Optimal. Leaf size=180 \[ \frac {\log \left (\sqrt [3]{x^3 (-a-b)+a b x^2+x^4}-\sqrt [3]{d} x\right )}{d^{2/3}}-\frac {\log \left (\sqrt [3]{d} x \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+\left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+d^{2/3} x^2\right )}{2 d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x}{2 \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+\sqrt [3]{d} x}\right )}{d^{2/3}} \]
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Rubi [F] time = 6.86, 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 {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b-(a+b+d) x+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b-(a+b+d) x+x^2\right )} \, dx &=\frac {\left (x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {-a b+x^2}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \left (a b-(a+b+d) x+x^2\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \left (\frac {1}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3}}-\frac {2 a b-(a+b+d) x}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \left (a b+(-a-b-d) x+x^2\right )}\right ) \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {\left (x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3}} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}-\frac {\left (x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {2 a b-(a+b+d) x}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \left (a b+(-a-b-d) x+x^2\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=-\frac {\left (x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \left (\frac {-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )}+\frac {-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )}\right ) \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (x^{4/3} (-b+x)^{2/3} \left (1-\frac {x}{a}\right )^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-b+x)^{2/3} \left (1-\frac {x}{a}\right )^{2/3}} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=-\frac {\left (\left (-a-b-d-\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}-\frac {\left (\left (-a-b-d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}+\frac {\left (x^{4/3} \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3}} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ &=\frac {3 x^2 \left (1-\frac {x}{a}\right )^{2/3} \left (1-\frac {x}{b}\right )^{2/3} F_1\left (\frac {2}{3};\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {x}{a},\frac {x}{b}\right )}{2 \left ((a-x) (b-x) x^2\right )^{2/3}}-\frac {\left (\left (-a-b-d-\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \left (-a-b-d-\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}-\frac {\left (\left (-a-b-d+\sqrt {a^2-2 a (b-d)+(b+d)^2}\right ) x^{4/3} (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-a+x)^{2/3} (-b+x)^{2/3} \left (-a-b-d+\sqrt {a^2-2 a b+b^2+2 a d+2 b d+d^2}+2 x\right )} \, dx}{\left (x^2 (-a+x) (-b+x)\right )^{2/3}}\\ \end {align*}
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Mathematica [F] time = 5.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (-a b+x^2\right )}{\left (x^2 (-a+x) (-b+x)\right )^{2/3} \left (a b-(a+b+d) x+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.54, size = 180, normalized size = 1.00 \begin {gather*} \frac {\log \left (\sqrt [3]{x^3 (-a-b)+a b x^2+x^4}-\sqrt [3]{d} x\right )}{d^{2/3}}-\frac {\log \left (\sqrt [3]{d} x \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+\left (x^3 (-a-b)+a b x^2+x^4\right )^{2/3}+d^{2/3} x^2\right )}{2 d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x}{2 \sqrt [3]{x^3 (-a-b)+a b x^2+x^4}+\sqrt [3]{d} x}\right )}{d^{2/3}} \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 {{\left (a b - x^{2}\right )} x}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {2}{3}} {\left (a b - {\left (a + b + d\right )} x + x^{2}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.59, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (-a b +x^{2}\right )}{\left (x^{2} \left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (a b -\left (a +b +d \right ) x +x^{2}\right )}\, dx \end {gather*}
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 {{\left (a b - x^{2}\right )} x}{\left ({\left (a - x\right )} {\left (b - x\right )} x^{2}\right )^{\frac {2}{3}} {\left (a b - {\left (a + b + d\right )} x + x^{2}\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 {x\,\left (a\,b-x^2\right )}{{\left (x^2\,\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (x^2+\left (-a-b-d\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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