Optimal. Leaf size=369 \[ -\frac {i \left (\sqrt {3}-i\right ) \log \left (\sqrt [3]{-1} \left (a^2-2 a x+x^2\right )-d^{2/3} \left (x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3\right )^{2/3}+(-1)^{2/3} \sqrt [3]{d} (a-x) \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}\right )}{4 d^{2/3}}+\frac {\left (1+i \sqrt {3}\right ) \log \left (\sqrt [3]{d} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+(-1)^{2/3} (a-x)\right )}{2 d^{2/3}}+\frac {\sqrt {\frac {1}{2} \left (-3+3 i \sqrt {3}\right )} \tan ^{-1}\left (\frac {3 \sqrt [3]{d} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{x^2 (-a-b-c)+x (a b+a c+b c)-a b c+x^3}+\sqrt {3} a-3 i a-\sqrt {3} x+3 i x}\right )}{d^{2/3}} \]
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Rubi [F] time = 10.15, 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 (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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\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 (a^2-b c d+(-2 a+b d+c 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 (a^2-b c d+(-2 a+b d+c 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-\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}}}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x} \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 {-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}}}{\sqrt [3]{-a+x} \sqrt [3]{-b+x} \sqrt [3]{-c+x} \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}{\sqrt [3]{(-a+x) (-b+x) (-c+x)}}\\ &=\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 ) \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 (-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}{\sqrt [3]{(-a+x) (-b+x) (-c+x)}}+\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 ) \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 (-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}{\sqrt [3]{(-a+x) (-b+x) (-c+x)}}\\ \end {align*}
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Mathematica [F] time = 9.93, 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 (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 = 6.34, size = 455, normalized size = 1.23 \begin {gather*} \frac {\sqrt {\frac {1}{2} \left (-3+3 i \sqrt {3}\right )} \tan ^{-1}\left (\frac {3 \sqrt [3]{d} \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}{-3 i a+\sqrt {3} a+3 i x-\sqrt {3} x+\sqrt {3} \sqrt [3]{d} \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}}\right )}{d^{2/3}}+\frac {\left (1+i \sqrt {3}\right ) \log \left (-a+\sqrt {3} (i a-i x)+x+2 \sqrt [3]{d} \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right )}{2 d^{2/3}}-\frac {i \left (-i+\sqrt {3}\right ) \log \left (-a^2+2 a x-x^2+\sqrt [3]{d} (a-x) \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}+2 d^{2/3} \left (-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3\right )^{2/3}+\sqrt {3} \left (-i a^2+2 i a x-i x^2+\sqrt [3]{d} (-i a+i x) \sqrt [3]{-a b c+(a b+a c+b c) x+(-a-b-c) x^2+x^3}\right )\right )}{4 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 {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 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 = 0.22, size = 0, normalized size = 0.00 \[\int \frac {a b +a c -2 b c +\left (-2 a +b +c \right ) x}{\left (\left (-a +x \right ) \left (-b +x \right ) \left (-c +x \right )\right )^{\frac {1}{3}} \left (a^{2}-b c d +\left (b d +c d -2 a \right ) x +\left (1-d \right ) x^{2}\right )}\, dx\]
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 (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 {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\,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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