Optimal. Leaf size=273 \[ -\frac {\log \left (a^2+d^{2/3} \left (x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3\right )^{2/3}+\sqrt [3]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3} \left (\sqrt [3]{d} x-a \sqrt [3]{d}\right )-2 a x+x^2\right )}{4 \sqrt [3]{d} (a-b)}+\frac {\log \left (\sqrt [3]{d} \sqrt [3]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}+a-x\right )}{2 \sqrt [3]{d} (a-b)}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a-\sqrt {3} x}{-2 \sqrt [3]{d} \sqrt [3]{x \left (2 a b+b^2\right )-a b^2+x^2 (-a-2 b)+x^3}+a-x}\right )}{2 \sqrt [3]{d} (a-b)} \]
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Rubi [A] time = 1.13, antiderivative size = 513, normalized size of antiderivative = 1.88, number of steps used = 10, number of rules used = 6, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {6719, 24, 911, 105, 59, 91} \begin {gather*} -\frac {(x-a)^{2/3} (x-b)^{4/3} \log \left (2 \left (1-\sqrt {d}\right ) \left (a+b \sqrt {d}\right )-2 (1-d) x\right )}{4 \sqrt [3]{d} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {(x-a)^{2/3} (x-b)^{4/3} \log \left (2 (1-d) x-2 \left (\sqrt {d}+1\right ) \left (a-b \sqrt {d}\right )\right )}{4 \sqrt [3]{d} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {3 (x-a)^{2/3} (x-b)^{4/3} \log \left (-\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{x-b}\right )}{4 \sqrt [3]{d} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {3 (x-a)^{2/3} (x-b)^{4/3} \log \left (\sqrt [6]{d} \sqrt [3]{x-b}-\sqrt [3]{x-a}\right )}{4 \sqrt [3]{d} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {\sqrt {3} (x-a)^{2/3} (x-b)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{x-a}}\right )}{2 \sqrt [3]{d} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {\sqrt {3} (x-a)^{2/3} (x-b)^{4/3} \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{d} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 24
Rule 59
Rule 91
Rule 105
Rule 911
Rule 6719
Rubi steps
\begin {align*} \int \frac {a b-(a+b) x+x^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (a^2-b^2 d-2 (a-b d) x+(1-d) x^2\right )} \, dx &=\frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {a b-(a+b) x+x^2}{(-a+x)^{2/3} (-b+x)^{4/3} \left (a^2-b^2 d-2 (a-b d) x+(1-d) x^2\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {\sqrt [3]{-a+x}}{\sqrt [3]{-b+x} \left (a^2-b^2 d-2 (a-b d) x+(1-d) x^2\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \left (\frac {(1-d) \sqrt [3]{-a+x}}{(a-b) \sqrt {d} \sqrt [3]{-b+x} \left (2 a-2 (a-b) \sqrt {d}-2 b d-2 (1-d) x\right )}+\frac {(1-d) \sqrt [3]{-a+x}}{(a-b) \sqrt {d} \sqrt [3]{-b+x} \left (-2 a-2 (a-b) \sqrt {d}+2 b d+2 (1-d) x\right )}\right ) \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left ((1-d) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {\sqrt [3]{-a+x}}{\sqrt [3]{-b+x} \left (2 a-2 (a-b) \sqrt {d}-2 b d-2 (1-d) x\right )} \, dx}{(a-b) \sqrt {d} \left ((-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left ((1-d) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {\sqrt [3]{-a+x}}{\sqrt [3]{-b+x} \left (-2 a-2 (a-b) \sqrt {d}+2 b d+2 (1-d) x\right )} \, dx}{(a-b) \sqrt {d} \left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left ((1-d) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (-2 a-2 (a-b) \sqrt {d}+2 b d+2 (1-d) x\right )} \, dx}{\left (1-\sqrt {d}\right ) \left ((-a+x) (-b+x)^2\right )^{2/3}}-\frac {\left ((1-d) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (2 a-2 (a-b) \sqrt {d}-2 b d-2 (1-d) x\right )} \, dx}{\left (1+\sqrt {d}\right ) \left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\sqrt {3} (-a+x)^{2/3} (-b+x)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{-b+x}}{\sqrt {3} \sqrt [3]{-a+x}}\right )}{2 (a-b) \sqrt [3]{d} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {\sqrt {3} (-a+x)^{2/3} (-b+x)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [3]{-b+x}}{\sqrt {3} \sqrt [3]{-a+x}}\right )}{2 (a-b) \sqrt [3]{d} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {(-a+x)^{2/3} (-b+x)^{4/3} \log \left (2 \left (1-\sqrt {d}\right ) \left (a+b \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) \sqrt [3]{d} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {(-a+x)^{2/3} (-b+x)^{4/3} \log \left (-2 \left (1+\sqrt {d}\right ) \left (a-b \sqrt {d}\right )+2 (1-d) x\right )}{4 (a-b) \sqrt [3]{d} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {3 (-a+x)^{2/3} (-b+x)^{4/3} \log \left (-\sqrt [3]{-a+x}-\sqrt [6]{d} \sqrt [3]{-b+x}\right )}{4 (a-b) \sqrt [3]{d} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {3 (-a+x)^{2/3} (-b+x)^{4/3} \log \left (-\sqrt [3]{-a+x}+\sqrt [6]{d} \sqrt [3]{-b+x}\right )}{4 (a-b) \sqrt [3]{d} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.28, size = 88, normalized size = 0.32 \begin {gather*} -\frac {3 \sqrt [3]{(x-a) (b-x)^2} \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} (b-x)}{x-a}\right )+\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} (x-b)}{x-a}\right )\right )}{4 (a-b) (x-a)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 6.37, size = 273, normalized size = 1.00 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a-\sqrt {3} x}{a-x-2 \sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}}\right )}{2 (a-b) \sqrt [3]{d}}+\frac {\log \left (a-x+\sqrt [3]{d} \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}\right )}{2 (a-b) \sqrt [3]{d}}-\frac {\log \left (a^2-2 a x+x^2+\left (-a \sqrt [3]{d}+\sqrt [3]{d} x\right ) \sqrt [3]{-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3}+d^{2/3} \left (-a b^2+\left (2 a b+b^2\right ) x+(-a-2 b) x^2+x^3\right )^{2/3}\right )}{4 (a-b) \sqrt [3]{d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 740, normalized size = 2.71 \begin {gather*} \left [-\frac {\sqrt {3} d \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \log \left (-\frac {b^{2} d + {\left (d + 2\right )} x^{2} + 2 \, a^{2} - 2 \, {\left (b d + 2 \, a\right )} x + \sqrt {3} {\left (2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (a - x\right )} \left (-d\right )^{\frac {2}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} d - {\left (b^{2} d - 2 \, b d x + d x^{2}\right )} \left (-d\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (-d\right )^{\frac {1}{3}}}{d}} - 3 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} \left (-d\right )^{\frac {2}{3}}}{b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x}\right ) + \left (-d\right )^{\frac {2}{3}} \log \left (\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (a - x\right )} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} \left (-d\right )^{\frac {1}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) - 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {1}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}}}{b^{2} - 2 \, b x + x^{2}}\right )}{4 \, {\left (a - b\right )} d}, \frac {2 \, \sqrt {3} d \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}} \arctan \left (-\frac {\sqrt {3} {\left ({\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {1}{3}} - 2 \, {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}}\right )} \sqrt {-\frac {\left (-d\right )^{\frac {1}{3}}}{d}}}{3 \, {\left (b^{2} - 2 \, b x + x^{2}\right )}}\right ) - \left (-d\right )^{\frac {2}{3}} \log \left (\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {2}{3}} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {1}{3}} {\left (a - x\right )} - {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}} \left (-d\right )^{\frac {1}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) + 2 \, \left (-d\right )^{\frac {2}{3}} \log \left (\frac {{\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d\right )^{\frac {1}{3}} + {\left (-a b^{2} - {\left (a + 2 \, b\right )} x^{2} + x^{3} + {\left (2 \, a b + b^{2}\right )} x\right )}^{\frac {2}{3}}}{b^{2} - 2 \, b x + x^{2}}\right )}{4 \, {\left (a - b\right )} d}\right ] \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 - {\left (a + b\right )} x + x^{2}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - 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 = 0.16, size = 0, normalized size = 0.00 \[\int \frac {a b -\left (a +b \right ) x +x^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (a^{2}-b^{2} d -2 \left (-b d +a \right ) x +\left (1-d \right ) x^{2}\right )}\, dx\]
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 - {\left (a + b\right )} x + x^{2}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - 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 {x^2+\left (-a-b\right )\,x+a\,b}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (b^2\,d+2\,x\,\left (a-b\,d\right )-a^2+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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