Optimal. Leaf size=249 \[ \frac {\log \left (\sqrt [3]{x^3-a x^2}-\sqrt [6]{d} x\right )}{2 a \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{x^3-a x^2}+\sqrt [6]{d} x\right )}{2 a \sqrt [3]{d}}-\frac {\log \left (-\sqrt [6]{d} x \sqrt [3]{x^3-a x^2}+\left (x^3-a x^2\right )^{2/3}+\sqrt [3]{d} x^2\right )}{4 a \sqrt [3]{d}}-\frac {\log \left (\sqrt [6]{d} x \sqrt [3]{x^3-a x^2}+\left (x^3-a x^2\right )^{2/3}+\sqrt [3]{d} x^2\right )}{4 a \sqrt [3]{d}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} x^2}{2 \left (x^3-a x^2\right )^{2/3}+\sqrt [3]{d} x^2}\right )}{2 a \sqrt [3]{d}} \]
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Rubi [A] time = 0.90, antiderivative size = 408, normalized size of antiderivative = 1.64, number of steps used = 10, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {1593, 6719, 911, 105, 59, 91} \begin {gather*} -\frac {x^{4/3} (x-a)^{2/3} \log \left (2 a \left (1-\sqrt {d}\right )-2 (1-d) x\right )}{4 a \sqrt [3]{d} \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 (\sqrt {d}+1\right )\right )}{4 a \sqrt [3]{d} \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}-\sqrt [6]{d} \sqrt [3]{x}\right )}{4 a \sqrt [3]{d} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {3 x^{4/3} (x-a)^{2/3} \log \left (\sqrt [6]{d} \sqrt [3]{x}-\sqrt [3]{x-a}\right )}{4 a \sqrt [3]{d} \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 [6]{d} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-a}}\right )}{2 a \sqrt [3]{d} \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 [6]{d} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{x-a}}+\frac {1}{\sqrt {3}}\right )}{2 a \sqrt [3]{d} \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 1593
Rule 6719
Rubi steps
\begin {align*} \int \frac {-a x+x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2-2 a x+(1-d) x^2\right )} \, dx &=\int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2-2 a 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^2-2 a 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-2 a \sqrt {d}-2 (1-d) x\right )}+\frac {(1-d) \sqrt [3]{-a+x}}{a \sqrt {d} \sqrt [3]{x} \left (-2 a-2 a \sqrt {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-2 a \sqrt {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-2 a \sqrt {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-2 a \sqrt {d}+2 (1-d) x\right )} \, dx}{\left (1-\sqrt {d}\right ) \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-2 a \sqrt {d}-2 (1-d) x\right )} \, dx}{\left (1+\sqrt {d}\right ) \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 [6]{d} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-a+x}}\right )}{2 a \sqrt [3]{d} \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 [6]{d} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-a+x}}\right )}{2 a \sqrt [3]{d} \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 )-2 (1-d) x\right )}{4 a \sqrt [3]{d} \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 )+2 (1-d) x\right )}{4 a \sqrt [3]{d} \left (-\left ((a-x) x^2\right )\right )^{2/3}}+\frac {3 x^{4/3} (-a+x)^{2/3} \log \left (-\sqrt [6]{d} \sqrt [3]{x}-\sqrt [3]{-a+x}\right )}{4 a \sqrt [3]{d} \left (-\left ((a-x) x^2\right )\right )^{2/3}}+\frac {3 x^{4/3} (-a+x)^{2/3} \log \left (\sqrt [6]{d} \sqrt [3]{x}-\sqrt [3]{-a+x}\right )}{4 a \sqrt [3]{d} \left (-\left ((a-x) x^2\right )\right )^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 68, normalized size = 0.27 \begin {gather*} -\frac {3 x^2 \left (\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} x}{a-x}\right )+\, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {d} x}{x-a}\right )\right )}{4 a \left (x^2 (x-a)\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.64, size = 247, normalized size = 0.99 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {x^2}{\sqrt {3}}+\frac {2 \left (-a x^2+x^3\right )^{2/3}}{\sqrt {3} \sqrt [3]{d}}}{x^2}\right )}{2 a \sqrt [3]{d}}+\frac {\log \left (-\sqrt [6]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{2 a \sqrt [3]{d}}+\frac {\log \left (\sqrt [6]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{2 a \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{d} x^2-\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{4 a \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{d} x^2+\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\left (-a x^2+x^3\right )^{2/3}\right )}{4 a \sqrt [3]{d}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 357, normalized size = 1.43 \begin {gather*} \left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (-\frac {{\left (d + 2\right )} x^{2} + 2 \, a^{2} - 4 \, a x - \sqrt {3} {\left (d^{\frac {4}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )} d^{\frac {2}{3}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}} - 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {2}{3}}}{{\left (d - 1\right )} x^{2} - a^{2} + 2 \, a x}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {1}{3}}}{x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{4 \, a d}, \frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}\right )}}{3 \, d^{\frac {1}{3}} x^{2}}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} {\left (a - x\right )} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d^{\frac {1}{3}}}{x^{2}}\right ) + 2 \, d^{\frac {2}{3}} \log \left (-\frac {d^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{4 \, a d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 100, normalized size = 0.40 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + d^{\frac {1}{3}}\right )}}{3 \, d^{\frac {1}{3}}}\right )}{2 \, a d^{\frac {1}{3}}} - \frac {\log \left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + d^{\frac {1}{3}} {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + d^{\frac {2}{3}}\right )}{4 \, a d^{\frac {1}{3}}} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} - d^{\frac {1}{3}} \right |}\right )}{2 \, a d^{\frac {1}{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 \[\int \frac {-a x +x^{2}}{\left (x^{2} \left (-a +x \right )\right )^{\frac {2}{3}} \left (a^{2}-2 a x +\left (1-d \right ) x^{2}\right )}\, dx\]
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 - x^{2}}{\left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}} {\left ({\left (d - 1\right )} x^{2} - a^{2} + 2 \, a 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-x^2}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (-a^2+2\,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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