Optimal. Leaf size=142 \[ \sqrt [3]{3} \log \left (3^{2/3} \sqrt [3]{2 x^3+x}-3 x\right )+3^{5/6} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{2 x^3+x}+\sqrt [3]{3} x}\right )+\frac {3 \sqrt [3]{2 x^3+x} \left (10 x^2+1\right )}{8 x^3}-\frac {1}{2} \sqrt [3]{3} \log \left (3^{2/3} \sqrt [3]{2 x^3+x} x+\sqrt [3]{3} \left (2 x^3+x\right )^{2/3}+3 x^2\right ) \]
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Rubi [A] time = 0.42, antiderivative size = 242, normalized size of antiderivative = 1.70, number of steps used = 13, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {2056, 580, 583, 12, 466, 465, 494, 292, 31, 634, 617, 204, 628} \begin {gather*} \frac {15 \sqrt [3]{2 x^3+x}}{4 x}+\frac {3 \sqrt [3]{2 x^3+x}}{8 x^3}+\frac {\sqrt [3]{3} \sqrt [3]{2 x^3+x} \log \left (1-\frac {\sqrt [3]{3} x^{2/3}}{\sqrt [3]{2 x^2+1}}\right )}{\sqrt [3]{x} \sqrt [3]{2 x^2+1}}-\frac {\sqrt [3]{3} \sqrt [3]{2 x^3+x} \log \left (\frac {3^{2/3} x^{4/3}}{\left (2 x^2+1\right )^{2/3}}+\frac {\sqrt [3]{3} x^{2/3}}{\sqrt [3]{2 x^2+1}}+1\right )}{2 \sqrt [3]{x} \sqrt [3]{2 x^2+1}}+\frac {3^{5/6} \sqrt [3]{2 x^3+x} \tan ^{-1}\left (\frac {2 x^{2/3}}{\sqrt [6]{3} \sqrt [3]{2 x^2+1}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x} \sqrt [3]{2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 204
Rule 292
Rule 465
Rule 466
Rule 494
Rule 580
Rule 583
Rule 617
Rule 628
Rule 634
Rule 2056
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right ) \sqrt [3]{x+2 x^3}}{x^4 \left (-1+x^2\right )} \, dx &=\frac {\sqrt [3]{x+2 x^3} \int \frac {\left (1+x^2\right ) \sqrt [3]{1+2 x^2}}{x^{11/3} \left (-1+x^2\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=\frac {3 \sqrt [3]{x+2 x^3}}{8 x^3}-\frac {\left (3 \sqrt [3]{x+2 x^3}\right ) \int \frac {-\frac {20}{3}-\frac {28 x^2}{3}}{x^{5/3} \left (-1+x^2\right ) \left (1+2 x^2\right )^{2/3}} \, dx}{8 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=\frac {3 \sqrt [3]{x+2 x^3}}{8 x^3}+\frac {15 \sqrt [3]{x+2 x^3}}{4 x}-\frac {\left (9 \sqrt [3]{x+2 x^3}\right ) \int -\frac {32 \sqrt [3]{x}}{3 \left (-1+x^2\right ) \left (1+2 x^2\right )^{2/3}} \, dx}{16 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=\frac {3 \sqrt [3]{x+2 x^3}}{8 x^3}+\frac {15 \sqrt [3]{x+2 x^3}}{4 x}+\frac {\left (6 \sqrt [3]{x+2 x^3}\right ) \int \frac {\sqrt [3]{x}}{\left (-1+x^2\right ) \left (1+2 x^2\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=\frac {3 \sqrt [3]{x+2 x^3}}{8 x^3}+\frac {15 \sqrt [3]{x+2 x^3}}{4 x}+\frac {\left (18 \sqrt [3]{x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right ) \left (1+2 x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=\frac {3 \sqrt [3]{x+2 x^3}}{8 x^3}+\frac {15 \sqrt [3]{x+2 x^3}}{4 x}+\frac {\left (9 \sqrt [3]{x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right ) \left (1+2 x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=\frac {3 \sqrt [3]{x+2 x^3}}{8 x^3}+\frac {15 \sqrt [3]{x+2 x^3}}{4 x}+\frac {\left (9 \sqrt [3]{x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {x}{-1+3 