Optimal. Leaf size=857 \[ \frac {(b-x)^{2/3} \sqrt [3]{-a+x} \left (\frac {3 a \sqrt [3]{b-x} (-a+x)^{2/3}}{2 (a-b)^2 (-b+x)}-\frac {3 b \sqrt [3]{b-x} (-a+x)^{2/3}}{2 (a-b)^2 (-b+x)}-\frac {\sqrt {3} a (-1+d) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {\sqrt {3} b c (-1+d) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {\sqrt {3} (a-b d) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {\sqrt {3} c (a-b d) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {a (-1+d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {b c (-1+d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 \sqrt [3]{d}}-\frac {c (a-b d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 \sqrt [3]{d}}+\frac {(-a+b d) \log \left (\sqrt [3]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{(a-b)^2 \sqrt [3]{d}}+\frac {a (-1+d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}+\frac {b c (-1+d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}+\frac {(a-b d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}+\frac {c (a-b d) \log \left (d^{2/3} (b-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{2 (a-b)^2 \sqrt [3]{d}}\right )}{\sqrt [3]{(b-x)^2 (-a+x)}} \]
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Rubi [A]
time = 1.07, 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 = {6851, 160, 12,
93} \begin {gather*} -\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} (c+d) \text {ArcTan}\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 {\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 {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 93
Rule 160
Rule 6851
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 [A]
time = 0.28, size = 282, normalized size = 0.33 \begin {gather*} \frac {3 a \sqrt [3]{d}-3 \sqrt [3]{d} x+2 \sqrt {3} (c+d) (b-x)^{2/3} \sqrt [3]{-a+x} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}}{\sqrt {3}}\right )+(c+d) (b-x)^{2/3} \sqrt [3]{-a+x} \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 c (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (1+\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 d (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (1+\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )}{2 (a-b) \sqrt [3]{d} \sqrt [3]{(b-x)^2 (-a+x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, 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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 976, normalized size = 1.14 \begin {gather*} \left [-\frac {\sqrt {3} {\left (b^{2} c d + b^{2} d^{2} + {\left (c d + d^{2}\right )} x^{2} - 2 \, {\left (b c d + b d^{2}\right )} x\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (-\frac {b^{2} d + {\left (d + 2\right )} x^{2} + 2 \, a b + 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 (b - x\right )} d^{\frac {2}{3}} - 2 \, {\left (b d + a + b\right )} x + \sqrt {3} {\left ({\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 (b d - d x\right )} - {\left (b^{2} d - 2 \, b d x + d x^{2}\right )} d^{\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}} d^{\frac {2}{3}}\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}}}{b^{2} d + {\left (d - 1\right )} x^{2} - a b - {\left (2 \, b d - a - b\right )} x}\right ) - {\left (b^{2} c + b^{2} d + {\left (c + d\right )} x^{2} - 2 \, {\left (b c + b d\right )} x\right )} d^{\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 (b - x\right )} d^{\frac {1}{3}} - {\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\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}}}{b^{2} - 2 \, b x + x^{2}}\right ) + 2 \, {\left (b^{2} c + b^{2} d + {\left (c + d\right )} x^{2} - 2 \, {\left (b c + b d\right )} x\right )} d^{\frac {2}{3}} \log \left (-\frac {{\left (b - x\right )} d^{\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 {1}{3}}}{b - x}\right ) + 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}{2 \, {\left ({\left (a - b\right )} d x^{2} - 2 \, {\left (a b - b^{2}\right )} d x + {\left (a b^{2} - b^{3}\right )} d\right )}}, \frac {{\left (b^{2} c + b^{2} d + {\left (c + d\right )} x^{2} - 2 \, {\left (b c + b d\right )} x\right )} d^{\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 (b - x\right )} d^{\frac {1}{3}} - {\left (b^{2} - 2 \, b x + x^{2}\right )} d^{\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}}}{b^{2} - 2 \, b x + x^{2}}\right ) - 2 \, {\left (b^{2} c + b^{2} d + {\left (c + d\right )} x^{2} - 2 \, {\left (b c + b d\right )} x\right )} d^{\frac {2}{3}} \log \left (-\frac {{\left (b - x\right )} d^{\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 {1}{3}}}{b - x}\right ) - \frac {2 \, \sqrt {3} {\left (b^{2} c d + b^{2} d^{2} + {\left (c d + d^{2}\right )} x^{2} - 2 \, {\left (b c d + b d^{2}\right )} x\right )} \arctan \left (\frac {\sqrt {3} {\left ({\left (b - x\right )} d^{\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}}\right )}}{3 \, {\left (b - x\right )} d^{\frac {1}{3}}}\right )}{d^{\frac {1}{3}}} - 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}{2 \, {\left ({\left (a - b\right )} d x^{2} - 2 \, {\left (a b - b^{2}\right )} d x + {\left (a b^{2} - b^{3}\right )} d\right )}}\right ] \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}{\sqrt [3]{\left (- a + x\right ) \left (- b + x\right )^{2}} \left (- b + x\right ) \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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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