Optimal. Leaf size=216 \[ -\frac {\log \left (\left (x (-a-b)+a b+x^2\right )^{2/3} \left (d^{2/3} x-b d^{2/3}\right )+\sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{4/3}+b^2 d-2 b d x+d x^2\right )}{2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [6]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}+b \sqrt {d}-\sqrt {d} x\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (x (-a-b)+a b+x^2\right )^{2/3}}{\left (x (-a-b)+a b+x^2\right )^{2/3}-2 b \sqrt [3]{d}+2 \sqrt [3]{d} x}\right )}{\sqrt [3]{d}} \]
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Rubi [C] time = 1.66, antiderivative size = 209, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 5, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.096, Rules used = {6719, 1586, 6728, 137, 136} \begin {gather*} -\frac {3 (a-x)^2 \left (-\frac {b-x}{a-b}\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};\frac {a-x}{a-b},-\frac {2 (a-x)}{\sqrt {d} \left (\sqrt {d}-\sqrt {4 a-4 b+d}\right )}\right )}{4 d ((a-x) (b-x))^{2/3}}-\frac {3 (a-x)^2 \left (-\frac {b-x}{a-b}\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};\frac {a-x}{a-b},-\frac {2 (a-x)}{\sqrt {d} \left (\sqrt {d}+\sqrt {4 a-4 b+d}\right )}\right )}{4 d ((a-x) (b-x))^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 136
Rule 137
Rule 1586
Rule 6719
Rule 6728
Rubi steps
\begin {align*} \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx &=\frac {\left ((-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {-a (a-2 b)-2 b x+x^2}{(-a+x)^{2/3} (-b+x)^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x} (a-2 b+x)}{(-b+x)^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \left (\frac {\left (1+\frac {\sqrt {4 a-4 b+d}}{\sqrt {d}}\right ) \sqrt [3]{-a+x}}{(-b+x)^{2/3} \left (-2 a-d-\sqrt {d} \sqrt {4 a-4 b+d}+2 x\right )}+\frac {\left (1-\frac {\sqrt {4 a-4 b+d}}{\sqrt {d}}\right ) \sqrt [3]{-a+x}}{(-b+x)^{2/3} \left (-2 a-d+\sqrt {d} \sqrt {4 a-4 b+d}+2 x\right )}\right ) \, dx}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left (\left (1-\frac {\sqrt {4 a-4 b+d}}{\sqrt {d}}\right ) (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{(-b+x)^{2/3} \left (-2 a-d+\sqrt {d} \sqrt {4 a-4 b+d}+2 x\right )} \, dx}{((-a+x) (-b+x))^{2/3}}+\frac {\left (\left (1+\frac {\sqrt {4 a-4 b+d}}{\sqrt {d}}\right ) (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{(-b+x)^{2/3} \left (-2 a-d-\sqrt {d} \sqrt {4 a-4 b+d}+2 x\right )} \, dx}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left (\left (1-\frac {\sqrt {4 a-4 b+d}}{\sqrt {d}}\right ) (-a+x)^{2/3} \left (\frac {-b+x}{a-b}\right )^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{\left (-2 a-d+\sqrt {d} \sqrt {4 a-4 b+d}+2 x\right ) \left (-\frac {b}{a-b}+\frac {x}{a-b}\right )^{2/3}} \, dx}{((-a+x) (-b+x))^{2/3}}+\frac {\left (\left (1+\frac {\sqrt {4 a-4 b+d}}{\sqrt {d}}\right ) (-a+x)^{2/3} \left (\frac {-b+x}{a-b}\right )^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{\left (-2 a-d-\sqrt {d} \sqrt {4 a-4 b+d}+2 x\right ) \left (-\frac {b}{a-b}+\frac {x}{a-b}\right )^{2/3}} \, dx}{((-a+x) (-b+x))^{2/3}}\\ &=-\frac {3 (a-x)^2 \left (-\frac {b-x}{a-b}\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};\frac {a-x}{a-b},-\frac {2 (a-x)}{\sqrt {d} \left (\sqrt {d}-\sqrt {4 a-4 b+d}\right )}\right )}{4 d ((a-x) (b-x))^{2/3}}-\frac {3 (a-x)^2 \left (-\frac {b-x}{a-b}\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};\frac {a-x}{a-b},-\frac {2 (a-x)}{\sqrt {d} \left (\sqrt {d}+\sqrt {4 a-4 b+d}\right )}\right )}{4 d ((a-x) (b-x))^{2/3}}\\ \end {align*}
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Mathematica [F] time = 1.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a (a-2 b)-2 b x+x^2}{((-a+x) (-b+x))^{2/3} \left (a^2+b d-(2 a+d) x+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.63, size = 216, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \left (a b+(-a-b) x+x^2\right )^{2/3}}{-2 b \sqrt [3]{d}+2 \sqrt [3]{d} x+\left (a b+(-a-b) x+x^2\right )^{2/3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (b \sqrt {d}-\sqrt {d} x+\sqrt [6]{d} \left (a b+(-a-b) x+x^2\right )^{2/3}\right )}{\sqrt [3]{d}}-\frac {\log \left (b^2 d-2 b d x+d x^2+\left (-b d^{2/3}+d^{2/3} x\right ) \left (a b+(-a-b) x+x^2\right )^{2/3}+\sqrt [3]{d} \left (a b+(-a-b) x+x^2\right )^{4/3}\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 {{\left (a - 2 \, b\right )} a + 2 \, b x - x^{2}}{{\left (a^{2} + b d - {\left (2 \, a + d\right )} x + x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\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 = 0.23, size = 0, normalized size = 0.00 \[\int \frac {-a \left (a -2 b \right )-2 b x +x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )\right )^{\frac {2}{3}} \left (a^{2}+b d -\left (2 a +d \right ) x +x^{2}\right )}\, dx\]
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 {{\left (a - 2 \, b\right )} a + 2 \, b x - x^{2}}{{\left (a^{2} + b d - {\left (2 \, a + d\right )} x + x^{2}\right )} \left ({\left (a - x\right )} {\left (b - x\right )}\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 = -0.00 \begin {gather*} -\int \frac {-x^2+2\,b\,x+a\,\left (a-2\,b\right )}{{\left (\left (a-x\right )\,\left (b-x\right )\right )}^{2/3}\,\left (b\,d-x\,\left (2\,a+d\right )+a^2+x^2\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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