Optimal. Leaf size=493 \[ \frac {(a+b) \log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6 \sqrt [3]{2}}-\frac {(a+b) \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}-\frac {(a+b) \log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{4 \sqrt [3]{2}}+\frac {(a+b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {(a+b) \tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {(a+b) \log \left ((1-x) (x+1)^2\right )}{12 \sqrt [3]{2}}-\frac {(a-c) \log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac {(a-c) \log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2 \sqrt [3]{2}}-\frac {(a-c) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {(b+c) \log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac {(b+c) \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac {(b+c) \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {1}{2} c \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {c \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [C] time = 0.85, antiderivative size = 576, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {6728, 239, 2148} \[ -\frac {\log \left (2\ 2^{2/3} \sqrt [3]{1-x^3}+2 x-i \sqrt {3}+1\right ) \left (3 i b-\sqrt {3} \left (2 a+b-i \sqrt {3} c-c\right )\right )}{4 \sqrt [3]{2} \left (\sqrt {3}+i\right )}-\frac {\log \left (2\ 2^{2/3} \sqrt [3]{1-x^3}+2 x+i \sqrt {3}+1\right ) \left (\sqrt {3} \left (2 a+b+i \sqrt {3} c-c\right )+3 i b\right )}{4 \sqrt [3]{2} \left (-\sqrt {3}+i\right )}-\frac {\tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (2 x-i \sqrt {3}+1\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt {3}}\right ) \left (2 a-i \sqrt {3} b+b-\left (1+i \sqrt {3}\right ) c\right )}{2 \sqrt [3]{2} \left (\sqrt {3}+i\right )}+\frac {\tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (2 x+i \sqrt {3}+1\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt {3}}\right ) \left (2 a+i \sqrt {3} b+b+i \sqrt {3} c-c\right )}{2 \sqrt [3]{2} \left (-\sqrt {3}+i\right )}+\frac {\log \left (-\left (-2 x-i \sqrt {3}+1\right )^2 \left (2 x-i \sqrt {3}+1\right )\right ) \left (3 i b-\sqrt {3} \left (2 a+b-i \sqrt {3} c-c\right )\right )}{12 \sqrt [3]{2} \left (\sqrt {3}+i\right )}+\frac {\log \left (-\left (-2 x+i \sqrt {3}+1\right )^2 \left (2 x+i \sqrt {3}+1\right )\right ) \left (\sqrt {3} \left (2 a+b+i \sqrt {3} c-c\right )+3 i b\right )}{12 \sqrt [3]{2} \left (-\sqrt {3}+i\right )}+\frac {1}{2} c \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {c \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 239
Rule 2148
Rule 6728
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx &=\int \left (\frac {c}{\sqrt [3]{1-x^3}}+\frac {a-c+(b+c) x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}}\right ) \, dx\\ &=c \int \frac {1}{\sqrt [3]{1-x^3}} \, dx+\int \frac {a-c+(b+c) x}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx\\ &=-\frac {c \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} c \log \left (x+\sqrt [3]{1-x^3}\right )+\int \left (\frac {b-\frac {i (2 a+b-c)}{\sqrt {3}}+c}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}}+\frac {b+\frac {i (2 a+b-c)}{\sqrt {3}}+c}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}}\right ) \, dx\\ &=-\frac {c \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} c \log \left (x+\sqrt [3]{1-x^3}\right )+\frac {1}{3} \left (3 b-i \sqrt {3} (2 a+b-c)+3 c\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx+\frac {1}{3} \left (3 b+i \sqrt {3} (2 a+b-c)+3 c\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt [3]{1-x^3}} \, dx\\ &=-\frac {c \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (2 a+b-i \sqrt {3} b-c-i \sqrt {3} c\right ) \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1-i \sqrt {3}+2 x\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt {3}}\right )}{2 \sqrt [3]{2} \left (i+\sqrt {3}\right )}+\frac {\left (2 a+b+i \sqrt {3} b-c+i \sqrt {3} c\right ) \tan ^{-1}\left (\frac {2-\frac {\sqrt [3]{2} \left (1+i \sqrt {3}+2 x\right )}{\sqrt [3]{1-x^3}}}{2 \sqrt {3}}\right )}{2 \sqrt [3]{2} \left (i-\sqrt {3}\right )}+\frac {\left (3 i b-\sqrt {3} \left (2 a+b-c-i \sqrt {3} c\right )\right ) \log \left (-\left (1-i \sqrt {3}-2 x\right )^2 \left (1-i \sqrt {3}+2 x\right )\right )}{12 \sqrt [3]{2} \left (i+\sqrt {3}\right )}+\frac {\left (3 i b+\sqrt {3} \left (2 a+b-c+i \sqrt {3} c\right )\right ) \log \left (-\left (1+i \sqrt {3}-2 x\right )^2 \left (1+i \sqrt {3}+2 x\right )\right )}{12 \sqrt [3]{2} \left (i-\sqrt {3}\right )}+\frac {1}{2} c \log \left (x+\sqrt [3]{1-x^3}\right )-\frac {\left (3 i b-\sqrt {3} \left (2 a+b-c-i \sqrt {3} c\right )\right ) \log \left (1-i \sqrt {3}+2 x+2\ 2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2} \left (i+\sqrt {3}\right )}-\frac {\left (3 i b+\sqrt {3} \left (2 a+b-c+i \sqrt {3} c\right )\right ) \log \left (1+i \sqrt {3}+2 x+2\ 2^{2/3} \sqrt [3]{1-x^3}\right )}{4 \sqrt [3]{2} \left (i-\sqrt {3}\right )}\\ \end {align*}
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Mathematica [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \frac {a+b x+c x^2}{\left (1-x+x^2\right ) \sqrt [3]{1-x^3}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {c x^{2} + b x + a}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {c \,x^{2}+b x +a}{\left (x^{2}-x +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {c x^{2} + b x + a}{{\left (-x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {c\,x^2+b\,x+a}{{\left (1-x^3\right )}^{1/3}\,\left (x^2-x+1\right )} \,d x \]
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 \[ \int \frac {a + b x + c x^{2}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} - x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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