Integrand size = 52, antiderivative size = 294 \[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{\sqrt [6]{d} x^2-2 \left (-a x^2+x^3\right )^{2/3}}\right )}{4 a d^{5/6}}-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x^2}{\sqrt [6]{d} x^2+2 \left (-a x^2+x^3\right )^{2/3}}\right )}{4 a d^{5/6}}+\frac {\text {arctanh}\left (\frac {\sqrt [6]{d} \left (-a x^2+x^3\right )^{2/3}}{\sqrt [3]{d} x^2+(-a+x) \sqrt [3]{-a x^2+x^3}}\right )}{4 a d^{5/6}}-\frac {\log \left (-\sqrt [12]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{4 a d^{5/6}}-\frac {\log \left (\sqrt [12]{d} x+\sqrt [3]{-a x^2+x^3}\right )}{4 a d^{5/6}}+\frac {\log \left (\sqrt [6]{d} x^2+\left (-a x^2+x^3\right )^{2/3}\right )}{4 a d^{5/6}} \]
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\[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=\int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{-a+x}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-a+x} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx}{\sqrt [3]{x^2 (-a+x)}} \\ & = \frac {\left (3 x^{2/3} \sqrt [3]{-a+x}\right ) \text {Subst}\left (\int \frac {x^9}{\sqrt [3]{-a+x^3} \left (-a^4+4 a^3 x^3-6 a^2 x^6+4 a x^9+(-1+d) x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x^2 (-a+x)}} \\ \end{align*}
Time = 1.19 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.34 \[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=\frac {x^{2/3} \sqrt [3]{-a+x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x^{2/3}}{\sqrt [6]{d} x^{2/3}-2 (-a+x)^{2/3}}\right )-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [6]{d} x^{2/3}}{\sqrt [6]{d} x^{2/3}+2 (-a+x)^{2/3}}\right )-2 \log \left (-\sqrt [12]{d} \sqrt [3]{x}+\sqrt [3]{-a+x}\right )-2 \log \left (\sqrt [12]{d} \sqrt [3]{x}+\sqrt [3]{-a+x}\right )+2 \log \left (\sqrt [6]{d} x^{2/3}+(-a+x)^{2/3}\right )+\log \left (\sqrt [6]{d} x^{2/3}-\sqrt [12]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )+\log \left (\sqrt [6]{d} x^{2/3}+\sqrt [12]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )-\log \left (\sqrt [6]{d} x^{2/3}-\sqrt {3} \sqrt [12]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )-\log \left (\sqrt [6]{d} x^{2/3}+\sqrt {3} \sqrt [12]{d} \sqrt [3]{x} \sqrt [3]{-a+x}+(-a+x)^{2/3}\right )\right )}{8 a d^{5/6} \sqrt [3]{x^2 (-a+x)}} \]
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Time = 0.93 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.82
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{6}} x^{2}-2 \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}\right )}{3 d^{\frac {1}{6}} x^{2}}\right )-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (d^{\frac {1}{6}} x^{2}+2 \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}\right )}{3 d^{\frac {1}{6}} x^{2}}\right )-\ln \left (\frac {d^{\frac {1}{6}} x^{2}+\left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )+\ln \left (\frac {d^{\frac {1}{6}} x^{2}-\left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}}{x^{2}}\right )+\frac {\ln \left (\frac {-d^{\frac {1}{6}} \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}+\left (-a +x \right ) \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}+d^{\frac {1}{3}} x^{2}}{x^{2}}\right )}{2}-\frac {\ln \left (\frac {d^{\frac {1}{6}} \left (-x^{2} \left (a -x \right )\right )^{\frac {2}{3}}+\left (-a +x \right ) \left (-x^{2} \left (a -x \right )\right )^{\frac {1}{3}}+d^{\frac {1}{3}} x^{2}}{x^{2}}\right )}{2}}{4 d^{\frac {5}{6}} a}\) | \(240\) |
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Time = 0.28 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.18 \[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=\frac {1}{8} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} a d x^{2} + a d x^{2}\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{2 \, x^{2}}\right ) - \frac {1}{8} \, {\left (\sqrt {-3} + 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} a d x^{2} + a d x^{2}\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{2 \, x^{2}}\right ) + \frac {1}{8} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {{\left (\sqrt {-3} a d x^{2} - a d x^{2}\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{2 \, x^{2}}\right ) - \frac {1}{8} \, {\left (\sqrt {-3} - 1\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {{\left (\sqrt {-3} a d x^{2} - a d x^{2}\right )} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - 2 \, {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{2 \, x^{2}}\right ) + \frac {1}{4} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (\frac {a d x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} + {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{4} \, \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {a d x^{2} \left (\frac {1}{a^{6} d^{5}}\right )^{\frac {1}{6}} - {\left (-a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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Timed out. \[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=\int { \frac {x^{3}}{{\left ({\left (d - 1\right )} x^{4} - a^{4} + 4 \, a^{3} x - 6 \, a^{2} x^{2} + 4 \, a x^{3}\right )} \left (-{\left (a - x\right )} x^{2}\right )^{\frac {1}{3}}} \,d x } \]
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Time = 0.35 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.71 \[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=-\frac {\sqrt {3} \log \left (\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{8 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} \log \left (-\sqrt {3} \left (-d\right )^{\frac {1}{6}} {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {a}{x} + 1\right )}^{\frac {4}{3}} + \left (-d\right )^{\frac {1}{3}}\right )}{8 \, a d} - \frac {\arctan \left (\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} + 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{4 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (-\frac {\sqrt {3} \left (-d\right )^{\frac {1}{6}} - 2 \, {\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{4 \, a \left (-d\right )^{\frac {5}{6}}} - \frac {\arctan \left (\frac {{\left (-\frac {a}{x} + 1\right )}^{\frac {2}{3}}}{\left (-d\right )^{\frac {1}{6}}}\right )}{2 \, a \left (-d\right )^{\frac {5}{6}}} \]
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Timed out. \[ \int \frac {x^3}{\sqrt [3]{x^2 (-a+x)} \left (-a^4+4 a^3 x-6 a^2 x^2+4 a x^3+(-1+d) x^4\right )} \, dx=\int \frac {x^3}{{\left (-x^2\,\left (a-x\right )\right )}^{1/3}\,\left (-a^4+4\,a^3\,x-6\,a^2\,x^2+4\,a\,x^3+\left (d-1\right )\,x^4\right )} \,d x \]
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