Integrand size = 37, antiderivative size = 151 \[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}}\right )}{\sqrt [3]{d}}+\frac {\log \left (a x^2-a \sqrt [3]{d} \sqrt [3]{-a x^2+x^3}\right )}{\sqrt [3]{d}}-\frac {\log \left (a^2 x^4+a^2 \sqrt [3]{d} x^2 \sqrt [3]{-a x^2+x^3}+a^2 d^{2/3} \left (-a x^2+x^3\right )^{2/3}\right )}{2 \sqrt [3]{d}} \]
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\[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=\int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{4/3} (-a+x)^{2/3}\right ) \int \frac {x^{5/3} (-4 a+3 x)}{(-a+x)^{2/3} \left (a d-d x+x^4\right )} \, dx}{\left (x^2 (-a+x)\right )^{2/3}} \\ & = \frac {\left (3 x^{4/3} (-a+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^7 \left (-4 a+3 x^3\right )}{\left (-a+x^3\right )^{2/3} \left (a d-d x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x)\right )^{2/3}} \\ & = \frac {\left (3 x^{4/3} (-a+x)^{2/3}\right ) \text {Subst}\left (\int \left (-\frac {4 a x^7}{\left (-a+x^3\right )^{2/3} \left (a d-d x^3+x^{12}\right )}+\frac {3 x^{10}}{\left (-a+x^3\right )^{2/3} \left (a d-d x^3+x^{12}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x)\right )^{2/3}} \\ & = \frac {\left (9 x^{4/3} (-a+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^{10}}{\left (-a+x^3\right )^{2/3} \left (a d-d x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x)\right )^{2/3}}-\frac {\left (12 a x^{4/3} (-a+x)^{2/3}\right ) \text {Subst}\left (\int \frac {x^7}{\left (-a+x^3\right )^{2/3} \left (a d-d x^3+x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\left (x^2 (-a+x)\right )^{2/3}} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.03 \[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=\frac {x^{4/3} (-a+x)^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{4/3}}{x^{4/3}+2 \sqrt [3]{d} \sqrt [3]{-a+x}}\right )+2 \log \left (a \left (x^{4/3}-\sqrt [3]{d} \sqrt [3]{-a+x}\right )\right )-\log \left (a^2 \left (x^{8/3}+\sqrt [3]{d} x^{4/3} \sqrt [3]{-a+x}+d^{2/3} (-a+x)^{2/3}\right )\right )\right )}{2 \sqrt [3]{d} \left (x^2 (-a+x)\right )^{2/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.50
method | result | size |
pseudoelliptic | \(\frac {a^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{12}-3 d \,\textit {\_Z}^{9}+3 d \,\textit {\_Z}^{6}-d \,\textit {\_Z}^{3}+a^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (-\left (a -x \right ) x^{2}\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{8}-2 \textit {\_R}^{5}+\textit {\_R}^{2}}\right )}{d}\) | \(76\) |
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Time = 0.25 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.24 \[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=\left [\frac {\sqrt {3} d \sqrt {-\frac {1}{d^{\frac {2}{3}}}} \log \left (-\frac {x^{4} - 3 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {1}{3}} x^{2} - 2 \, a d + 2 \, d x + \sqrt {3} {\left (d^{\frac {1}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x^{2} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d\right )} \sqrt {-\frac {1}{d^{\frac {2}{3}}}}}{x^{4} + a d - d x}\right ) + 2 \, d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x^{2}}\right ) - d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{4}}\right )}{2 \, d}, -\frac {2 \, \sqrt {3} d^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (d^{\frac {1}{3}} x^{2} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}}\right )}}{3 \, d^{\frac {1}{3}} x^{2}}\right ) - 2 \, d^{\frac {2}{3}} \log \left (\frac {d^{\frac {2}{3}} x^{2} - {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d}{x^{2}}\right ) + d^{\frac {2}{3}} \log \left (\frac {d^{\frac {1}{3}} x^{4} + {\left (-a x^{2} + x^{3}\right )}^{\frac {1}{3}} d^{\frac {2}{3}} x^{2} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}} d}{x^{4}}\right )}{2 \, d}\right ] \]
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Timed out. \[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=\int { -\frac {{\left (4 \, a - 3 \, x\right )} x^{3}}{{\left (x^{4} + a d - d x\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {2}{3}}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.56 \[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=-\sqrt {3} \left (-\frac {1}{d}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \sqrt {3} a {\left | d \right |}^{\frac {2}{3}} - \sqrt {3} d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}{d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a {\left | d \right |}^{\frac {2}{3}} - d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}}\right ) - \frac {1}{2} \, \left (-\frac {1}{d}\right )^{\frac {1}{3}} \log \left (\frac {3}{4} \, a^{2} d^{\frac {4}{3}} + \frac {1}{4} \, {\left (2 \, d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} - a d^{\frac {2}{3}} - 2 \, d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2}\right ) + \frac {1}{2} \, \left (-\frac {1}{d}\right )^{\frac {1}{3}} \log \left ({\left (\sqrt {3} d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \sqrt {3} a {\left | d \right |}^{\frac {2}{3}} - \sqrt {3} d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2} + {\left (d {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + a {\left | d \right |}^{\frac {2}{3}} - d {\left (-\frac {a}{x} + 1\right )}^{\frac {1}{3}}\right )}^{2}\right ) \]
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Timed out. \[ \int \frac {x^3 (-4 a+3 x)}{\left (x^2 (-a+x)\right )^{2/3} \left (a d-d x+x^4\right )} \, dx=\int -\frac {x^3\,\left (4\,a-3\,x\right )}{{\left (-x^2\,\left (a-x\right )\right )}^{2/3}\,\left (x^4-d\,x+a\,d\right )} \,d x \]
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