Integrand size = 34, antiderivative size = 147 \[ \int \frac {2 d+c x^4}{\sqrt [4]{-b+a x^4} \left (-2 f+e x^8\right )} \, dx=\frac {\text {RootSum}\left [b^2 e-2 a^2 f+4 a f \text {$\#$1}^4-2 f \text {$\#$1}^8\&,\frac {-b c \log (x)-2 a d \log (x)+b c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+2 a d \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+2 d \log (x) \text {$\#$1}^4-2 d \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-\text {$\#$1}^5}\&\right ]}{16 f} \]
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Leaf count is larger than twice the leaf count of optimal. \(459\) vs. \(2(147)=294\).
Time = 0.60 (sec) , antiderivative size = 459, normalized size of antiderivative = 3.12, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {6857, 385, 218, 214, 211} \[ \int \frac {2 d+c x^4}{\sqrt [4]{-b+a x^4} \left (-2 f+e x^8\right )} \, dx=-\frac {\left (c \sqrt {f}+\sqrt {2} d \sqrt {e}\right ) \arctan \left (\frac {x \sqrt [4]{\sqrt {2} a \sqrt {f}-b \sqrt {e}}}{\sqrt [8]{2} \sqrt [8]{f} \sqrt [4]{a x^4-b}}\right )}{4\ 2^{3/8} \sqrt {e} f^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {f}-b \sqrt {e}}}-\frac {\left (\sqrt {2} d \sqrt {e}-c \sqrt {f}\right ) \arctan \left (\frac {x \sqrt [4]{\sqrt {2} a \sqrt {f}+b \sqrt {e}}}{\sqrt [8]{2} \sqrt [8]{f} \sqrt [4]{a x^4-b}}\right )}{4\ 2^{3/8} \sqrt {e} f^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {f}+b \sqrt {e}}}-\frac {\left (c \sqrt {f}+\sqrt {2} d \sqrt {e}\right ) \text {arctanh}\left (\frac {x \sqrt [4]{\sqrt {2} a \sqrt {f}-b \sqrt {e}}}{\sqrt [8]{2} \sqrt [8]{f} \sqrt [4]{a x^4-b}}\right )}{4\ 2^{3/8} \sqrt {e} f^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {f}-b \sqrt {e}}}-\frac {\left (\sqrt {2} d \sqrt {e}-c \sqrt {f}\right ) \text {arctanh}\left (\frac {x \sqrt [4]{\sqrt {2} a \sqrt {f}+b \sqrt {e}}}{\sqrt [8]{2} \sqrt [8]{f} \sqrt [4]{a x^4-b}}\right )}{4\ 2^{3/8} \sqrt {e} f^{7/8} \sqrt [4]{\sqrt {2} a \sqrt {f}+b \sqrt {e}}} \]
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Rule 211
Rule 214
Rule 218
Rule 385
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 d \sqrt {e}+\sqrt {2} c \sqrt {f}}{2 \sqrt {2} \sqrt {e} \sqrt {f} \sqrt [4]{-b+a x^4} \left (\sqrt {2} \sqrt {f}-\sqrt {e} x^4\right )}+\frac {-2 d \sqrt {e}+\sqrt {2} c \sqrt {f}}{2 \sqrt {2} \sqrt {e} \sqrt {f} \sqrt [4]{-b+a x^4} \left (\sqrt {2} \sqrt {f}+\sqrt {e} x^4\right )}\right ) \, dx \\ & = \frac {1}{2} \left (\frac {c}{\sqrt {e}}-\frac {\sqrt {2} d}{\sqrt {f}}\right ) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (\sqrt {2} \sqrt {f}+\sqrt {e} x^4\right )} \, dx-\frac {1}{2} \left (\frac {c}{\sqrt {e}}+\frac {\sqrt {2} d}{\sqrt {f}}\right ) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (\sqrt {2} \sqrt {f}-\sqrt {e} x^4\right )} \, dx \\ & = \frac {1}{2} \left (\frac {c}{\sqrt {e}}-\frac {\sqrt {2} d}{\sqrt {f}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2} \sqrt {f}-\left (b \sqrt {e}+\sqrt {2} a \sqrt {f}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )-\frac {1}{2} \left (\frac {c}{\sqrt {e}}+\frac {\sqrt {2} d}{\sqrt {f}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2} \sqrt {f}-\left (-b \sqrt {e}+\sqrt {2} a \sqrt {f}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right ) \\ & = \frac {\left (\frac {c}{\sqrt {e}}-\frac {\sqrt {2} d}{\sqrt {f}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{f}-\sqrt {b \sqrt {e}+\sqrt {2} a \sqrt {f}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{2} \sqrt [4]{f}}+\frac {\left (\frac {c}{\sqrt {e}}-\frac {\sqrt {2} d}{\sqrt {f}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{f}+\sqrt {b \sqrt {e}+\sqrt {2} a \sqrt {f}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{2} \sqrt [4]{f}}-\frac {\left (\frac {c}{\sqrt {e}}+\frac {\sqrt {2} d}{\sqrt {f}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{f}-\sqrt {-b \sqrt {e}+\sqrt {2} a \sqrt {f}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{2} \sqrt [4]{f}}-\frac {\left (\frac {c}{\sqrt {e}}+\frac {\sqrt {2} d}{\sqrt {f}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2} \sqrt [4]{f}+\sqrt {-b \sqrt {e}+\sqrt {2} a \sqrt {f}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt [4]{2} \sqrt [4]{f}} \\ & = -\frac {\left (\frac {c}{\sqrt {e}}+\frac {\sqrt {2} d}{\sqrt {f}}\right ) \arctan \left (\frac {\sqrt [4]{-b \sqrt {e}+\sqrt {2} a \sqrt {f}} x}{\sqrt [8]{2} \sqrt [8]{f} \sqrt [4]{-b+a x^4}}\right )}{4\ 2^{3/8} \sqrt [4]{-b \sqrt {e}+\sqrt {2} a \sqrt {f}} f^{3/8}}+\frac {\left (\frac {c}{\sqrt {e}}-\frac {\sqrt {2} d}{\sqrt {f}}\right ) \arctan \left (\frac {\sqrt [4]{b \sqrt {e}+\sqrt {2} a \sqrt {f}} x}{\sqrt [8]{2} \sqrt [8]{f} \sqrt [4]{-b+a x^4}}\right )}{4\ 2^{3/8} \sqrt [4]{b \sqrt {e}+\sqrt {2} a \sqrt {f}} f^{3/8}}-\frac {\left (\frac {c}{\sqrt {e}}+\frac {\sqrt {2} d}{\sqrt {f}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{-b \sqrt {e}+\sqrt {2} a \sqrt {f}} x}{\sqrt [8]{2} \sqrt [8]{f} \sqrt [4]{-b+a x^4}}\right )}{4\ 2^{3/8} \sqrt [4]{-b \sqrt {e}+\sqrt {2} a \sqrt {f}} f^{3/8}}+\frac {\left (\frac {c}{\sqrt {e}}-\frac {\sqrt {2} d}{\sqrt {f}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{b \sqrt {e}+\sqrt {2} a \sqrt {f}} x}{\sqrt [8]{2} \sqrt [8]{f} \sqrt [4]{-b+a x^4}}\right )}{4\ 2^{3/8} \sqrt [4]{b \sqrt {e}+\sqrt {2} a \sqrt {f}} f^{3/8}} \\ \end{align*}
Time = 1.83 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {2 d+c x^4}{\sqrt [4]{-b+a x^4} \left (-2 f+e x^8\right )} \, dx=\frac {\text {RootSum}\left [b^2 e-2 a^2 f+4 a f \text {$\#$1}^4-2 f \text {$\#$1}^8\&,\frac {b c \log (x)+2 a d \log (x)-b c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-2 a d \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-2 d \log (x) \text {$\#$1}^4+2 d \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+\text {$\#$1}^5}\&\right ]}{16 f} \]
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Time = 0.38 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.57
method | result | size |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 f \,\textit {\_Z}^{8}-4 a f \,\textit {\_Z}^{4}+2 a^{2} f -b^{2} e \right )}{\sum }\frac {\left (-2 d \,\textit {\_R}^{4}+2 a d +b c \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}-b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (-\textit {\_R}^{4}+a \right )}}{16 f}\) | \(84\) |
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Timed out. \[ \int \frac {2 d+c x^4}{\sqrt [4]{-b+a x^4} \left (-2 f+e x^8\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 27.95 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.18 \[ \int \frac {2 d+c x^4}{\sqrt [4]{-b+a x^4} \left (-2 f+e x^8\right )} \, dx=\int \frac {c x^{4} + 2 d}{\sqrt [4]{a x^{4} - b} \left (e x^{8} - 2 f\right )}\, dx \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.23 \[ \int \frac {2 d+c x^4}{\sqrt [4]{-b+a x^4} \left (-2 f+e x^8\right )} \, dx=\int { \frac {c x^{4} + 2 \, d}{{\left (e x^{8} - 2 \, f\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.23 \[ \int \frac {2 d+c x^4}{\sqrt [4]{-b+a x^4} \left (-2 f+e x^8\right )} \, dx=\int { \frac {c x^{4} + 2 \, d}{{\left (e x^{8} - 2 \, f\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}} \,d x } \]
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Not integrable
Time = 5.98 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.24 \[ \int \frac {2 d+c x^4}{\sqrt [4]{-b+a x^4} \left (-2 f+e x^8\right )} \, dx=\int -\frac {c\,x^4+2\,d}{{\left (a\,x^4-b\right )}^{1/4}\,\left (2\,f-e\,x^8\right )} \,d x \]
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