Optimal. Leaf size=857 \[ \frac {(b-x)^{2/3} \sqrt [3]{x-a} \left (-\frac {\sqrt {3} (d-1) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [3]{d} \sqrt [3]{b-x}}\right ) a}{(a-b)^2 \sqrt [3]{d}}-\frac {(d-1) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right ) a}{(a-b)^2 \sqrt [3]{d}}+\frac {(d-1) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right ) a}{2 (a-b)^2 \sqrt [3]{d}}+\frac {3 \sqrt [3]{b-x} (x-a)^{2/3} a}{2 (a-b)^2 (x-b)}-\frac {\sqrt {3} c (a-b d) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [3]{d} \sqrt [3]{b-x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {\sqrt {3} (a-b d) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [3]{d} \sqrt [3]{b-x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {\sqrt {3} b c (d-1) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [3]{d} \sqrt [3]{b-x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {c (a-b d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{(a-b)^2 \sqrt [3]{d}}+\frac {(b d-a) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {b c (d-1) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{x-a}\right )}{(a-b)^2 \sqrt [3]{d}}+\frac {c (a-b d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}+\frac {(a-b d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}+\frac {b c (d-1) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}-\frac {3 b \sqrt [3]{b-x} (x-a)^{2/3}}{2 (a-b)^2 (x-b)}\right )}{\sqrt [3]{(b-x)^2 (x-a)}} \]
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Rubi [A] time = 1.44, antiderivative size = 283, normalized size of antiderivative = 0.33, number of steps used = 4, number of rules used = 4, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.078, Rules used = {6719, 155, 12, 91} \begin {gather*} \frac {\sqrt [3]{x-a} (x-b)^{2/3} (c+d) \log (a-b d-(1-d) x)}{2 \sqrt [3]{d} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} (c+d) \log \left (\frac {\sqrt [3]{x-a}}{\sqrt [3]{d}}-\sqrt [3]{x-b}\right )}{2 \sqrt [3]{d} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} (c+d) \tan ^{-1}\left (\frac {2 \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{d} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {3 (a-x)}{2 (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}} \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}{(-b+x) \sqrt [3]{(-a+x) (-b+x)^2} (a-b d+(-1+d) x)} \, dx &=\frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {-a-b c+(1+c) x}{\sqrt [3]{-a+x} (-b+x)^{5/3} (a-b d+(-1+d) x)} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {3 (a-x)}{2 (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (3 \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {2 (a-b)^2 (c+d)}{3 \sqrt [3]{-a+x} (-b+x)^{2/3} (a-b d+(-1+d) x)} \, dx}{2 (a-b)^2 \sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {3 (a-x)}{2 (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left ((c+d) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{-a+x} (-b+x)^{2/3} (a-b d+(-1+d) x)} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {3 (a-x)}{2 (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt {3} (c+d) \sqrt [3]{-a+x} (-b+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{-b+x}}\right )}{(a-b) \sqrt [3]{d} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {(c+d) \sqrt [3]{-a+x} (-b+x)^{2/3} \log (a-b d-(1-d) x)}{2 (a-b) \sqrt [3]{d} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 (c+d) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (\frac {\sqrt [3]{-a+x}}{\sqrt [3]{d}}-\sqrt [3]{-b+x}\right )}{2 (a-b) \sqrt [3]{d} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 65, normalized size = 0.08 \begin {gather*} \frac {3 \left (2 (x-b) (c+d) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {d (b-x)}{a-x}\right )+a-x\right )}{2 (a-b) \sqrt [3]{(x-a) (b-x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 11.66, size = 242, normalized size = 0.28 \begin {gather*} \frac {(b-x)^{2/3} \sqrt [3]{-a+x} \left (-\frac {3 (-a+x)^{2/3}}{2 (a-b) (b-x)^{2/3}}+\frac {\sqrt {3} (c+d) \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) \sqrt [3]{d}}+\frac {(c+d) \log \left (1+\frac {d^{2/3} (b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )}{2 (a-b) \sqrt [3]{d}}+\frac {(-c-d) \log \left (1+\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )}{(a-b) \sqrt [3]{d}}\right )}{\sqrt [3]{(b-x)^2 (-a+x)}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 976, normalized size = 1.14
result too large to display
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 {1}{3}} {\left (b d - {\left (d - 1\right )} x - a\right )} {\left (b - x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {-a -b c +\left (1+c \right ) x}{\left (-b +x \right ) \left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (a -b d +\left (-1+d \right ) x \right )}\, dx\]
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 {b c - {\left (c + 1\right )} x + a}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (b d - {\left (d - 1\right )} x - a\right )} {\left (b - 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\,c-x\,\left (c+1\right )}{\left (b-x\right )\,{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\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(-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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