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 = 0.30, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}
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Mathematica [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {x + 1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {x +1}{\left (x^{2}+x +1\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {x + 1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (x^{2} + x + 1\right )}}\,{d x} \]
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 \[ \int \frac {x+1}{{\left (b\,x^3+a\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \]
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 \[ \int \frac {x + 1}{\sqrt [3]{a + b x^{3}} \left (x^{2} + x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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