Optimal. Leaf size=311 \[ -\frac {\log \left (a^2 d^{2/3}+\left (\sqrt [3]{d} x-a \sqrt [3]{d}\right ) \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+\left (x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3\right )^{2/3}-2 a d^{2/3} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+a \sqrt [3]{d}-\sqrt [3]{d} x\right )}{\sqrt [3]{d}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}}{\sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x}\right )}{\sqrt [3]{d}} \]
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Rubi [F] time = 9.85, 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 b-a c+2 b c+(2 a-b-c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {-a b-a c+2 b c+(2 a-b-c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x}\right ) \int \frac {-a b-a c+2 b c+(2 a-b-c) x}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x}\right ) \int \left (\frac {2 a-b-c+\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x} \left (b+c-2 a d-\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}+2 (-1+d) x\right )}+\frac {2 a-b-c-\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x} \left (b+c-2 a d+\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}+2 (-1+d) x\right )}\right ) \, dx}{\sqrt [3]{(-a+x) (-b+x) (-c+x)}}\\ &=\frac {\left (\left (2 a-b-c-\sqrt {b^2+c^2+4 a^2 d-4 a c d-2 b (c+2 a d-2 c d)}\right ) \sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x}\right ) \int \frac {1}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x} \left (b+c-2 a d+\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}+2 (-1+d) x\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x) (-c+x)}}+\frac {\left (\left (2 a-b-c+\sqrt {b^2+c^2+4 a^2 d-4 a c d-2 b (c+2 a d-2 c d)}\right ) \sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x}\right ) \int \frac {1}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x} \left (b+c-2 a d-\sqrt {b^2-2 b c+c^2+4 a^2 d-4 a b d-4 a c d+4 b c d}+2 (-1+d) x\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x) (-c+x)}}\\ \end {align*}
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Mathematica [F] time = 9.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a b-a c+2 b c+(2 a-b-c) x}{\sqrt [3]{(-a+x) (-b+x) (-c+x)} \left (-b c+a^2 d+(b+c-2 a d) x+(-1+d) x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.15, size = 311, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{-2 a \sqrt [3]{d}+2 \sqrt [3]{d} x+\sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a \sqrt [3]{d}-\sqrt [3]{d} x+\sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 d^{2/3}-2 a d^{2/3} x+d^{2/3} x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+\left (-a b c+(a b+a c+b c) x+(-a-b-c) 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 {a b + a c - 2 \, b c - {\left (2 \, a - b - c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {1}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b c - {\left (2 \, a d - b - c\right )} x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {-a b -a c +2 b c +\left (2 a -b -c \right ) x}{\left (\left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )\right )^{\frac {1}{3}} \left (-b c +a^{2} d +\left (-2 a d +b +c \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]
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 {a b + a c - 2 \, b c - {\left (2 \, a - b - c\right )} x}{\left (-{\left (a - x\right )} {\left (b - x\right )} {\left (c - x\right )}\right )^{\frac {1}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b c - {\left (2 \, a d - b - c\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 {a\,b+a\,c-2\,b\,c+x\,\left (b-2\,a+c\right )}{{\left (-\left (a-x\right )\,\left (b-x\right )\,\left (c-x\right )\right )}^{1/3}\,\left (x\,\left (b+c-2\,a\,d\right )-b\,c+a^2\,d+x^2\,\left (d-1\right )\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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