Optimal. Leaf size=154 \[ \frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}-\frac {\log \left (\sqrt [3]{a+b} x-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}} \]
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Rubi [A]
time = 0.13, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2183, 384, 455,
57, 631, 210, 31} \begin {gather*} \frac {\text {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 {\text {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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 210
Rule 384
Rule 455
Rule 631
Rule 2183
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 = 10.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {1+x}{\left (x^{2}+x +1\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {x + 1}{\sqrt [3]{a + b x^{3}} \left (x^{2} + x + 1\right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {x+1}{{\left (b\,x^3+a\right )}^{1/3}\,\left (x^2+x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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