Optimal. Leaf size=105 \[ \log \left (\sqrt [3]{a x^2+b x+c}+x\right )-\frac {1}{2} \log \left (-x \sqrt [3]{a x^2+b x+c}+\left (a x^2+b x+c\right )^{2/3}+x^2\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a x^2+b x+c}}{\sqrt [3]{a x^2+b x+c}-2 x}\right ) \]
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Rubi [F] time = 1.13, 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 = {} \begin {gather*} \int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx &=\int \left (\frac {3 c}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )}+\frac {2 b x}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )}+\frac {a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )}\right ) \, dx\\ &=a \int \frac {x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx+(2 b) \int \frac {x}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx+(3 c) \int \frac {1}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.55, size = 105, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{c+b x+a x^2}}{-2 x+\sqrt [3]{c+b x+a x^2}}\right )+\log \left (x+\sqrt [3]{c+b x+a x^2}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{c+b x+a x^2}+\left (c+b x+a x^2\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{2} + 2 \, b x + 3 \, c}{{\left (a x^{2} + x^{3} + b x + c\right )} {\left (a x^{2} + b x + c\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}+2 b x +3 c}{\left (a \,x^{2}+b x +c \right )^{\frac {1}{3}} \left (a \,x^{2}+x^{3}+b x +c \right )}\, dx\]
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*} \int \frac {a x^{2} + 2 \, b x + 3 \, c}{{\left (a x^{2} + x^{3} + b x + c\right )} {\left (a x^{2} + b x + c\right )}^{\frac {1}{3}}}\,{d x} \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 {a\,x^2+2\,b\,x+3\,c}{{\left (a\,x^2+b\,x+c\right )}^{1/3}\,\left (x^3+a\,x^2+b\,x+c\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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