Integrand size = 40, antiderivative size = 243 \[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}}\right )}{2 a d^{2/3}}+\frac {\log \left (x-\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}\right )}{2 a d^{2/3}}+\frac {\log \left (x+\sqrt [6]{d} \sqrt [3]{-a x^2+x^3}\right )}{2 a d^{2/3}}-\frac {\log \left (x^2-\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}\right )}{4 a d^{2/3}}-\frac {\log \left (x^2+\sqrt [6]{d} x \sqrt [3]{-a x^2+x^3}+\sqrt [3]{d} \left (-a x^2+x^3\right )^{2/3}\right )}{4 a d^{2/3}} \]
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Time = 0.58 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.67, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6851, 925, 129, 495, 337, 503} \[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} x^{4/3} (x-a)^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x-a}}}{\sqrt {3}}\right )}{2 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {\sqrt {3} x^{4/3} (x-a)^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{x-a}}+1}{\sqrt {3}}\right )}{2 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}-\frac {x^{4/3} (x-a)^{2/3} \log \left (a \left (1-\sqrt {d}\right ) \sqrt {d}-(1-d) x\right )}{4 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}-\frac {x^{4/3} (x-a)^{2/3} \log \left (a \sqrt {d} \left (\sqrt {d}+1\right )+(1-d) x\right )}{4 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {3 x^{4/3} (x-a)^{2/3} \log \left (\sqrt [3]{x}-\sqrt [6]{d} \sqrt [3]{x-a}\right )}{4 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}}+\frac {3 x^{4/3} (x-a)^{2/3} \log \left (\sqrt [6]{d} \sqrt [3]{x-a}+\sqrt [3]{x}\right )}{4 a d^{2/3} \left (-\left (x^2 (a-x)\right )\right )^{2/3}} \]
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Rule 129
Rule 337
Rule 495
Rule 503
Rule 925
Rule 6851
Rubi steps \begin{align*} \text {integral}& = \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 (3 (1-d) x^{4/3} (-a+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{-a+x^3}}{-2 a \sqrt {d}-2 a d-2 (1-d) x^3} \, 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 \sqrt [3]{-a+x^3}}{-2 a \sqrt {d}+2 a d+2 (1-d) x^3} \, 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 \sqrt {d}-2 a 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 \sqrt {d}+2 a 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 [3]{x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}}{\sqrt {3}}\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} \arctan \left (\frac {1+\frac {2 \sqrt [3]{x}}{\sqrt [6]{d} \sqrt [3]{-a+x}}}{\sqrt {3}}\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 (a \left (1-\sqrt {d}\right ) \sqrt {d}-(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 (a \left (1+\sqrt {d}\right ) \sqrt {d}+(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 (\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 (\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*}
Time = 0.42 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.89 \[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=-\frac {x^{4/3} (-a+x)^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{d} (-a+x)^{2/3}}\right )-2 \log \left (\sqrt [3]{x}-\sqrt [6]{d} \sqrt [3]{-a+x}\right )-2 \log \left (\sqrt [3]{x}+\sqrt [6]{d} \sqrt [3]{-a+x}\right )+\log \left (x^{2/3}-\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )+\log \left (x^{2/3}+\sqrt [6]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+\sqrt [3]{d} (-a+x)^{2/3}\right )\right )}{4 a d^{2/3} \left (x^2 (-a+x)\right )^{2/3}} \]
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Time = 0.78 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.58
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}+2 \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}\right )}{3 \left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}}\right )+2 \ln \left (\frac {-\left (\frac {1}{d}\right )^{\frac {1}{3}} x^{2}+\left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )-\ln \left (\frac {\left (\frac {1}{d}\right )^{\frac {1}{3}} \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}+\left (-a +x \right ) \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}+\left (\frac {1}{d}\right )^{\frac {2}{3}} x^{2}}{x^{2}}\right )}{4 a d \left (\frac {1}{d}\right )^{\frac {1}{3}}}\) | \(141\) |
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Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.69 \[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\frac {2 \, \sqrt {3} {\left (d^{2}\right )}^{\frac {1}{6}} d \arctan \left (\frac {\sqrt {3} {\left ({\left (d^{2}\right )}^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d\right )} {\left (d^{2}\right )}^{\frac {1}{6}}}{3 \, d x^{2}}\right ) - {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (d^{2}\right )}^{\frac {2}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {1}{3}} d - {\left (a d^{2} - d^{2} x\right )} {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x^{2}}\right ) + 2 \, {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (d^{2}\right )}^{\frac {1}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{2}}\right )}{4 \, a d^{2}} \]
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\[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int \frac {x \left (- a + x\right )}{\left (x^{2} \left (- a + x\right )\right )^{\frac {2}{3}} \left (a^{2} d - 2 a d x + d x^{2} - x^{2}\right )}\, dx \]
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\[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int { -\frac {{\left (a - x\right )} x}{{\left (a^{2} d - 2 \, a d x + {\left (d - 1\right )} x^{2}\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}}} \,d x } \]
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Time = 0.36 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.42 \[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} d^{\frac {1}{3}} {\left (2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + \frac {1}{d^{\frac {1}{3}}}\right )}\right )}{2 \, a {\left | d \right |}^{\frac {2}{3}}} - \frac {\log \left ({\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{d^{\frac {1}{3}}} + \frac {1}{d^{\frac {2}{3}}}\right )}{4 \, a {\left | d \right |}^{\frac {2}{3}}} + \frac {\log \left ({\left | {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} - \frac {1}{d^{\frac {1}{3}}} \right |}\right )}{2 \, a d^{\frac {2}{3}}} \]
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Timed out. \[ \int \frac {x (-a+x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a^2 d-2 a d x+(-1+d) x^2\right )} \, dx=\int -\frac {x\,\left (a-x\right )}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (d\,a^2-2\,d\,a\,x+\left (d-1\right )\,x^2\right )} \,d x \]
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