Integrand size = 51, antiderivative size = 123 \[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-2 \sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b+a x^3}}{\sqrt {2}}}{x \sqrt [4]{-b+a x^3}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^3}}{x^2+\sqrt {-b+a x^3}}\right ) \]
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\[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4 b}{x^4 \sqrt [4]{-b+a x^3}}-\frac {a}{x \sqrt [4]{-b+a x^3}}+\frac {2 \left (4 b-a x^3\right )}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )}\right ) \, dx \\ & = 2 \int \frac {4 b-a x^3}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx-a \int \frac {1}{x \sqrt [4]{-b+a x^3}} \, dx+(4 b) \int \frac {1}{x^4 \sqrt [4]{-b+a x^3}} \, dx \\ & = 2 \int \left (\frac {4 b}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )}+\frac {a x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )}\right ) \, dx-\frac {1}{3} a \text {Subst}\left (\int \frac {1}{x \sqrt [4]{-b+a x}} \, dx,x,x^3\right )+\frac {1}{3} (4 b) \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{-b+a x}} \, dx,x,x^3\right ) \\ & = \frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-\frac {4}{3} \text {Subst}\left (\int \frac {x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )+\frac {1}{3} a \text {Subst}\left (\int \frac {1}{x \sqrt [4]{-b+a x}} \, dx,x,x^3\right )+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx \\ & = \frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}+\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )-\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )+\frac {4}{3} \text {Subst}\left (\int \frac {x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx \\ & = \frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )+\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^3}\right )-\frac {1}{3} a \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )-\frac {1}{3} a \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx-\frac {a \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}-\frac {a \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx \\ & = \frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-\frac {a \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}+\frac {a \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^3}+\sqrt {-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}+\frac {1}{3} a \text {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )+\frac {1}{3} a \text {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+\frac {a \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}+\frac {a \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^3}\right )}{3 \sqrt {2} \sqrt [4]{b}}-\frac {\left (\sqrt {2} a\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+\frac {\left (\sqrt {2} a\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx \\ & = \frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}+\frac {\sqrt {2} a \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}-\frac {\sqrt {2} a \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+\frac {\left (\sqrt {2} a\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}-\frac {\left (\sqrt {2} a\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^3}}{\sqrt [4]{b}}\right )}{3 \sqrt [4]{b}}+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx \\ & = \frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}+(2 a) \int \frac {x^3}{\sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx+(8 b) \int \frac {1}{\sqrt [4]{-b+a x^3} \left (b-a x^3-x^4\right )} \, dx \\ \end{align*}
Time = 1.52 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\frac {4 \left (-b+a x^3\right )^{3/4}}{3 x^3}-2 \sqrt {2} \arctan \left (\frac {-x^2+\sqrt {-b+a x^3}}{\sqrt {2} x \sqrt [4]{-b+a x^3}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-b+a x^3}}{x^2+\sqrt {-b+a x^3}}\right ) \]
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\[\int \frac {\left (a \,x^{3}-4 b \right ) \left (-a \,x^{3}+x^{4}+b \right )}{x^{4} \left (a \,x^{3}-b \right )^{\frac {1}{4}} \left (a \,x^{3}+x^{4}-b \right )}d x\]
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Timed out. \[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\int { -\frac {{\left (a x^{3} - x^{4} - b\right )} {\left (a x^{3} - 4 \, b\right )}}{{\left (a x^{3} + x^{4} - b\right )} {\left (a x^{3} - b\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
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\[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\int { -\frac {{\left (a x^{3} - x^{4} - b\right )} {\left (a x^{3} - 4 \, b\right )}}{{\left (a x^{3} + x^{4} - b\right )} {\left (a x^{3} - b\right )}^{\frac {1}{4}} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (-4 b+a x^3\right ) \left (b-a x^3+x^4\right )}{x^4 \sqrt [4]{-b+a x^3} \left (-b+a x^3+x^4\right )} \, dx=\int -\frac {\left (4\,b-a\,x^3\right )\,\left (x^4-a\,x^3+b\right )}{x^4\,{\left (a\,x^3-b\right )}^{1/4}\,\left (x^4+a\,x^3-b\right )} \,d x \]
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