Optimal. Leaf size=96 \[ -\frac {3 \sqrt [3]{x^3+x}}{2 x}-\frac {1}{2} \log \left (\sqrt [3]{x^3+x}-x\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )+\frac {1}{4} \log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.13, antiderivative size = 180, normalized size of antiderivative = 1.88, number of steps used = 11, number of rules used = 11, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {2020, 2032, 329, 275, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} -\frac {3 \sqrt [3]{x^3+x}}{2 x}-\frac {x^{2/3} \left (x^2+1\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2+1}}\right )}{2 \left (x^3+x\right )^{2/3}}+\frac {x^{2/3} \left (x^2+1\right )^{2/3} \log \left (\frac {x^{4/3}}{\left (x^2+1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2+1}}+1\right )}{4 \left (x^3+x\right )^{2/3}}-\frac {\sqrt {3} x^{2/3} \left (x^2+1\right )^{2/3} \tan ^{-1}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \left (x^3+x\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 329
Rule 331
Rule 618
Rule 628
Rule 634
Rule 2020
Rule 2032
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{x+x^3}}{x^2} \, dx &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\int \frac {x}{\left (x+x^3\right )^{2/3}} \, dx\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (x^{2/3} \left (1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3}} \, dx}{\left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 \left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (3 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}+\frac {\left (x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}-\frac {\left (x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}-\frac {x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}+\frac {\left (x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \left (x+x^3\right )^{2/3}}-\frac {\left (3 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}-\frac {x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}+\frac {x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \left (x+x^3\right )^{2/3}}+\frac {\left (3 x^{2/3} \left (1+x^2\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}\\ &=-\frac {3 \sqrt [3]{x+x^3}}{2 x}-\frac {\sqrt {3} x^{2/3} \left (1+x^2\right )^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \left (x+x^3\right )^{2/3}}-\frac {x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 \left (x+x^3\right )^{2/3}}+\frac {x^{2/3} \left (1+x^2\right )^{2/3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 \left (x+x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 40, normalized size = 0.42 \begin {gather*} -\frac {3 \sqrt [3]{x^3+x} \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};-x^2\right )}{2 x \sqrt [3]{x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 96, normalized size = 1.00 \begin {gather*} -\frac {3 \sqrt [3]{x^3+x}}{2 x}-\frac {1}{2} \log \left (\sqrt [3]{x^3+x}-x\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+x}+x}\right )+\frac {1}{4} \log \left (\sqrt [3]{x^3+x} x+\left (x^3+x\right )^{2/3}+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 95, normalized size = 0.99 \begin {gather*} -\frac {2 \, \sqrt {3} x \arctan \left (-\frac {196 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (539 \, x^{2} + 507\right )} - 1274 \, \sqrt {3} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{2205 \, x^{2} + 2197}\right ) + x \log \left (3 \, {\left (x^{3} + x\right )}^{\frac {1}{3}} x - 3 \, {\left (x^{3} + x\right )}^{\frac {2}{3}} + 1\right ) + 6 \, {\left (x^{3} + x\right )}^{\frac {1}{3}}}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 64, normalized size = 0.67 \begin {gather*} \frac {1}{2} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {3}{2} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + \frac {1}{4} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{2} \, \log \left ({\left | {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.27, size = 729, normalized size = 7.59 \begin {gather*} -\frac {3 \left (x \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{2 x}+\frac {\left (\frac {\RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}+11 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}+15 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-40 x^{4}+15 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}-48 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+28 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}-48 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+15 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}-2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}-70 x^{2}-48 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+17 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )-30}{x^{2}+1}\right )}{4}-\frac {\ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}+19 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}+15 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+10 x^{4}+15 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+28 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+15 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}+14 x^{2}+18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+9 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )+4}{x^{2}+1}\right ) \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )}{4}+\frac {\ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2} x^{4}+19 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{4}+15 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+10 x^{4}+15 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+28 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) x^{2}+18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {2}{3}}+15 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right ) \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+2 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )^{2}+14 x^{2}+18 \left (x^{6}+2 x^{4}+x^{2}\right )^{\frac {1}{3}}+9 \RootOf \left (\textit {\_Z}^{2}-2 \textit {\_Z} +4\right )+4}{x^{2}+1}\right )}{2}\right ) \left (x \left (x^{2}+1\right )\right )^{\frac {1}{3}} \left (x^{2} \left (x^{2}+1\right )^{2}\right )^{\frac {1}{3}}}{x \left (x^{2}+1\right )} \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 (x^{3} + x\right )}^{\frac {1}{3}}}{x^{2}}\,{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 (x^3+x\right )}^{1/3}}{x^2} \,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 (x^{2} + 1\right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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