left (1+2 x^2\right )^2}}{8\ 2^{2/3}}+\frac {\log \left (1+2 x^2\right )}{8\ 2^{2/3}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2^{2/3}}+x} \, dx,x,\sqrt [3]{x^2+x^4}\right )}{16\ 2^{2/3}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2 \sqrt [3]{2}}-\frac {x}{2^{2/3}}+x^2} \, dx,x,\sqrt [3]{x^2+x^4}\right )}{32 \sqrt [3]{2}}\\ &=\frac {3 \sqrt [3]{-1+\left (1+2 x^2\right )^2}}{8\ 2^{2/3}}+\frac {\log \left (1+2 x^2\right )}{8\ 2^{2/3}}-\frac {3 \log \left (1+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{16\ 2^{2/3}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2\ 2^{2/3} \sqrt [3]{x^2+x^4}\right )}{8\ 2^{2/3}}\\ &=\frac {3 \sqrt [3]{-1+\left (1+2 x^2\right )^2}}{8\ 2^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1-2\ 2^{2/3} \sqrt [3]{x^2+x^4}}{\sqrt {3}}\right )}{8\ 2^{2/3}}+\frac {\log \left (1+2 x^2\right )}{8\ 2^{2/3}}-\frac {3 \log \left (1+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{16\ 2^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 57, normalized size = 0.42 \begin {gather*} \frac {3 \sqrt [3]{x^4+x^2} \left (\sqrt [3]{x^2+1}-F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-x^2,-2 x^2\right )\right )}{8 \sqrt [3]{x^2+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.25, size = 136, normalized size = 1.00 \begin {gather*} \frac {3}{8} \sqrt [3]{x^2+x^4}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2\ 2^{2/3} \sqrt [3]{x^2+x^4}}{\sqrt {3}}\right )}{8\ 2^{2/3}}-\frac {\log \left (1+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{8\ 2^{2/3}}+\frac {\log \left (-1+2^{2/3} \sqrt [3]{x^2+x^4}-2 \sqrt [3]{2} \left (x^2+x^4\right )^{2/3}\right )}{16\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 128, normalized size = 0.94 \begin {gather*} \frac {1}{16} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} {\left (2 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} - 4^{\frac {1}{3}}\right )}\right ) - \frac {1}{64} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}\right ) + \frac {1}{32} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} + 4 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}\right ) + \frac {3}{8} \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x}{2 \, x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 22.48, size = 1334, normalized size = 9.81
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1334\) |
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^{4} + x^{2}\right )}^{\frac {1}{3}} x}{2 \, x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,{\left (x^4+x^2\right )}^{1/3}}{2\,x^2+1} \,d x \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 {x \sqrt [3]{x^{2} \left (x^{2} + 1\right )}}{2 x^{2} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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