Optimal. Leaf size=849 \[ \frac {(b-x)^{4/3} (x-a)^{2/3} \left (\frac {\sqrt {3} (d-1) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [3]{d} \sqrt [3]{b-x}}\right ) a}{(a-b)^2 d^{2/3}}-\frac {(d-1) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right ) a}{(a-b)^2 d^{2/3}}+\frac {(d-1) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right ) a}{2 (a-b)^2 d^{2/3}}-\frac {3 (b-x)^{2/3} \sqrt [3]{x-a} a}{(a-b)^2 (x-b)}+\frac {\sqrt {3} c (a-b d) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [3]{d} \sqrt [3]{b-x}}\right )}{(a-b)^2 d^{2/3}}+\frac {\sqrt {3} (a-b d) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [3]{d} \sqrt [3]{b-x}}\right )}{(a-b)^2 d^{2/3}}+\frac {\sqrt {3} b c (d-1) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [3]{d} \sqrt [3]{b-x}}\right )}{(a-b)^2 d^{2/3}}-\frac {c (a-b d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{(a-b)^2 d^{2/3}}+\frac {(b d-a) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{(a-b)^2 d^{2/3}}-\frac {b c (d-1) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{(a-b)^2 d^{2/3}}+\frac {c (a-b d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{2 (a-b)^2 d^{2/3}}+\frac {(a-b d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{2 (a-b)^2 d^{2/3}}+\frac {b c (d-1) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{2 (a-b)^2 d^{2/3}}+\frac {3 b (b-x)^{2/3} \sqrt [3]{x-a}}{(a-b)^2 (x-b)}\right )}{\left ((b-x)^2 (x-a)\right )^{2/3}} \]
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Rubi [A] time = 0.88, antiderivative size = 286, normalized size of antiderivative = 0.34, number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6719, 155, 12, 91} \begin {gather*} \frac {(x-a)^{2/3} (x-b)^{4/3} (c+d) \log (a-b d-(1-d) x)}{2 d^{2/3} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {3 (x-a)^{2/3} (x-b)^{4/3} (c+d) \log \left (\sqrt [3]{d} \sqrt [3]{x-b}-\sqrt [3]{x-a}\right )}{2 d^{2/3} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {\sqrt {3} (x-a)^{2/3} (x-b)^{4/3} (c+d) \tan ^{-1}\left (\frac {2 \sqrt [3]{d} \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{d^{2/3} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {3 (a-x) (b-x)}{(a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 91
Rule 155
Rule 6719
Rubi steps
\begin {align*} \int \frac {-a-b c+(1+c) x}{\left ((-a+x) (-b+x)^2\right )^{2/3} (a-b d+(-1+d) x)} \, dx &=\frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {-a-b c+(1+c) x}{(-a+x)^{2/3} (-b+x)^{4/3} (a-b d+(-1+d) x)} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=-\frac {3 (a-x) (b-x)}{(a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {\left (3 (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {(a-b)^2 (c+d)}{3 (-a+x)^{2/3} \sqrt [3]{-b+x} (a-b d+(-1+d) x)} \, dx}{(a-b)^2 \left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=-\frac {3 (a-x) (b-x)}{(a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {\left ((c+d) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{(-a+x)^{2/3} \sqrt [3]{-b+x} (a-b d+(-1+d) x)} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=-\frac {3 (a-x) (b-x)}{(a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {\sqrt {3} (c+d) (-a+x)^{2/3} (-b+x)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{-b+x}}{\sqrt {3} \sqrt [3]{-a+x}}\right )}{(a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {(c+d) (-a+x)^{2/3} (-b+x)^{4/3} \log (a-b d-(1-d) x)}{2 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {3 (c+d) (-a+x)^{2/3} (-b+x)^{4/3} \log \left (-\sqrt [3]{-a+x}+\sqrt [3]{d} \sqrt [3]{-b+x}\right )}{2 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 72, normalized size = 0.08 \begin {gather*} \frac {3 (x-b) \left ((x-b) (c+d) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {d (b-x)}{a-x}\right )+2 (a-x)\right )}{2 (a-b) \left ((x-a) (b-x)^2\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.04, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 389, normalized size = 0.46 \begin {gather*} -\frac {2 \, \sqrt {3} {\left (b c d + b d^{2} - {\left (c d + d^{2}\right )} x\right )} {\left (d^{2}\right )}^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} {\left ({\left (b d - d x\right )} {\left (d^{2}\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 {1}{3}} {\left (d^{2}\right )}^{\frac {2}{3}}\right )}}{3 \, {\left (b d^{2} - d^{2} x\right )}}\right ) + {\left (b c + b d - {\left (c + d\right )} x\right )} {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\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 (d^{2}\right )}^{\frac {2}{3}} {\left (b - 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}} d - {\left (b^{2} d - 2 \, b d x + d x^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) - 2 \, {\left (b c + b d - {\left (c + d\right )} x\right )} {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (d^{2}\right )}^{\frac {2}{3}} {\left (b - 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 {1}{3}} d}{b - x}\right ) + 6 \, {\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}} d^{2}}{2 \, {\left ({\left (a - b\right )} d^{2} x - {\left (a b - b^{2}\right )} d^{2}\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 {b c - {\left (c + 1\right )} x + a}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b d - {\left (d - 1\right )} x - a\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.42, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-a -b c +\left (1+c \right ) x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (a -b d +\left (-1+d \right ) x \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 {b c - {\left (c + 1\right )} x + a}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b d - {\left (d - 1\right )} x - a\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\,c-x\,\left (c+1\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (a-b\,d+x\,\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {- a - b c + c x + x}{\left (\left (- a + x\right ) \left (- b + x\right )^{2}\right )^{\frac {2}{3}} \left (a - b d + d x - x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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