Optimal. Leaf size=1655 \[ \frac {(b-x)^{2/3} \sqrt [3]{-a+x} \left (-\frac {\sqrt {3} a \left (-1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} b c \left (-1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} \left (a-b \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} c \left (a-b \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\sqrt {3} a \left (1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\sqrt {3} b c \left (1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} \left (a+b \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} c \left (a+b \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {a \left (1+\sqrt {d}\right ) \log \left (-\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {b c \left (1+\sqrt {d}\right ) \log \left (-\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-a-b \sqrt {d}\right ) \log \left (-\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}-\frac {c \left (a+b \sqrt {d}\right ) \log \left (-\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}-\frac {a \left (-1+\sqrt {d}\right ) \log \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}-\frac {b c \left (-1+\sqrt {d}\right ) \log \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}-\frac {c \left (a-b \sqrt {d}\right ) \log \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-a+b \sqrt {d}\right ) \log \left (\sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {a \left (-1+\sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {b c \left (-1+\sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (a-b \sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {c \left (a-b \sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}-\frac {a \left (1+\sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}-\frac {b c \left (1+\sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (a+b \sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {c \left (a+b \sqrt {d}\right ) \log \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}\right )}{\sqrt [3]{(b-x)^2 (-a+x)}} \]
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Rubi [A]
time = 1.43, antiderivative size = 561, normalized size of antiderivative = 0.34, number of steps
used = 5, number of rules used = 3, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6851, 6860, 93}
\begin {gather*} -\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} \left (c-\sqrt {d}\right ) \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-a}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-b}}\right )}{2 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} \left (c+\sqrt {d}\right ) \text {ArcTan}\left (\frac {2 \sqrt [3]{x-a}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{2 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (c+\sqrt {d}\right ) \log \left (2 \left (\sqrt {d}+1\right ) \left (a-b \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt [3]{x-a} (x-b)^{2/3} \left (c-\sqrt {d}\right ) \log \left (2 \left (1-\sqrt {d}\right ) \left (a+b \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} \left (c-\sqrt {d}\right ) \log \left (-\frac {\sqrt [3]{x-a}}{\sqrt [6]{d}}-\sqrt [3]{x-b}\right )}{4 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} \left (c+\sqrt {d}\right ) \log \left (\frac {\sqrt [3]{x-a}}{\sqrt [6]{d}}-\sqrt [3]{x-b}\right )}{4 d^{2/3} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 6851
Rule 6860
Rubi steps
\begin {align*} \int \frac {-a-b c+(1+c) x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, 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)^{2/3} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \left (\frac {1+c+\frac {-c-d}{\sqrt {d}}}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (-2 (a-b) \sqrt {d}+2 (a-b d)+2 (-1+d) x\right )}+\frac {1+c-\frac {-c-d}{\sqrt {d}}}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (2 (a-b) \sqrt {d}+2 (a-b d)+2 (-1+d) x\right )}\right ) \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {\left (\left (1+c-\frac {-c-d}{\sqrt {d}}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (2 (a-b) \sqrt {d}+2 (a-b d)+2 (-1+d) x\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}+\frac {\left (\left (1+c-\frac {c+d}{\sqrt {d}}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (-2 (a-b) \sqrt {d}+2 (a-b d)+2 (-1+d) x\right )} \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=-\frac {\sqrt {3} \left (c-\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{-b+x}}\right )}{2 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt {3} \left (c+\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{-b+x}}\right )}{2 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (c+\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (2 \left (1+\sqrt {d}\right ) \left (a-b \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\left (c-\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (2 \left (1-\sqrt {d}\right ) \left (a+b \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \left (c-\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (-\frac {\sqrt [3]{-a+x}}{\sqrt [6]{d}}-\sqrt [3]{-b+x}\right )}{4 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \left (c+\sqrt {d}\right ) \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (\frac {\sqrt [3]{-a+x}}{\sqrt [6]{d}}-\sqrt [3]{-b+x}\right )}{4 (a-b) d^{2/3} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\\ \end {align*}
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Mathematica [A]
time = 0.43, size = 501, normalized size = 0.30 \begin {gather*} \frac {(b-x)^{2/3} \sqrt [3]{-a+x} \left (2 \sqrt {3} \left (c+\sqrt {d}\right ) \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}}{\sqrt {3}}\right )+2 \sqrt {3} \left (c-\sqrt {d}\right ) \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}}{\sqrt {3}}\right )+c \log \left (1+\frac {\sqrt [3]{d} (b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+\sqrt {d} \log \left (1+\frac {\sqrt [3]{d} (b-x)^{2/3}}{(-a+x)^{2/3}}-\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 c \log \left (-1+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+2 \sqrt {d} \log \left (-1+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 c \log \left (1+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-2 \sqrt {d} \log \left (1+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )+c \log \left (1+\frac {\sqrt [3]{d} (b-x)^{2/3}}{(-a+x)^{2/3}}+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )-\sqrt {d} \log \left (1+\frac {\sqrt [3]{d} (b-x)^{2/3}}{(-a+x)^{2/3}}+\frac {\sqrt [6]{d} \sqrt [3]{b-x}}{\sqrt [3]{-a+x}}\right )\right )}{4 (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 = 0.05, size = 0, normalized size = 0.00 \[\int \frac {-a -b c +\left (1+c \right ) x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (-a^{2}+b^{2} d +2 \left (-b d +a \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 11598 vs.
\(2 (1277) = 2554\).
time = 1.08, size = 11598, normalized size = 7.01 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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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 (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\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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