Integrand size = 39, antiderivative size = 65 \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\frac {1}{6} \text {RootSum}\left [a^6+4 a b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b+a^3 x^3+a x^6}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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\[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt [3]{-b+a^3 x^3+a x^6}}-\frac {2 b}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}}\right ) \, dx \\ & = -\left ((2 b) \int \frac {1}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx\right )+\int \frac {1}{\sqrt [3]{-b+a^3 x^3+a x^6}} \, dx \\ & = -\left ((2 b) \int \left (\frac {1}{2 \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^3\right ) \sqrt [3]{-b+a^3 x^3+a x^6}}+\frac {1}{2 \sqrt {b} \left (\sqrt {b}+\sqrt {-a} x^3\right ) \sqrt [3]{-b+a^3 x^3+a x^6}}\right ) \, dx\right )+\frac {\left (\sqrt [3]{1+\frac {2 a x^3}{a^3-\sqrt {a} \sqrt {a^5+4 b}}} \sqrt [3]{1+\frac {2 a x^3}{a^3+\sqrt {a} \sqrt {a^5+4 b}}}\right ) \int \frac {1}{\sqrt [3]{1+\frac {2 a x^3}{a^3-\sqrt {a^6+4 a b}}} \sqrt [3]{1+\frac {2 a x^3}{a^3+\sqrt {a^6+4 a b}}}} \, dx}{\sqrt [3]{-b+a^3 x^3+a x^6}} \\ & = \frac {x \sqrt [3]{1+\frac {2 \sqrt {a} x^3}{a^{5/2}-\sqrt {a^5+4 b}}} \sqrt [3]{1+\frac {2 \sqrt {a} x^3}{a^{5/2}+\sqrt {a^5+4 b}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 a x^3}{a^3-\sqrt {a^6+4 a b}},-\frac {2 a x^3}{a^3+\sqrt {a^6+4 a b}}\right )}{\sqrt [3]{-b+a^3 x^3+a x^6}}-\sqrt {b} \int \frac {1}{\left (\sqrt {b}-\sqrt {-a} x^3\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx-\sqrt {b} \int \frac {1}{\left (\sqrt {b}+\sqrt {-a} x^3\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx \\ \end{align*}
Time = 1.50 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\frac {1}{6} \text {RootSum}\left [a^6+4 a b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b+a^3 x^3+a x^6}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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Time = 2.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a^{3} \textit {\_Z}^{3}+a^{6}+4 a b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (a \,x^{6}+a^{3} x^{3}-b \right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )}{6}\) | \(58\) |
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Exception generated. \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 11.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.48 \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int \frac {a x^{6} - b}{\left (a x^{6} + b\right ) \sqrt [3]{a^{3} x^{3} + a x^{6} - b}}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.60 \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int { \frac {a x^{6} - b}{{\left (a x^{6} + a^{3} x^{3} - b\right )}^{\frac {1}{3}} {\left (a x^{6} + b\right )}} \,d x } \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.60 \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int { \frac {a x^{6} - b}{{\left (a x^{6} + a^{3} x^{3} - b\right )}^{\frac {1}{3}} {\left (a x^{6} + b\right )}} \,d x } \]
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Not integrable
Time = 5.49 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.60 \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int -\frac {b-a\,x^6}{\left (a\,x^6+b\right )\,{\left (a^3\,x^3+a\,x^6-b\right )}^{1/3}} \,d x \]
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