Optimal. Leaf size=481 \[ \frac {\left (1+i \sqrt {3}\right ) (x-b)^{2/3} \left (-\sqrt [6]{d} \sqrt [3]{a-x} \sqrt [3]{x-b}+\sqrt [3]{d} (a-x)^{2/3}+(x-b)^{2/3}\right ) \left (\sqrt [6]{d} \sqrt [3]{a-x} \sqrt [3]{x-b}+\sqrt [3]{d} (a-x)^{2/3}+(x-b)^{2/3}\right ) \left (\sqrt [3]{a-x} (x-b)^{2/3}+a \left (-\sqrt [3]{d}\right )+\sqrt [3]{d} x\right ) \left (\frac {\left (\sqrt {3}-3 i\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-b}}{\sqrt [3]{x-b}-2 \sqrt [6]{d} \sqrt [3]{a-x}}\right )}{4 d^{5/6} (a-b)}+\frac {\left (-\sqrt {3}+3 i\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-b}}{2 \sqrt [6]{d} \sqrt [3]{a-x}+\sqrt [3]{x-b}}\right )}{4 d^{5/6} (a-b)}+\frac {\left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {(a-x)^{2/3} \sqrt [3]{x-b}}{\sqrt [6]{d} (x-a)}\right )}{2 d^{5/6} (a-b)}+\frac {i \left (\sqrt {3}+i\right ) \tanh ^{-1}\left (\frac {\sqrt [3]{d} (a-x)^{2/3}+(x-b)^{2/3}}{\sqrt [6]{d} \sqrt [3]{a-x} \sqrt [3]{x-b}}\right )}{4 d^{5/6} (a-b)}\right )}{2 \sqrt [3]{(x-a) (b-x)^2} \left (a^2 (-d)+2 a d x+b^2-2 b x-(d-1) x^2\right )} \]
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Rubi [A] time = 1.06, antiderivative size = 513, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 5, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.096, Rules used = {6719, 911, 105, 59, 91} \begin {gather*} -\frac {\sqrt [3]{x-a} (x-b)^{2/3} \log \left (2 \left (\sqrt {d}+1\right ) \left (b-a \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt [3]{x-a} (x-b)^{2/3} \log \left (2 (1-d) x-2 \left (1-\sqrt {d}\right ) \left (a \sqrt {d}+b\right )\right )}{4 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} \log \left (-\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x-b}\right )}{4 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {3 \sqrt [3]{x-a} (x-b)^{2/3} \log \left (\sqrt [6]{d} \sqrt [3]{x-a}-\sqrt [3]{x-b}\right )}{4 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x-b}}\right )}{2 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt {3} \sqrt [3]{x-a} (x-b)^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \sqrt [3]{x-a}}{\sqrt {3} \sqrt [3]{x-b}}+\frac {1}{\sqrt {3}}\right )}{2 d^{5/6} (a-b) \sqrt [3]{-\left ((a-x) (b-x)^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 91
Rule 105
Rule 911
Rule 6719
Rubi steps
\begin {align*} \int \frac {-a+x}{\sqrt [3]{(-a+x) (-b+x)^2} \left (-b^2+a^2 d+2 (b-a d) x+(-1+d) x^2\right )} \, dx &=\frac {\left (\sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {(-a+x)^{2/3}}{(-b+x)^{2/3} \left (-b^2+a^2 d+2 (b-a 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+d) (-a+x)^{2/3}}{(a-b) \sqrt {d} (-b+x)^{2/3} \left (2 b-2 (a-b) \sqrt {d}-2 a d-2 (1-d) x\right )}+\frac {(-1+d) (-a+x)^{2/3}}{(a-b) \sqrt {d} (-b+x)^{2/3} \left (-2 b-2 (a-b) \sqrt {d}+2 a d+2 (1-d) x\right )}\right ) \, dx}{\sqrt [3]{(-a+x) (-b+x)^2}}\\ &=-\frac {\left ((1-d) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {(-a+x)^{2/3}}{(-b+x)^{2/3} \left (2 b-2 (a-b) \sqrt {d}-2 a d-2 (1-d) x\right )} \, dx}{(a-b) \sqrt {d} \sqrt [3]{(-a+x) (-b+x)^2}}-\frac {\left ((1-d) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {(-a+x)^{2/3}}{(-b+x)^{2/3} \left (-2 b-2 (a-b) \sqrt {d}+2 a d+2 (1-d) x\right )} \, dx}{(a-b) \sqrt {d} \sqrt [3]{(-a+x) (-b+x)^2}}\\ &=\frac {\left ((1-d) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (2 b-2 (a-b) \sqrt {d}-2 a d-2 (1-d) x\right )} \, dx}{\left (1-\sqrt {d}\right ) \sqrt {d} \sqrt [3]{(-a+x) (-b+x)^2}}+\frac {\left ((1-d) \sqrt [3]{-a+x} (-b+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{-a+x} (-b+x)^{2/3} \left (-2 b-2 (a-b) \sqrt {d}+2 a d+2 (1-d) x\right )} \, dx}{\left (1+\sqrt {d}\right ) \sqrt {d} \sqrt [3]{(-a+x) (-b+x)^2}}\\ &=-\frac {\sqrt {3} \sqrt [3]{-a+x} (-b+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [3]{-b+x}}\right )}{2 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt {3} \sqrt [3]{-a+x} (-b+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [3]{-a+x}}{\sqrt {3} \sqrt [3]{-b+x}}\right )}{2 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {\sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (2 \left (1+\sqrt {d}\right ) \left (b-a \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {\sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (-2 \left (1-\sqrt {d}\right ) \left (b+a \sqrt {d}\right )+2 (1-d) x\right )}{4 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}-\frac {3 \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (-\sqrt [6]{d} \sqrt [3]{-a+x}-\sqrt [3]{-b+x}\right )}{4 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}+\frac {3 \sqrt [3]{-a+x} (-b+x)^{2/3} \log \left (\sqrt [6]{d} \sqrt [3]{-a+x}-\sqrt [3]{-b+x}\right )}{4 (a-b) d^{5/6} \sqrt [3]{-\left ((a-x) (b-x)^2\right )}}\\ \end {align*}
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Mathematica [C] time = 0.28, size = 89, normalized size = 0.19 \begin {gather*} -\frac {3 (x-b) \left (\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {b-x}{\sqrt {d} (x-a)}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {x-b}{\sqrt {d} (x-a)}\right )\right )}{2 d (a-b) \sqrt [3]{(x-a) (b-x)^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 15.40, size = 481, normalized size = 1.00 \begin {gather*} \frac {\left (1+i \sqrt {3}\right ) (-b+x)^{2/3} \left (\sqrt [3]{d} (a-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{a-x} \sqrt [3]{-b+x}+(-b+x)^{2/3}\right ) \left (\sqrt [3]{d} (a-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{a-x} \sqrt [3]{-b+x}+(-b+x)^{2/3}\right ) \left (-a \sqrt [3]{d}+\sqrt [3]{d} x+\sqrt [3]{a-x} (-b+x)^{2/3}\right ) \left (\frac {\left (-3 i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-b+x}}{-2 \sqrt [6]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}}\right )}{4 (a-b) d^{5/6}}+\frac {\left (3 i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-b+x}}{2 \sqrt [6]{d} \sqrt [3]{a-x}+\sqrt [3]{-b+x}}\right )}{4 (a-b) d^{5/6}}+\frac {\left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {(a-x)^{2/3} \sqrt [3]{-b+x}}{\sqrt [6]{d} (-a+x)}\right )}{2 (a-b) d^{5/6}}+\frac {i \left (i+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [3]{d} (a-x)^{2/3}+(-b+x)^{2/3}}{\sqrt [6]{d} \sqrt [3]{a-x} \sqrt [3]{-b+x}}\right )}{4 (a-b) d^{5/6}}\right )}{2 \sqrt [3]{(b-x)^2 (-a+x)} \left (b^2-a^2 d-2 b x+2 a d x-(-1+d) x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.95, size = 2613, normalized size = 5.43
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 {a - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b^{2} - 2 \, {\left (a d - b\right )} x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {-a +x}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {1}{3}} \left (-b^{2}+a^{2} d +2 \left (-a d +b \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]
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*} -\int \frac {a - x}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {1}{3}} {\left (a^{2} d + {\left (d - 1\right )} x^{2} - b^{2} - 2 \, {\left (a d - b\right )} 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-x}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{1/3}\,\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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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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