Optimal. Leaf size=262 \[ -\frac {\log \left (a^2+d^{2/3} \left (x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3\right )^{2/3}+\left (\sqrt [3]{d} x-a \sqrt [3]{d}\right ) \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}-2 a x+x^2\right )}{2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+a-x\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a-\sqrt {3} x}{-2 \sqrt [3]{d} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+a-x}\right )}{\sqrt [3]{d}} \]
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Rubi [F] time = 12.90, 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 {a (a b+a c-2 b c)-2 \left (a^2-b c\right ) x+(2 a-b-c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (a^2-b c d+(-2 a+b d+c 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 {a (a b+a c-2 b c)-2 \left (a^2-b c\right ) x+(2 a-b-c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx &=\frac {\left ((-a+x)^{2/3} (-b+x)^{2/3} (-c+x)^{2/3}\right ) \int \frac {a (a b+a c-2 b c)-2 \left (a^2-b c\right ) x+(2 a-b-c) x^2}{(-a+x)^{2/3} (-b+x)^{2/3} (-c+x)^{2/3} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{2/3} (-c+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x} (-a b-a c+2 b c+(2 a-b-c) x)}{(-b+x)^{2/3} (-c+x)^{2/3} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{2/3} (-c+x)^{2/3}\right ) \int \left (\frac {\left (2 a-b-c+\frac {\sqrt {4 a^2-4 a b-4 a c+4 b c+b^2 d-2 b c d+c^2 d}}{\sqrt {d}}\right ) \sqrt [3]{-a+x}}{(-b+x)^{2/3} (-c+x)^{2/3} \left (-2 a+b d+c d-\sqrt {d} \sqrt {4 a^2-4 a b-4 a c+4 b c+b^2 d-2 b c d+c^2 d}+2 (1-d) x\right )}+\frac {\left (2 a-b-c-\frac {\sqrt {4 a^2-4 a b-4 a c+4 b c+b^2 d-2 b c d+c^2 d}}{\sqrt {d}}\right ) \sqrt [3]{-a+x}}{(-b+x)^{2/3} (-c+x)^{2/3} \left (-2 a+b d+c d+\sqrt {d} \sqrt {4 a^2-4 a b-4 a c+4 b c+b^2 d-2 b c d+c^2 d}+2 (1-d) x\right )}\right ) \, dx}{((-a+x) (-b+x) (-c+x))^{2/3}}\\ &=\frac {\left (\left (2 a-b-c-\frac {\sqrt {4 a^2-4 a (b+c)+2 b c (2-d)+b^2 d+c^2 d}}{\sqrt {d}}\right ) (-a+x)^{2/3} (-b+x)^{2/3} (-c+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{(-b+x)^{2/3} (-c+x)^{2/3} \left (-2 a+b d+c d+\sqrt {d} \sqrt {4 a^2-4 a b-4 a c+4 b c+b^2 d-2 b c d+c^2 d}+2 (1-d) x\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{2/3}}+\frac {\left (\left (2 a-b-c+\frac {\sqrt {4 a^2-4 a (b+c)+2 b c (2-d)+b^2 d+c^2 d}}{\sqrt {d}}\right ) (-a+x)^{2/3} (-b+x)^{2/3} (-c+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{(-b+x)^{2/3} (-c+x)^{2/3} \left (-2 a+b d+c d-\sqrt {d} \sqrt {4 a^2-4 a b-4 a c+4 b c+b^2 d-2 b c d+c^2 d}+2 (1-d) x\right )} \, dx}{((-a+x) (-b+x) (-c+x))^{2/3}}\\ \end {align*}
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Mathematica [F] time = 3.50, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a (a b+a c-2 b c)-2 \left (a^2-b c\right ) x+(2 a-b-c) x^2}{((-a+x) (-b+x) (-c+x))^{2/3} \left (a^2-b c d+(-2 a+b d+c d) x+(1-d) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 7.20, size = 262, normalized size = 1.00 \begin {gather*} -\frac {\log \left (a^2+d^{2/3} \left (x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3\right )^{2/3}+\left (\sqrt [3]{d} x-a \sqrt [3]{d}\right ) \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}-2 a x+x^2\right )}{2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+a-x\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a-\sqrt {3} x}{-2 \sqrt [3]{d} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+a-x}\right )}{\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 (2 \, a - b - c\right )} x^{2} + {\left (a b + a c - 2 \, b c\right )} a - 2 \, {\left (a^{2} - b c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {2}{3}} {\left (b c d + {\left (d - 1\right )} x^{2} - a^{2} - {\left (b d + c d - 2 \, a\right )} x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.82, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a \left (a b +a c -2 b c \right )-2 \left (a^{2}-b c \right ) x +\left (2 a -b -c \right ) x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )\right )^{\frac {2}{3}} \left (a^{2}-b c d +\left (b d +c d -2 a \right ) x +\left (1-d \right ) 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 (2 \, a - b - c\right )} x^{2} + {\left (a b + a c - 2 \, b c\right )} a - 2 \, {\left (a^{2} - b c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {2}{3}} {\left (b c d + {\left (d - 1\right )} x^{2} - a^{2} - {\left (b d + c d - 2 \, a\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 = -0.00 \begin {gather*} \int \frac {2\,x\,\left (b\,c-a^2\right )-x^2\,\left (b-2\,a+c\right )+a\,\left (a\,b+a\,c-2\,b\,c\right )}{{\left (-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )\right )}^{2/3}\,\left (x\,\left (b\,d-2\,a+c\,d\right )+a^2-x^2\,\left (d-1\right )-b\,c\,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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