Optimal. Leaf size=233 \[ \frac {\log \left (a^2 d^{2/3}+\sqrt [3]{x^2 (-a-b)+a b x+x^3} \left (\sqrt [3]{d} x-a \sqrt [3]{d}\right )+\left (x^2 (-a-b)+a b x+x^3\right )^{2/3}-2 a d^{2/3} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{x^2 (-a-b)+a b x+x^3}+a \sqrt [3]{d}-\sqrt [3]{d} x\right )}{\sqrt [3]{d}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^2 (-a-b)+a b x+x^3}}{\sqrt [3]{x^2 (-a-b)+a b x+x^3}-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x}\right )}{\sqrt [3]{d}} \]
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Rubi [F] time = 3.69, 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 {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}}\\ &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \left (\frac {-2 a+b-\sqrt {b^2+4 a^2 d-4 a b d}}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (b-2 a d-\sqrt {b^2+4 a^2 d-4 a b d}+2 (-1+d) x\right )}+\frac {-2 a+b+\sqrt {b^2+4 a^2 d-4 a b d}}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (b-2 a d+\sqrt {b^2+4 a^2 d-4 a b d}+2 (-1+d) x\right )}\right ) \, dx}{\sqrt [3]{x (-a+x) (-b+x)}}\\ &=\frac {\left (\left (-2 a+b-\sqrt {b^2+4 a^2 d-4 a b d}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (b-2 a d-\sqrt {b^2+4 a^2 d-4 a b d}+2 (-1+d) x\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}}+\frac {\left (\left (-2 a+b+\sqrt {b^2+4 a^2 d-4 a b d}\right ) \sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (b-2 a d+\sqrt {b^2+4 a^2 d-4 a b d}+2 (-1+d) x\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}}\\ \end {align*}
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Mathematica [F] time = 4.91, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a b+(-2 a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (a^2 d+(b-2 a d) x+(-1+d) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.95, size = 233, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a b x+(-a-b) x^2+x^3}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{\sqrt [3]{d}}+\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{a b x+(-a-b) x^2+x^3}+\left (a b x+(-a-b) x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{d}} \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 b - {\left (2 \, a - b\right )} x}{{\left (a^{2} d + {\left (d - 1\right )} x^{2} - {\left (2 \, a d - b\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\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 = 0.16, size = 0, normalized size = 0.00 \[\int \frac {a b +\left (-2 a +b \right ) x}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (a^{2} d +\left (-2 a d +b \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]
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 b - {\left (2 \, a - b\right )} x}{{\left (a^{2} d + {\left (d - 1\right )} x^{2} - {\left (2 \, a d - b\right )} x\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\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 = -0.00 \begin {gather*} \int \frac {a\,b-x\,\left (2\,a-b\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (a^2\,d+x\,\left (b-2\,a\,d\right )+x^2\,\left (d-1\right )\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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