Integrand size = 36, antiderivative size = 205 \[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x-2 \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x}{\sqrt [6]{d} x+2 \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} x}{\sqrt [3]{-a x^2+x^3}}\right )}{a d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} x^2+\frac {\left (-a x^2+x^3\right )^{2/3}}{\sqrt [6]{d}}}{x \sqrt [3]{-a x^2+x^3}}\right )}{2 a d^{5/6}} \]
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Time = 0.64 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.94, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6851, 925, 129, 494, 337, 503} \[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} x^{4/3} (x-a)^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{x-a}}}{\sqrt {3}}\right )}{2 a d^{5/6} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}-\frac {\sqrt {3} x^{4/3} (x-a)^{2/3} \arctan \left (\frac {\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{x-a}}+1}{\sqrt {3}}\right )}{2 a d^{5/6} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}-\frac {x^{4/3} (x-a)^{2/3} \log \left (a \left (1-\sqrt {d}\right )-(1-d) x\right )}{4 a d^{5/6} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {x^{4/3} (x-a)^{2/3} \log \left (a \left (\sqrt {d}+1\right )-(1-d) x\right )}{4 a d^{5/6} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}-\frac {3 x^{4/3} (x-a)^{2/3} \log \left (\sqrt [6]{d} \sqrt [3]{x}-\sqrt [3]{x-a}\right )}{4 a d^{5/6} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {3 x^{4/3} (x-a)^{2/3} \log \left (\sqrt [3]{x-a}+\sqrt [6]{d} \sqrt [3]{x}\right )}{4 a d^{5/6} \left (-\left (x^2 (a-x)\right )\right )^{2/3}} \]
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Rule 129
Rule 337
Rule 494
Rule 503
Rule 925
Rule 6851
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{4/3} (-a+x)^{2/3}\right ) \int \frac {x^{2/3}}{(-a+x)^{2/3} \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) x^{2/3}}{a \sqrt {d} (-a+x)^{2/3} \left (2 a-2 a \sqrt {d}-2 (1-d) x\right )}+\frac {(-1+d) x^{2/3}}{a \sqrt {d} (-a+x)^{2/3} \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 {x^{2/3}}{(-a+x)^{2/3} \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 {x^{2/3}}{(-a+x)^{2/3} \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 (3 (1-d) x^{4/3} (-a+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^4}{\left (-a+x^3\right )^{2/3} \left (2 a-2 a \sqrt {d}-2 (1-d) x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{a \sqrt {d} \left (x^2 (-a+x)\right )^{2/3}}-\frac {\left (3 (1-d) x^{4/3} (-a+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^4}{\left (-a+x^3\right )^{2/3} \left (-2 a-2 a \sqrt {d}+2 (1-d) x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{a \sqrt {d} \left (x^2 (-a+x)\right )^{2/3}} \\ & = -\frac {\left (3 (1-d) x^{4/3} (-a+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x}{\left (-a+x^3\right )^{2/3} \left (-2 a-2 a \sqrt {d}+2 (1-d) x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (1-\sqrt {d}\right ) \sqrt {d} \left (x^2 (-a+x)\right )^{2/3}}-\frac {\left (3 (1-d) x^{4/3} (-a+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x}{\left (-a+x^3\right )^{2/3} \left (2 a-2 a \sqrt {d}-2 (1-d) x^3\right )} \, dx,x,\sqrt [3]{x}\right )}{\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} \arctan \left (\frac {1-\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{-a+x}}}{\sqrt {3}}\right )}{2 a d^{5/6} \left (-\left ((a-x) x^2\right )\right )^{2/3}}-\frac {\sqrt {3} x^{4/3} (-a+x)^{2/3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{-a+x}}}{\sqrt {3}}\right )}{2 a d^{5/6} \left (-\left ((a-x) x^2\right )\right )^{2/3}}-\frac {x^{4/3} (-a+x)^{2/3} \log \left (a \left (1-\sqrt {d}\right )-(1-d) x\right )}{4 a d^{5/6} \left (-\left ((a-x) x^2\right )\right )^{2/3}}+\frac {x^{4/3} (-a+x)^{2/3} \log \left (a \left (1+\sqrt {d}\right )-(1-d) x\right )}{4 a d^{5/6} \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 d^{5/6} \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 d^{5/6} \left (-\left ((a-x) x^2\right )\right )^{2/3}} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.97 \[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=\frac {x^{4/3} (-a+x)^{2/3} \left (\sqrt {3} \left (\arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x}-2 \sqrt [3]{-a+x}}\right )-\arctan \left (\frac {\sqrt {3} \sqrt [6]{d} \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x}+2 \sqrt [3]{-a+x}}\right )\right )+2 \text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{-a+x}}\right )+\text {arctanh}\left (\frac {\sqrt [6]{d} \sqrt [3]{x}}{\sqrt [3]{-a+x}}+\frac {\sqrt [3]{-a+x}}{\sqrt [6]{d} \sqrt [3]{x}}\right )\right )}{2 a d^{5/6} \left (x^2 (-a+x)\right )^{2/3}} \]
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Time = 1.24 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.09
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{6}} x +2 \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{6}} x}\right )-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{6}} x -2 \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}\right )}{3 d^{\frac {1}{6}} x}\right )-2 \ln \left (\frac {d^{\frac {1}{6}} x -\left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {d^{\frac {1}{3}} x^{2}+d^{\frac {1}{6}} \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}} x +\left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {d^{\frac {1}{3}} x^{2}-d^{\frac {1}{6}} \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}} x +\left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )+2 \ln \left (\frac {d^{\frac {1}{6}} x +\left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}}{x}\right )}{4 d^{\frac {5}{6}} a}\) | \(223\) |
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Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (159) = 318\).
Time = 0.25 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.59 \[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=\frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} a d x + a d x\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} a d x + a d x\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} a d x - a d x\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} a d x - a d x\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{2 \, x}\right ) + \frac {1}{2} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {a d x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{2} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {a d x \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) \]
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\[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=\int \frac {x^{2}}{\left (x^{2} \left (- a + x\right )\right )^{\frac {2}{3}} \left (- a^{2} + 2 a x + d x^{2} - x^{2}\right )}\, dx \]
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\[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=\int { \frac {x^{2}}{\left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}} {\left ({\left (d - 1\right )} x^{2} - a^{2} + 2 \, a x\right )}} \,d x } \]
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none
Time = 0.30 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.02 \[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \log \left (\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{4 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} \log \left (-\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{4 \, a d} - \frac {\arctan \left (\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{2 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (-\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} - 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{2 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (\frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{a \left (-d\right )^{\frac {5}{6}}} \]
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Timed out. \[ \int \frac {x^2}{\left (x^2 (-a+x)\right )^{2/3} \left (-a^2+2 a x+(-1+d) x^2\right )} \, dx=\int \frac {x^2}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (-a^2+2\,a\,x+\left (d-1\right )\,x^2\right )} \,d x \]
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