Optimal. Leaf size=166 \[ -\frac {\log \left (d^{2/3} \left (x^2 (-a-b)+a b x+x^3\right )^{2/3}+\sqrt [3]{d} x \sqrt [3]{x^2 (-a-b)+a b x+x^3}+x^2\right )}{2 \sqrt [3]{d}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{x^2 (-a-b)+a b x+x^3}\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{d} \sqrt [3]{x^2 (-a-b)+a b x+x^3}+x}\right )}{\sqrt [3]{d}} \]
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Rubi [F] time = 6.08, 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 x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a b d-(a+b) 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 {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx &=\int \frac {x (-2 a b+(a+b) x)}{(x (-a+x) (-b+x))^{2/3} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx\\ &=\frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{x} (-2 a b+(a+b) x)}{(-a+x)^{2/3} (-b+x)^{2/3} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx}{(x (-a+x) (-b+x))^{2/3}}\\ &=\frac {\left (x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \left (\frac {\left (a+b+\frac {\sqrt {4 a b+a^2 d-2 a b d+b^2 d}}{\sqrt {d}}\right ) \sqrt [3]{x}}{(-a+x)^{2/3} (-b+x)^{2/3} \left (-((a+b) d)-\sqrt {d} \sqrt {4 a b+a^2 d-2 a b d+b^2 d}+2 (-1+d) x\right )}+\frac {\left (a+b-\frac {\sqrt {4 a b+a^2 d-2 a b d+b^2 d}}{\sqrt {d}}\right ) \sqrt [3]{x}}{(-a+x)^{2/3} (-b+x)^{2/3} \left (-((a+b) d)+\sqrt {d} \sqrt {4 a b+a^2 d-2 a b d+b^2 d}+2 (-1+d) x\right )}\right ) \, dx}{(x (-a+x) (-b+x))^{2/3}}\\ &=\frac {\left (\left (a+b-\frac {\sqrt {2 a b (2-d)+a^2 d+b^2 d}}{\sqrt {d}}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(-a+x)^{2/3} (-b+x)^{2/3} \left (-((a+b) d)+\sqrt {d} \sqrt {4 a b+a^2 d-2 a b d+b^2 d}+2 (-1+d) x\right )} \, dx}{(x (-a+x) (-b+x))^{2/3}}+\frac {\left (\left (a+b+\frac {\sqrt {2 a b (2-d)+a^2 d+b^2 d}}{\sqrt {d}}\right ) x^{2/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{x}}{(-a+x)^{2/3} (-b+x)^{2/3} \left (-((a+b) d)-\sqrt {d} \sqrt {4 a b+a^2 d-2 a b d+b^2 d}+2 (-1+d) x\right )} \, dx}{(x (-a+x) (-b+x))^{2/3}}\\ \end {align*}
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Mathematica [F] time = 10.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 a b x+(a+b) x^2}{(x (-a+x) (-b+x))^{2/3} \left (a b d-(a+b) d x+(-1+d) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.79, size = 166, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (x-\sqrt [3]{d} \sqrt [3]{a b x+(-a-b) x^2+x^3}\right )}{\sqrt [3]{d}}-\frac {\log \left (x^2+\sqrt [3]{d} x \sqrt [3]{a b x+(-a-b) x^2+x^3}+d^{2/3} \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 x - {\left (a + b\right )} x^{2}}{{\left (a b d - {\left (a + b\right )} d x + {\left (d - 1\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {-2 a b x +\left (a +b \right ) x^{2}}{\left (x \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (a b d -\left (a +b \right ) d x +\left (-1+d \right ) x^{2}\right )}\, dx\]
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 x - {\left (a + b\right )} x^{2}}{{\left (a b d - {\left (a + b\right )} d x + {\left (d - 1\right )} x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )} x\right )^{\frac {2}{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 {x^2\,\left (a+b\right )-2\,a\,b\,x}{{\left (x\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (\left (d-1\right )\,x^2-d\,\left (a+b\right )\,x+a\,b\,d\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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