Optimal. Leaf size=173 \[ \frac {\log \left (\sqrt [3]{d} x \sqrt [3]{x^2 (-a-b)+a b x+x^3}+\left (x^2 (-a-b)+a b x+x^3\right )^{2/3}+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{x^2 (-a-b)+a b x+x^3}-\sqrt [3]{d} x\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x}{2 \sqrt [3]{x^2 (-a-b)+a b x+x^3}+\sqrt [3]{d} x}\right )}{\sqrt [3]{d}} \]
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Rubi [F] time = 4.00, 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 {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) 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 {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x}\right ) \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (-a b+(a+b) 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 {a+b-\sqrt {a^2-2 a b+b^2+4 a b d}}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a+b-\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )}+\frac {a+b+\sqrt {a^2-2 a b+b^2+4 a b d}}{\sqrt [3]{x} \sqrt [3]{-a+x} \sqrt [3]{-b+x} \left (a+b+\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )}\right ) \, dx}{\sqrt [3]{x (-a+x) (-b+x)}}\\ &=\frac {\left (\left (a+b-\sqrt {a^2-2 a b+b^2+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 (a+b-\sqrt {a^2-2 a b+b^2+4 a b d}+2 (-1+d) x\right )} \, dx}{\sqrt [3]{x (-a+x) (-b+x)}}+\frac {\left (\left (a+b+\sqrt {a^2-2 a b+b^2+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 (a+b+\sqrt {a^2-2 a b+b^2+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 = 4.93, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 a b+(a+b) x}{\sqrt [3]{x (-a+x) (-b+x)} \left (-a b+(a+b) x+(-1+d) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.49, size = 173, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x}{\sqrt [3]{d} x+2 \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (-\sqrt [3]{d} x+\sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{\sqrt [3]{d}}+\frac {\log \left (d^{2/3} x^2+\sqrt [3]{d} x \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 {2 \, a b - {\left (a + b\right )} x}{\left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}} {\left ({\left (d - 1\right )} x^{2} - a b + {\left (a + b\right )} x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {-2 a b +\left (a +b \right ) x}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {1}{3}} \left (-a b +\left (a +b \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]
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 {2 \, a b - {\left (a + b\right )} x}{\left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {1}{3}} {\left ({\left (d - 1\right )} x^{2} - a b + {\left (a + b\right )} x\right )}}\,{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 {2\,a\,b-x\,\left (a+b\right )}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{1/3}\,\left (\left (d-1\right )\,x^2+\left (a+b\right )\,x-a\,b\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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