Optimal. Leaf size=154 \[ \frac{\log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}-\frac{\log \left (x \sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}+\frac{\tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a+b}}+\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a+b}} \]
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Rubi [F] time = 0.299582, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx &=\int \left (\frac{1-\frac{i}{\sqrt{3}}}{\left (1-i \sqrt{3}+2 x\right ) \sqrt [3]{a+b x^3}}+\frac{1+\frac{i}{\sqrt{3}}}{\left (1+i \sqrt{3}+2 x\right ) \sqrt [3]{a+b x^3}}\right ) \, dx\\ &=\frac{1}{3} \left (3-i \sqrt{3}\right ) \int \frac{1}{\left (1-i \sqrt{3}+2 x\right ) \sqrt [3]{a+b x^3}} \, dx+\frac{1}{3} \left (3+i \sqrt{3}\right ) \int \frac{1}{\left (1+i \sqrt{3}+2 x\right ) \sqrt [3]{a+b x^3}} \, dx\\ \end{align*}
Mathematica [F] time = 0.28751, size = 0, normalized size = 0. \[ \int \frac{1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{\frac{1+x}{{x}^{2}+x+1}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (x^{2} + x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1}{\sqrt [3]{a + b x^{3}} \left (x^{2} + x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (x^{2} + x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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