Optimal. Leaf size=201 \[ -\frac {\log \left (d^{2/3} \left (x (-a-b)+a b+x^2\right )^{4/3}+\left (x (-a-b)+a b+x^2\right )^{2/3} \left (\sqrt [3]{d} x-b \sqrt [3]{d}\right )+b^2-2 b x+x^2\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}+b-x\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}}{\sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}-2 b+2 x}\right )}{d^{2/3}} \]
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Rubi [C] time = 1.56, antiderivative size = 191, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 5, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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 d (a-x)}{1-\sqrt {4 a d-4 b d+1}}\right )}{4 ((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 d (a-x)}{\sqrt {4 a d-4 b d+1}+1}\right )}{4 ((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 (b+a^2 d-(1+2 a d) x+d 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 (b+a^2 d-(1+2 a d) x+d 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 (b+a^2 d-(1+2 a d) x+d 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+\sqrt {1+4 a d-4 b d}\right ) \sqrt [3]{-a+x}}{(-b+x)^{2/3} \left (-1-2 a d-\sqrt {1+4 a d-4 b d}+2 d x\right )}+\frac {\left (1-\sqrt {1+4 a d-4 b d}\right ) \sqrt [3]{-a+x}}{(-b+x)^{2/3} \left (-1-2 a d+\sqrt {1+4 a d-4 b d}+2 d x\right )}\right ) \, dx}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left (\left (1-\sqrt {1+4 a d-4 b d}\right ) (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{(-b+x)^{2/3} \left (-1-2 a d+\sqrt {1+4 a d-4 b d}+2 d x\right )} \, dx}{((-a+x) (-b+x))^{2/3}}+\frac {\left (\left (1+\sqrt {1+4 a d-4 b d}\right ) (-a+x)^{2/3} (-b+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{(-b+x)^{2/3} \left (-1-2 a d-\sqrt {1+4 a d-4 b d}+2 d x\right )} \, dx}{((-a+x) (-b+x))^{2/3}}\\ &=\frac {\left (\left (1-\sqrt {1+4 a d-4 b d}\right ) (-a+x)^{2/3} \left (\frac {-b+x}{a-b}\right )^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{\left (-\frac {b}{a-b}+\frac {x}{a-b}\right )^{2/3} \left (-1-2 a d+\sqrt {1+4 a d-4 b d}+2 d x\right )} \, dx}{((-a+x) (-b+x))^{2/3}}+\frac {\left (\left (1+\sqrt {1+4 a d-4 b d}\right ) (-a+x)^{2/3} \left (\frac {-b+x}{a-b}\right )^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{\left (-\frac {b}{a-b}+\frac {x}{a-b}\right )^{2/3} \left (-1-2 a d-\sqrt {1+4 a d-4 b d}+2 d x\right )} \, 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 d (a-x)}{1-\sqrt {1+4 a d-4 b d}}\right )}{4 ((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 d (a-x)}{1+\sqrt {1+4 a d-4 b d}}\right )}{4 ((a-x) (b-x))^{2/3}}\\ \end {align*}
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Mathematica [F] time = 1.07, 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 (b+a^2 d-(1+2 a d) x+d x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.77, size = 201, normalized size = 1.00 \begin {gather*} -\frac {\log \left (d^{2/3} \left (x (-a-b)+a b+x^2\right )^{4/3}+\left (x (-a-b)+a b+x^2\right )^{2/3} \left (\sqrt [3]{d} x-b \sqrt [3]{d}\right )+b^2-2 b x+x^2\right )}{2 d^{2/3}}+\frac {\log \left (\sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}+b-x\right )}{d^{2/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}}{\sqrt [3]{d} \left (x (-a-b)+a b+x^2\right )^{2/3}-2 b+2 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 - 2 \, b\right )} a + 2 \, b x - x^{2}}{{\left (a^{2} d + d x^{2} - {\left (2 \, a d + 1\right )} x + b\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 = 0.69, size = 0, normalized size = 0.00 \begin {gather*} \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 (b +a^{2} d -\left (2 a d +1\right ) x +d \,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 - 2 \, b\right )} a + 2 \, b x - x^{2}}{{\left (a^{2} d + d x^{2} - {\left (2 \, a d + 1\right )} x + b\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-x\,\left (2\,a\,d+1\right )+a^2\,d+d\,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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