Optimal. Leaf size=1387 \[ \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{a-x} (-b+x)^{2/3} \left (\sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}\right ) \left (d^{2/3} (a-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{a-x} \sqrt [3]{-b+x}+(-b+x)^{2/3}\right ) \left (\frac {3 \left (a \sqrt [3]{-b+x}-i \sqrt {3} a \sqrt [3]{-b+x}\right )}{2 (a-b)^2 \sqrt [3]{a-x}}+\frac {3 i \left (i b \sqrt [3]{-b+x}+\sqrt {3} b \sqrt [3]{-b+x}\right )}{2 (a-b)^2 \sqrt [3]{a-x}}+\frac {3 \left (a c \sqrt [3]{-b+x}-i \sqrt {3} a c \sqrt [3]{-b+x}\right )}{2 (a-b)^2 \sqrt [3]{a-x}}+\frac {3 i \left (i a c \sqrt [3]{-b+x}+\sqrt {3} a c \sqrt [3]{-b+x}\right )}{2 (a-b)^2 \sqrt [3]{a-x}}+\frac {\left (3 i b-\sqrt {3} b-3 i a d+\sqrt {3} a d\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-b+x}}{-2 \sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-3 i b+\sqrt {3} b+3 i b d-\sqrt {3} b d\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-b+x}}{-2 \sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-3 i a c+\sqrt {3} a c+3 i a c d-\sqrt {3} a c d\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-b+x}}{-2 \sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (3 i b c-\sqrt {3} b c-3 i a c d+\sqrt {3} a c d\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-b+x}}{-2 \sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (b-i \sqrt {3} b-a d+i \sqrt {3} a d\right ) \log \left (\sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-b+i \sqrt {3} b+b d-i \sqrt {3} b d\right ) \log \left (\sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-a c+i \sqrt {3} a c+a c d-i \sqrt {3} a c d\right ) \log \left (\sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (b c-i \sqrt {3} b c-a c d+i \sqrt {3} a c d\right ) \log \left (\sqrt [3]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-b+i \sqrt {3} b+a d-i \sqrt {3} a d\right ) \log \left (d^{2/3} (a-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{a-x} \sqrt [3]{-b+x}+(-b+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (b-i \sqrt {3} b-b d+i \sqrt {3} b d\right ) \log \left (d^{2/3} (a-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{a-x} \sqrt [3]{-b+x}+(-b+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (-b c+i \sqrt {3} b c+a c d-i \sqrt {3} a c d\right ) \log \left (d^{2/3} (a-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{a-x} \sqrt [3]{-b+x}+(-b+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (a c-i \sqrt {3} a c-a c d+i \sqrt {3} a c d\right ) \log \left (d^{2/3} (a-x)^{2/3}-\sqrt [3]{d} \sqrt [3]{a-x} \sqrt [3]{-b+x}+(-b+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}\right )}{2 \sqrt [3]{-\left ((a-x) (b-x)^2\right )} (-b+a d+x-d x)} \]
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Rubi [A]
time = 1.08, antiderivative size = 280, normalized size of antiderivative = 0.20, 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]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt [3]{x-a} (x-b)^{2/3} (c+d) \log (-a d+b-(1-d) x)}{2 d^{2/3} (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 (\sqrt [3]{d} \sqrt [3]{x-a}-\sqrt [3]{x-b}\right )}{2 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 (b-x)}{(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 {-b-a c+(1+c) x}{(-a+x) \sqrt [3]{(-a+x) (-b+x)^2} (b-a d+(-1+d) x)} \, dx &=\frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {-b-a c+(1+c) x}{(-a+x)^{4/3} (-b+x)^{2/3} (b-a d+(-1+d) x)} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=-\frac {3 (b-x)}{(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 {(a-b)^2 (c+d)}{3 \sqrt [3]{-a+x} (-b+x)^{2/3} (b-a d+(-1+d) x)} \, dx}{(a-b)^2 \sqrt [3]{(-a+x) (-b+x)^2}}\\ &=-\frac {3 (b-x)}{(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} (b-a d+(-1+d) x)} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=-\frac {3 (b-x)}{(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]{d} \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [3]{-b+x}}\right )}{(a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {(c+d) \sqrt [3]{-a+x} (-b+x)^{2/3} \log (b-a d-(1-d) x)}{2 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {3 (c+d) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (\sqrt [3]{d} \sqrt [3]{-a+x}-\sqrt [3]{-b+x}\right )}{2 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 305, normalized size = 0.22 \begin {gather*} \frac {-6 b d^{2/3}+6 d^{2/3} x-2 \sqrt {3} (c+d) (b-x)^{2/3} \sqrt [3]{-a+x} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{b-x}}{\sqrt [3]{d} \sqrt [3]{-a+x}}}{\sqrt {3}}\right )+2 (c+d) (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (\sqrt [3]{d}+\frac {\sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-c (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (d^{2/3}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-d (b-x)^{2/3} \sqrt [3]{-a+x} \log \left (d^{2/3}+\frac {(b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [3]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )}{2 (a-b) d^{2/3} \sqrt [3]{(b-x)^2 (-a+x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {-b -a c +\left (1+c \right ) x}{\left (-a +x \right ) \left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (b -a d +\left (-1+d \right ) x \right )}\, dx\]
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 = 0.38, size = 473, normalized size = 0.34 \begin {gather*} \frac {2 \, \sqrt {3} {\left (a b c d + a b d^{2} + {\left (c d + d^{2}\right )} x^{2} - {\left ({\left (a + b\right )} c d + {\left (a + b\right )} d^{2}\right )} x\right )} \sqrt {-\left (-d^{2}\right )^{\frac {1}{3}}} \arctan \left (-\frac {\sqrt {3} {\left (\left (-d^{2}\right )^{\frac {1}{3}} {\left (b - x\right )} + 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}} d\right )} \sqrt {-\left (-d^{2}\right )^{\frac {1}{3}}}}{3 \, {\left (b d - d x\right )}}\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 {2}{3}} d^{2} - {\left (a b c + a b d + {\left (c + d\right )} x^{2} - {\left ({\left (a + b\right )} c + {\left (a + b\right )} 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 {2}{3}} d^{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 (b d - d x\right )} \left (-d^{2}\right )^{\frac {1}{3}} + {\left (b^{2} - 2 \, b x + x^{2}\right )} \left (-d^{2}\right )^{\frac {2}{3}}}{b^{2} - 2 \, b x + x^{2}}\right ) + 2 \, {\left (a b c + a b d + {\left (c + d\right )} x^{2} - {\left ({\left (a + b\right )} c + {\left (a + b\right )} d\right )} x\right )} \left (-d^{2}\right )^{\frac {2}{3}} \log \left (\frac {\left (-d^{2}\right )^{\frac {1}{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 )}{2 \, {\left ({\left (a - b\right )} d^{2} x^{2} - {\left (a^{2} - b^{2}\right )} d^{2} x + {\left (a^{2} b - a b^{2}\right )} d^{2}\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 c - b + c x + x}{\sqrt [3]{\left (- a + x\right ) \left (- b + x\right )^{2}} \left (- a + x\right ) \left (- a d + b + 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 {b+a\,c-x\,\left (c+1\right )}{\left (a-x\right )\,{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\left (b-a\,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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