Optimal. Leaf size=438 \[ \frac {(x-a)^{2/3} (x-b) \left (-\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}+\sqrt [3]{d} (b-x)^{2/3}\right ) \left (\sqrt [6]{d} \sqrt [3]{x-a} \sqrt [3]{b-x}+(x-a)^{2/3}+\sqrt [3]{d} (b-x)^{2/3}\right ) \left ((x-a)^{2/3} \sqrt [3]{b-x}+b \left (-\sqrt [3]{d}\right )+\sqrt [3]{d} x\right ) \left (\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}-2 \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 d^{5/6} (a-b)}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x-a}}{\sqrt [3]{x-a}+2 \sqrt [6]{d} \sqrt [3]{b-x}}\right )}{2 d^{5/6} (a-b)}+\frac {\tanh ^{-1}\left (\frac {\sqrt [3]{x-a} (b-x)^{2/3}}{\sqrt [6]{d} (x-b)}\right )}{d^{5/6} (a-b)}-\frac {\tanh ^{-1}\left (\frac {\frac {(x-a)^{2/3}}{\sqrt [6]{d}}+\sqrt [6]{d} (b-x)^{2/3}}{\sqrt [3]{x-a} \sqrt [3]{b-x}}\right )}{2 d^{5/6} (a-b)}\right )}{\left ((x-a) (b-x)^2\right )^{2/3} \left (-a^2+2 a x+b^2 d-2 b d x+(d-1) x^2\right )} \]
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Rubi [A] time = 1.32, antiderivative size = 513, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 5, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {6719, 911, 105, 59, 91} \begin {gather*} -\frac {(x-a)^{2/3} (x-b)^{4/3} \log \left (2 \left (1-\sqrt {d}\right ) \left (a+b \sqrt {d}\right )-2 (1-d) x\right )}{4 d^{5/6} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {(x-a)^{2/3} (x-b)^{4/3} \log \left (2 (1-d) x-2 \left (\sqrt {d}+1\right ) \left (a-b \sqrt {d}\right )\right )}{4 d^{5/6} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {3 (x-a)^{2/3} (x-b)^{4/3} \log \left (-\sqrt [3]{x-a}-\sqrt [6]{d} \sqrt [3]{x-b}\right )}{4 d^{5/6} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {3 (x-a)^{2/3} (x-b)^{4/3} \log \left (\sqrt [6]{d} \sqrt [3]{x-b}-\sqrt [3]{x-a}\right )}{4 d^{5/6} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {\sqrt {3} (x-a)^{2/3} (x-b)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{x-a}}\right )}{2 d^{5/6} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {\sqrt {3} (x-a)^{2/3} (x-b)^{4/3} \tan ^{-1}\left (\frac {2 \sqrt [6]{d} \sqrt [3]{x-b}}{\sqrt {3} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{2 d^{5/6} (a-b) \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}} \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 {(-b+x)^2}{\left ((-a+x) (-b+x)^2\right )^{2/3} \left (-a^2+b^2 d+2 (a-b d) x+(-1+d) x^2\right )} \, dx &=\frac {\left ((-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {(-b+x)^{2/3}}{(-a+x)^{2/3} \left (-a^2+b^2 d+2 (a-b 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+d) (-b+x)^{2/3}}{(a-b) \sqrt {d} (-a+x)^{2/3} \left (2 a-2 (a-b) \sqrt {d}-2 b d-2 (1-d) x\right )}+\frac {(-1+d) (-b+x)^{2/3}}{(a-b) \sqrt {d} (-a+x)^{2/3} \left (-2 a-2 (a-b) \sqrt {d}+2 b d+2 (1-d) x\right )}\right ) \, dx}{\left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=-\frac {\left ((1-d) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {(-b+x)^{2/3}}{(-a+x)^{2/3} \left (2 a-2 (a-b) \sqrt {d}-2 b d-2 (1-d) x\right )} \, dx}{(a-b) \sqrt {d} \left ((-a+x) (-b+x)^2\right )^{2/3}}-\frac {\left ((1-d) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {(-b+x)^{2/3}}{(-a+x)^{2/3} \left (-2 a-2 (a-b) \sqrt {d}+2 b d+2 (1-d) x\right )} \, dx}{(a-b) \sqrt {d} \left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=-\frac {\left ((1-d) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (-2 a-2 (a-b) \sqrt {d}+2 b d+2 (1-d) x\right )} \, dx}{\left (1-\sqrt {d}\right ) \sqrt {d} \left ((-a+x) (-b+x)^2\right )^{2/3}}-\frac {\left ((1-d) (-a+x)^{2/3} (-b+x)^{4/3}\right ) \int \frac {1}{(-a+x)^{2/3} \sqrt [3]{-b+x} \left (2 a-2 (a-b) \sqrt {d}-2 b d-2 (1-d) x\right )} \, dx}{\left (1+\sqrt {d}\right ) \sqrt {d} \left ((-a+x) (-b+x)^2\right )^{2/3}}\\ &=\frac {\sqrt {3} (-a+x)^{2/3} (-b+x)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [6]{d} \sqrt [3]{-b+x}}{\sqrt {3} \sqrt [3]{-a+x}}\right )}{2 (a-b) d^{5/6} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {\sqrt {3} (-a+x)^{2/3} (-b+x)^{4/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{d} \sqrt [3]{-b+x}}{\sqrt {3} \sqrt [3]{-a+x}}\right )}{2 (a-b) d^{5/6} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {(-a+x)^{2/3} (-b+x)^{4/3} \log \left (2 \left (1-\sqrt {d}\right ) \left (a+b \sqrt {d}\right )-2 (1-d) x\right )}{4 (a-b) d^{5/6} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {(-a+x)^{2/3} (-b+x)^{4/3} \log \left (-2 \left (1+\sqrt {d}\right ) \left (a-b \sqrt {d}\right )+2 (1-d) x\right )}{4 (a-b) d^{5/6} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}+\frac {3 (-a+x)^{2/3} (-b+x)^{4/3} \log \left (-\sqrt [3]{-a+x}-\sqrt [6]{d} \sqrt [3]{-b+x}\right )}{4 (a-b) d^{5/6} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}-\frac {3 (-a+x)^{2/3} (-b+x)^{4/3} \log \left (-\sqrt [3]{-a+x}+\sqrt [6]{d} \sqrt [3]{-b+x}\right )}{4 (a-b) d^{5/6} \left (-\left ((a-x) (b-x)^2\right )\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.28, size = 95, normalized size = 0.22 \begin {gather*} -\frac {3 (b-x)^2 \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} (b-x)}{x-a}\right )-\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} (x-b)}{x-a}\right )\right )}{4 \sqrt {d} (a-b) \left ((x-a) (b-x)^2\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 14.42, size = 438, normalized size = 1.00 \begin {gather*} \frac {(-a+x)^{2/3} (-b+x) \left (\sqrt [3]{d} (b-x)^{2/3}-\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right ) \left (\sqrt [3]{d} (b-x)^{2/3}+\sqrt [6]{d} \sqrt [3]{b-x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right ) \left (-b \sqrt [3]{d}+\sqrt [3]{d} x+\sqrt [3]{b-x} (-a+x)^{2/3}\right ) \left (\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{-2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b) d^{5/6}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-a+x}}{2 \sqrt [6]{d} \sqrt [3]{b-x}+\sqrt [3]{-a+x}}\right )}{2 (a-b) d^{5/6}}+\frac {\tanh ^{-1}\left (\frac {(b-x)^{2/3} \sqrt [3]{-a+x}}{\sqrt [6]{d} (-b+x)}\right )}{(a-b) d^{5/6}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{d} (b-x)^{2/3}+\frac {(-a+x)^{2/3}}{\sqrt [6]{d}}}{\sqrt [3]{b-x} \sqrt [3]{-a+x}}\right )}{2 (a-b) d^{5/6}}\right )}{\left ((b-x)^2 (-a+x)\right )^{2/3} \left (-a^2+b^2 d+2 a x-2 b d x+(-1+d) x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 2217, normalized size = 5.06
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 {{\left (b - x\right )}^{2}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\right )} x\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {\left (-b +x \right )^{2}}{\left (\left (-a +x \right ) \left (-b +x \right )^{2}\right )^{\frac {2}{3}} \left (-a^{2}+b^{2} d +2 \left (-b d +a \right ) x +\left (-1+d \right ) x^{2}\right )}\, dx\]
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*} \int \frac {{\left (b - x\right )}^{2}}{\left (-{\left (a - x\right )} {\left (b - x\right )}^{2}\right )^{\frac {2}{3}} {\left (b^{2} d + {\left (d - 1\right )} x^{2} - a^{2} - 2 \, {\left (b d - a\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 {{\left (b-x\right )}^2}{{\left (-\left (a-x\right )\,{\left (b-x\right )}^2\right )}^{2/3}\,\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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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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