Optimal. Leaf size=1886 \[ \frac {(b-x)^{4/3} (-a+x)^{2/3} \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right ) \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right ) \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right ) \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right ) \left (\frac {\sqrt {3} b \left (-1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{-2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\sqrt {3} a c \left (-1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{-2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} \left (-b+a \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{-2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} c \left (-b+a \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{-2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} b \left (1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}-\frac {\sqrt {3} a c \left (1+\sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\sqrt {3} \left (b+a \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\sqrt {3} c \left (b+a \sqrt {d}\right ) \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}{2 \sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b)^2 d^{2/3}}+\frac {b \left (1+\sqrt {d}\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {a c \left (1+\sqrt {d}\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-b-a \sqrt {d}\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}-\frac {c \left (b+a \sqrt {d}\right ) \log \left (\sqrt [3]{b-x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}-\frac {b \left (-1+\sqrt {d}\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}-\frac {a c \left (-1+\sqrt {d}\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {\left (-b+a \sqrt {d}\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {c \left (-b+a \sqrt {d}\right ) \log \left (\sqrt [3]{b-x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )}{2 (a-b)^2 d^{2/3}}+\frac {b \left (-1+\sqrt {d}\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {a c \left (-1+\sqrt {d}\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (b-a \sqrt {d}\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}-\frac {c \left (-b+a \sqrt {d}\right ) \log \left ((b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}-\frac {b \left (1+\sqrt {d}\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}-\frac {a c \left (1+\sqrt {d}\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {\left (b+a \sqrt {d}\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}+\frac {c \left (b+a \sqrt {d}\right ) \log \left ((b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )}{4 (a-b)^2 d^{2/3}}\right )}{\left ((b-x)^2 (-a+x)\right )^{2/3} \left (b^2-a^2 d-2 b x+2 a d x-(-1+d) x^2\right )} \]
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Rubi [A]
time = 1.66, antiderivative size = 561, normalized size of antiderivative = 0.30, number of steps
used = 5, number of rules used = 3, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {6851, 6860, 93}
\begin {gather*} \frac {\sqrt {3} (x-a)^{2/3} (x-b)^{4/3} \left (c-\sqrt {d}\right ) \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x-b}}{\sqrt {3} \sqrt [6]{d} \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} \left (c+\sqrt {d}\right ) \text {ArcTan}\left (\frac {2 \sqrt [3]{x-b}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{2 d^{2/3} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {(x-a)^{2/3} (x-b)^{4/3} \left (c+\sqrt {d}\right ) \log \left (2 \left (\sqrt {d}+1\right ) \left (b-a \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{2/3} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {(x-a)^{2/3} (x-b)^{4/3} \left (c-\sqrt {d}\right ) \log \left (2 \left (1-\sqrt {d}\right ) \left (a \sqrt {d}+b\right )-2 (1-d) x\right )}{4 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} \left (c-\sqrt {d}\right ) \log \left (-\sqrt [3]{x-a}-\frac {\sqrt [3]{x-b}}{\sqrt [6]{d}}\right )}{4 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} \left (c+\sqrt {d}\right ) \log \left (\frac {\sqrt [3]{x-b}}{\sqrt [6]{d}}-\sqrt [3]{x-a}\right )}{4 d^{2/3} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 6851
Rule 6860
Rubi steps
\begin {align*} \int \frac {(-b+x) (-b-a c+(1+c) x)}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx &=\frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {-b-a c+(1+c) x}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \left (\frac {1+c+\frac {c+d}{\sqrt {d}}}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (-2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )}+\frac {1+c-\frac {c+d}{\sqrt {d}}}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )}\right ) \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\left (\left (1+c-\frac {c+d}{\sqrt {d}}\right ) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}+\frac {\left (\left (1+c+\frac {c+d}{\sqrt {d}}\right ) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (-2 (a-b) \sqrt {d}+2 (b-a d)+2 (-1+d) x\right )} \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\sqrt {3} \left (c-\sqrt {d}\right ) (-a+x)^{2/3} (-b+x)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-b+x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {\sqrt {3} \left (c+\sqrt {d}\right ) (-a+x)^{2/3} (-b+x)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {\left (c+\sqrt {d}\right ) (-a+x)^{2/3} (-b+x)^{4/3} \log \left (2 \left (1+\sqrt {d}\right ) \left (b-a \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {\left (c-\sqrt {d}\right ) (-a+x)^{2/3} (-b+x)^{4/3} \log \left (2 \left (1-\sqrt {d}\right ) \left (b+a \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {3 \left (c-\sqrt {d}\right ) (-a+x)^{2/3} (-b+x)^{4/3} \log \left (-\sqrt [3]{-a+x}-\frac {\sqrt [3]{-b+x}}{\sqrt [6]{d}}\right )}{4 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {3 \left (c+\sqrt {d}\right ) (-a+x)^{2/3} (-b+x)^{4/3} \log \left (-\sqrt [3]{-a+x}+\frac {\sqrt [3]{-b+x}}{\sqrt [6]{d}}\right )}{4 (a-b) d^{2/3} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 501, normalized size = 0.27 \begin {gather*} -\frac {(b-x)^{4/3} (-a+x)^{2/3} \left (2 \sqrt {3} \left (c+\sqrt {d}\right ) \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}}{\sqrt {3}}\right )+2 \sqrt {3} \left (c-\sqrt {d}\right ) \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}}{\sqrt {3}}\right )-2 c \log \left (-1+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}\right )+2 \sqrt {d} \log \left (-1+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}\right )-2 c \log \left (1+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}\right )-2 \sqrt {d} \log \left (1+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}\right )+c \log \left (1-\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}+\frac {\sqrt [3]{d} (-a+x)^{2/3}}{(b-x)^{2/3}}\right )+\sqrt {d} \log \left (1-\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}+\frac {\sqrt [3]{d} (-a+x)^{2/3}}{(b-x)^{2/3}}\right )+c \log \left (1+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}+\frac {\sqrt [3]{d} (-a+x)^{2/3}}{(b-x)^{2/3}}\right )-\sqrt {d} \log \left (1+\frac {\sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt [3]{b-x}}+\frac {\sqrt [3]{d} (-a+x)^{2/3}}{(b-x)^{2/3}}\right )\right )}{4 (a-b) d^{2/3} \left ((b-x)^2 (-a+x)\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (-b +x \right ) \left (-b -a c +\left (1+c \right ) x \right )}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (-b^{2}+a^{2} d +2 \left (-a d +b \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]
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. 9468 vs.
\(2 (1441) = 2882\).
time = 0.79, size = 9468, normalized size = 5.02 \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 {\left (b-x\right )\,\left (b+a\,c-x\,\left (c+1\right )\right )}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\left (a^2\,d+2\,x\,\left (b-a\,d\right )-b^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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