x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+2 x^2}}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=\frac {3 \sqrt [3]{x+2 x^3}}{8 x^3}+\frac {15 \sqrt [3]{x+2 x^3}}{4 x}+\frac {\left (3^{2/3} \sqrt [3]{x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt [3]{3} x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+2 x^2}}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}-\frac {\left (3^{2/3} \sqrt [3]{x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {-1+\sqrt [3]{3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+2 x^2}}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=\frac {3 \sqrt [3]{x+2 x^3}}{8 x^3}+\frac {15 \sqrt [3]{x+2 x^3}}{4 x}+\frac {\sqrt [3]{3} \sqrt [3]{x+2 x^3} \log \left (1-\frac {\sqrt [3]{3} x^{2/3}}{\sqrt [3]{1+2 x^2}}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}-\frac {\left (\sqrt [3]{3} \sqrt [3]{x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{3}+2\ 3^{2/3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+2 x^2}}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}+\frac {\left (3\ 3^{2/3} \sqrt [3]{x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+2 x^2}}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=\frac {3 \sqrt [3]{x+2 x^3}}{8 x^3}+\frac {15 \sqrt [3]{x+2 x^3}}{4 x}+\frac {\sqrt [3]{3} \sqrt [3]{x+2 x^3} \log \left (1-\frac {\sqrt [3]{3} x^{2/3}}{\sqrt [3]{1+2 x^2}}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}-\frac {\sqrt [3]{3} \sqrt [3]{x+2 x^3} \log \left (1+\frac {3^{2/3} x^{4/3}}{\left (1+2 x^2\right )^{2/3}}+\frac {\sqrt [3]{3} x^{2/3}}{\sqrt [3]{1+2 x^2}}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}-\frac {\left (3 \sqrt [3]{3} \sqrt [3]{x+2 x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{3} x^{2/3}}{\sqrt [3]{1+2 x^2}}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ &=\frac {3 \sqrt [3]{x+2 x^3}}{8 x^3}+\frac {15 \sqrt [3]{x+2 x^3}}{4 x}+\frac {3^{5/6} \sqrt [3]{x+2 x^3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{3} x^{2/3}}{\sqrt [3]{1+2 x^2}}}{\sqrt {3}}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}+\frac {\sqrt [3]{3} \sqrt [3]{x+2 x^3} \log \left (1-\frac {\sqrt [3]{3} x^{2/3}}{\sqrt [3]{1+2 x^2}}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2}}-\frac {\sqrt [3]{3} \sqrt [3]{x+2 x^3} \log \left (1+\frac {3^{2/3} x^{4/3}}{\left (1+2 x^2\right )^{2/3}}+\frac {\sqrt [3]{3} x^{2/3}}{\sqrt [3]{1+2 x^2}}\right )}{2 \sqrt [3]{x} \sqrt [3]{1+2 x^2}}\\ \end {align*}
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Mathematica [C] time = 4.23, size = 103, normalized size = 0.73 \begin {gather*} \frac {3 \left (2 x^2+1\right ) \sqrt [3]{2 x^3+x} \left (2 \left (6 x^4-7 x^2+1\right ) \, _2F_1\left (1,1;\frac {2}{3};\frac {3 x^2}{x^2-1}\right )+27 \left (2 x^4+x^2\right ) \, _2F_1\left (2,2;\frac {5}{3};\frac {3 x^2}{x^2-1}\right )-\left (x^2-1\right )^2\right )}{8 x^3 \left (x^2-1\right )^2} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.38, size = 142, normalized size = 1.00 \begin {gather*} \sqrt [3]{3} \log \left (3^{2/3} \sqrt [3]{2 x^3+x}-3 x\right )+3^{5/6} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{2 x^3+x}+\sqrt [3]{3} x}\right )+\frac {3 \sqrt [3]{2 x^3+x} \left (10 x^2+1\right )}{8 x^3}-\frac {1}{2} \sqrt [3]{3} \log \left (3^{2/3} \sqrt [3]{2 x^3+x} x+\sqrt [3]{3} \left (2 x^3+x\right )^{2/3}+3 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.63, size = 263, normalized size = 1.85 \begin {gather*} \frac {8 \cdot 3^{\frac {5}{6}} x^{3} \arctan \left (\frac {6 \cdot 3^{\frac {5}{6}} {\left (8 \, x^{4} - 7 \, x^{2} - 1\right )} {\left (2 \, x^{3} + x\right )}^{\frac {2}{3}} - \sqrt {3} {\left (377 \, x^{6} + 300 \, x^{4} + 51 \, x^{2} + 1\right )} - 18 \cdot 3^{\frac {1}{6}} {\left (55 \, x^{5} + 25 \, x^{3} + x\right )} {\left (2 \, x^{3} + x\right )}^{\frac {1}{3}}}{3 \, {\left (487 \, x^{6} + 240 \, x^{4} + 3 \, x^{2} - 1\right )}}\right ) - 4 \cdot 3^{\frac {1}{3}} x^{3} \log \left (\frac {3 \cdot 3^{\frac {2}{3}} {\left (2 \, x^{3} + x\right )}^{\frac {2}{3}} {\left (8 \, x^{2} + 1\right )} + 3^{\frac {1}{3}} {\left (55 \, x^{4} + 25 \, x^{2} + 1\right )} + 9 \, {\left (7 \, x^{3} + 2 \, x\right )} {\left (2 \, x^{3} + x\right )}^{\frac {1}{3}}}{x^{4} - 2 \, x^{2} + 1}\right ) + 8 \cdot 3^{\frac {1}{3}} x^{3} \log \left (-\frac {3^{\frac {2}{3}} {\left (x^{2} - 1\right )} - 9 \cdot 3^{\frac {1}{3}} {\left (2 \, x^{3} + x\right )}^{\frac {1}{3}} x + 9 \, {\left (2 \, x^{3} + x\right )}^{\frac {2}{3}}}{x^{2} - 1}\right ) + 9 \, {\left (2 \, x^{3} + x\right )}^{\frac {1}{3}} {\left (10 \, x^{2} + 1\right )}}{24 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 90, normalized size = 0.63 \begin {gather*} -3^{\frac {5}{6}} \arctan \left (\frac {1}{3} \cdot 3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} + 2 \, {\left (\frac {1}{x^{2}} + 2\right )}^{\frac {1}{3}}\right )}\right ) + \frac {3}{8} \, {\left (\frac {1}{x^{2}} + 2\right )}^{\frac {4}{3}} - \frac {1}{2} \cdot 3^{\frac {1}{3}} \log \left (3^{\frac {2}{3}} + 3^{\frac {1}{3}} {\left (\frac {1}{x^{2}} + 2\right )}^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 2\right )}^{\frac {2}{3}}\right ) + 3^{\frac {1}{3}} \log \left ({\left | -3^{\frac {1}{3}} + {\left (\frac {1}{x^{2}} + 2\right )}^{\frac {1}{3}} \right |}\right ) + 3 \, {\left (\frac {1}{x^{2}} + 2\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 9.39, size = 1861, normalized size = 13.11 \begin {gather*} \text {Expression too large to display} \end {gather*}
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 (2 \, x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{{\left (x^{2} - 1\right )} x^{4}}\,{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 {{\left (2\,x^3+x\right )}^{1/3}\,\left (x^2+1\right )}{x^4\,\left (x^2-1\right )} \,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 {\sqrt [3]{x \left (2 x^{2} + 1\right )} \left (x^{2} + 1\right )}{x^{4} \left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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