Optimal. Leaf size=243 \[ \frac {\log \left (x-\sqrt [6]{d} \sqrt [3]{x^3-a x^2}\right )}{2 a d^{2/3}}+\frac {\log \left (\sqrt [6]{d} \sqrt [3]{x^3-a x^2}+x\right )}{2 a d^{2/3}}-\frac {\log \left (-\sqrt [6]{d} x \sqrt [3]{x^3-a x^2}+\sqrt [3]{d} \left (x^3-a x^2\right )^{2/3}+x^2\right )}{4 a d^{2/3}}-\frac {\log \left (\sqrt [6]{d} x \sqrt [3]{x^3-a x^2}+\sqrt [3]{d} \left (x^3-a x^2\right )^{2/3}+x^2\right )}{4 a d^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x^2}{2 \sqrt [3]{d} \left (x^3-a x^2\right )^{2/3}+x^2}\right )}{2 a d^{2/3}} \]
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Rubi [A] time = 0.84, antiderivative size = 418, normalized size of antiderivative = 1.72, number of steps used = 9, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6719, 911, 105, 59, 91} \begin {gather*} -\frac {x^{4/3} (x-a)^{2/3} \log \left (-2 a \sqrt {d} \left (\sqrt {d}+1\right )-2 (1-d) x\right )}{4 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}-\frac {x^{4/3} (x-a)^{2/3} \log \left (2 (1-d) x-2 a \left (1-\sqrt {d}\right ) \sqrt {d}\right )}{4 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {3 x^{4/3} (x-a)^{2/3} \log \left (-\sqrt [3]{x-a}-\frac {\sqrt [3]{x}}{\sqrt [6]{d}}\right )}{4 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {3 x^{4/3} (x-a)^{2/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [6]{d}}-\sqrt [3]{x-a}\right )}{4 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {\sqrt {3} x^{4/3} (x-a)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}\right )}{2 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {\sqrt {3} x^{4/3} (x-a)^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{2 a d^{2/3} \left (-\left (x^2 (a-x)\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 {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx &=\frac {\left (x^{4/3} (-a+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{\sqrt [3]{x} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx}{\left (x^2 (-a+x)\right )^{2/3}}\\ &=\frac {\left (x^{4/3} (-a+x)^{2/3}\right ) \int \left (\frac {(-1+d) \sqrt [3]{-a+x}}{a \sqrt {d} \sqrt [3]{x} \left (-2 a \sqrt {d}-2 a d-2 (1-d) x\right )}+\frac {(-1+d) \sqrt [3]{-a+x}}{a \sqrt {d} \sqrt [3]{x} \left (-2 a \sqrt {d}+2 a d+2 (1-d) x\right )}\right ) \, dx}{\left (x^2 (-a+x)\right )^{2/3}}\\ &=-\frac {\left ((1-d) x^{4/3} (-a+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{\sqrt [3]{x} \left (-2 a \sqrt {d}-2 a d-2 (1-d) x\right )} \, dx}{a \sqrt {d} \left (x^2 (-a+x)\right )^{2/3}}-\frac {\left ((1-d) x^{4/3} (-a+x)^{2/3}\right ) \int \frac {\sqrt [3]{-a+x}}{\sqrt [3]{x} \left (-2 a \sqrt {d}+2 a d+2 (1-d) x\right )} \, dx}{a \sqrt {d} \left (x^2 (-a+x)\right )^{2/3}}\\ &=\frac {\left ((1-d) x^{4/3} (-a+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-a+x)^{2/3} \left (-2 a \sqrt {d}-2 a d-2 (1-d) x\right )} \, dx}{\left (1-\sqrt {d}\right ) \sqrt {d} \left (x^2 (-a+x)\right )^{2/3}}+\frac {\left ((1-d) x^{4/3} (-a+x)^{2/3}\right ) \int \frac {1}{\sqrt [3]{x} (-a+x)^{2/3} \left (-2 a \sqrt {d}+2 a d+2 (1-d) x\right )} \, dx}{\left (1+\sqrt {d}\right ) \sqrt {d} \left (x^2 (-a+x)\right )^{2/3}}\\ &=\frac {\sqrt {3} x^{4/3} (-a+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 a d^{2/3} \left (-\left ((a-x) x^2\right )\right )^{2/3}}+\frac {\sqrt {3} x^{4/3} (-a+x)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{x}}{\sqrt {3} \sqrt [6]{d} \sqrt [3]{-a+x}}\right )}{2 a d^{2/3} \left (-\left ((a-x) x^2\right )\right )^{2/3}}-\frac {x^{4/3} (-a+x)^{2/3} \log \left (-2 a \left (1+\sqrt {d}\right ) \sqrt {d}-2 (1-d) x\right )}{4 a d^{2/3} \left (-\left ((a-x) x^2\right )\right )^{2/3}}-\frac {x^{4/3} (-a+x)^{2/3} \log \left (-2 a \left (1-\sqrt {d}\right ) \sqrt {d}+2 (1-d) x\right )}{4 a d^{2/3} \left (-\left ((a-x) x^2\right )\right )^{2/3}}+\frac {3 x^{4/3} (-a+x)^{2/3} \log \left (-\frac {\sqrt [3]{x}}{\sqrt [6]{d}}-\sqrt [3]{-a+x}\right )}{4 a d^{2/3} \left (-\left ((a-x) x^2\right )\right )^{2/3}}+\frac {3 x^{4/3} (-a+x)^{2/3} \log \left (\frac {\sqrt [3]{x}}{\sqrt [6]{d}}-\sqrt [3]{-a+x}\right )}{4 a d^{2/3} \left (-\left ((a-x) x^2\right )\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 71, normalized size = 0.29 \begin {gather*} -\frac {3 x^2 \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {x}{\sqrt {d} (a-x)}\right )+\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {x}{\sqrt {d} (x-a)}\right )\right )}{4 a d \left (x^2 (x-a)\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.60, size = 243, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}}\right )}{2 a d^{2/3}}+\frac {\log \left (x-\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}\right )}{2 a d^{2/3}}+\frac {\log \left (x+\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}\right )}{2 a d^{2/3}}-\frac {\log \left (x^2-\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}\right )}{4 a d^{2/3}}-\frac {\log \left (x^2+\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}\right )}{4 a d^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 167, normalized size = 0.69 \begin {gather*} \frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left ({\left (d^{2}\right )}^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d\right )} {\left (d^{2}\right )}^{\frac {1}{6}}}{3 \, d x^{2}}\right ) - {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (d^{2}\right )}^{\frac {2}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {1}{3}} d - {\left (a d^{2} - d^{2} x\right )} {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x^{2}}\right ) + 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (d^{2}\right )}^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right )}{4 \, a d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.64, size = 108, normalized size = 0.44 \begin {gather*} \frac {\sqrt {3} {\left | d \right |}^{\frac {4}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} d^{\frac {1}{3}} {\left (2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \frac {1}{d^{\frac {1}{3}}}\right )}\right )}{2 \, a d^{2}} - \frac {{\left | d \right |}^{\frac {4}{3}} \log \left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{d^{\frac {1}{3}}} + \frac {1}{d^{\frac {2}{3}}}\right )}{4 \, a d^{2}} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} - \frac {1}{d^{\frac {1}{3}}} \right |}\right )}{2 \, a d^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {x \left (-a +x \right )}{\left (x^{2} \left (-a +x \right )\right )^{\frac {2}{3}} \left (a^{2} d -2 a d x +\left (-1+d \right ) x^{2}\right )}\, dx\]
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 (a - x\right )} x}{{\left (a^{2} d - 2 \, a d x + {\left (d - 1\right )} x^{2}\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}}}\,{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 {x\,\left (a-x\right )}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (d\,a^2-2\,d\,a\,x+\left (d-1\right )\,x^2\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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