Integrand size = 23, antiderivative size = 154 \[ \int \frac {1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx=\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}-\frac {\log \left (\sqrt [3]{a+b} x-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}} \]
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Time = 0.14 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2183, 384, 455, 57, 631, 210, 31} \[ \int \frac {1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx=\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}-\frac {\log \left (x \sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}} \]
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Rule 31
Rule 57
Rule 210
Rule 384
Rule 455
Rule 631
Rule 2183
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}}-\frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}}\right ) \, dx \\ & = \int \frac {1}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx-\int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx \\ & = \frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\log \left (1-x^3\right )}{6 \sqrt [3]{a+b}}-\frac {\log \left (\sqrt [3]{a+b} x-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{(1-x) \sqrt [3]{a+b x}} \, dx,x,x^3\right ) \\ & = \frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}-\frac {\log \left (\sqrt [3]{a+b} x-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{(a+b)^{2/3}+\sqrt [3]{a+b} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )-\frac {\text {Subst}\left (\int \frac {1}{\sqrt [3]{a+b}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}} \\ & = \frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}-\frac {\log \left (\sqrt [3]{a+b} x-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}\right )}{\sqrt [3]{a+b}} \\ & = \frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}-\frac {\log \left (\sqrt [3]{a+b} x-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}} \\ \end{align*}
\[ \int \frac {1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx=\int \frac {1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx \]
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\[\int \frac {1+x}{\left (x^{2}+x +1\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}d x\]
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Timed out. \[ \int \frac {1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx=\text {Timed out} \]
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\[ \int \frac {1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx=\int \frac {x + 1}{\sqrt [3]{a + b x^{3}} \left (x^{2} + x + 1\right )}\, dx \]
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\[ \int \frac {1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx=\int { \frac {x + 1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}} \,d x } \]
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\[ \int \frac {1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx=\int { \frac {x + 1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx=\int \frac {x+1}{{\left (b\,x^3+a\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \]
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