Integrand size = 33, antiderivative size = 111 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-2 b+a^6 x^6\right )} \, dx=\frac {\text {RootSum}\left [2 a^{18}-a^6 b^{11}-12 a^{15} \text {$\#$1}^3+30 a^{12} \text {$\#$1}^6-40 a^9 \text {$\#$1}^9+30 a^6 \text {$\#$1}^{12}-12 a^3 \text {$\#$1}^{15}+2 \text {$\#$1}^{18}\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{12 b} \]
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Leaf count is larger than twice the leaf count of optimal. \(2286\) vs. \(2(111)=222\).
Time = 1.54 (sec) , antiderivative size = 2286, normalized size of antiderivative = 20.59, number of steps used = 13, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2081, 6857, 93} \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-2 b+a^6 x^6\right )} \, dx=\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [18]{2} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2-b^{11/6}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [18]{2} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2+b^{11/6}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [18]{2} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2-\sqrt [3]{-1} b^{11/6}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-\sqrt [3]{-1} b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [18]{2} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2+\sqrt [3]{-1} b^{11/6}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+\sqrt [3]{-1} b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [18]{2} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2-(-1)^{2/3} b^{11/6}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-(-1)^{2/3} b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [18]{2} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2+(-1)^{2/3} b^{11/6}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2\ 2^{17/18} \sqrt {3} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+(-1)^{2/3} b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (-a x-\sqrt [6]{2} \sqrt [6]{b}\right )}{12\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [6]{2} \sqrt [6]{b}-a x\right )}{12\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{-1} a x-\sqrt [6]{2} \sqrt [6]{b}\right )}{12\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+\sqrt [3]{-1} b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{-1} a x+\sqrt [6]{2} \sqrt [6]{b}\right )}{12\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-\sqrt [3]{-1} b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (-(-1)^{2/3} a x-\sqrt [6]{2} \sqrt [6]{b}\right )}{12\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-(-1)^{2/3} b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [6]{2} \sqrt [6]{b}-(-1)^{2/3} a x\right )}{12\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+(-1)^{2/3} b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [18]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2-b^{11/6}}}-\sqrt [3]{x}\right )}{4\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [18]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2+b^{11/6}}}-\sqrt [3]{x}\right )}{4\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [18]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2-\sqrt [3]{-1} b^{11/6}}}-\sqrt [3]{x}\right )}{4\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-\sqrt [3]{-1} b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [18]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2+\sqrt [3]{-1} b^{11/6}}}-\sqrt [3]{x}\right )}{4\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+\sqrt [3]{-1} b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [18]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2-(-1)^{2/3} b^{11/6}}}-\sqrt [3]{x}\right )}{4\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2-(-1)^{2/3} b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [18]{2} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{2} a^2+(-1)^{2/3} b^{11/6}}}-\sqrt [3]{x}\right )}{4\ 2^{17/18} \sqrt [3]{a} b \sqrt [3]{\sqrt [6]{2} a^2+(-1)^{2/3} b^{11/6}} \sqrt [3]{a^3 x^3+b^2 x^2}} \]
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Rule 93
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x} \left (-2 b+a^6 x^6\right )} \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (-\frac {1}{2 \sqrt {2} \sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {2} \sqrt {b}-a^3 x^3\right )}-\frac {1}{2 \sqrt {2} \sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {2} \sqrt {b}+a^3 x^3\right )}\right ) \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = -\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {2} \sqrt {b}-a^3 x^3\right )} \, dx}{2 \sqrt {2} \sqrt {b} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {2} \sqrt {b}+a^3 x^3\right )} \, dx}{2 \sqrt {2} \sqrt {b} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = -\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (-\frac {1}{3 \sqrt [3]{2} \sqrt [3]{b} x^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{b}-a x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 \sqrt [3]{2} \sqrt [3]{b} x^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 \sqrt [3]{2} \sqrt [3]{b} x^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{2 \sqrt {2} \sqrt {b} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (\frac {1}{3 \sqrt [3]{2} \sqrt [3]{b} x^{2/3} \left (\sqrt [6]{2} \sqrt [6]{b}-a x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{3 \sqrt [3]{2} \sqrt [3]{b} x^{2/3} \left (\sqrt [6]{2} \sqrt [6]{b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{3 \sqrt [3]{2} \sqrt [3]{b} x^{2/3} \left (\sqrt [6]{2} \sqrt [6]{b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{2 \sqrt {2} \sqrt {b} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{b}-a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} b^{5/6} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [6]{2} \sqrt [6]{b}-a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} b^{5/6} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} b^{5/6} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [6]{2} \sqrt [6]{b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} b^{5/6} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (-\sqrt [6]{2} \sqrt [6]{b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} b^{5/6} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [6]{2} \sqrt [6]{b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6\ 2^{5/6} b^{5/6} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-2 b+a^6 x^6\right )} \, dx=\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \text {RootSum}\left [2 a^{18}-a^6 b^{11}-12 a^{15} \text {$\#$1}^3+30 a^{12} \text {$\#$1}^6-40 a^9 \text {$\#$1}^9+30 a^6 \text {$\#$1}^{12}-12 a^3 \text {$\#$1}^{15}+2 \text {$\#$1}^{18}\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{12 b \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]
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Time = 0.46 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{18}-12 a^{3} \textit {\_Z}^{15}+30 a^{6} \textit {\_Z}^{12}-40 a^{9} \textit {\_Z}^{9}+30 a^{12} \textit {\_Z}^{6}-12 a^{15} \textit {\_Z}^{3}+2 a^{18}-a^{6} b^{11}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{12 b}\) | \(98\) |
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Exception generated. \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-2 b+a^6 x^6\right )} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 4.53 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.24 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-2 b+a^6 x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a^{6} x^{6} - 2 b\right )}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-2 b+a^6 x^6\right )} \, dx=\int { \frac {1}{{\left (a^{6} x^{6} - 2 \, b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
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Not integrable
Time = 0.73 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-2 b+a^6 x^6\right )} \, dx=\int { \frac {1}{{\left (a^{6} x^{6} - 2 \, b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (-2 b+a^6 x^6\right )} \, dx=-\int \frac {1}{\left (2\,b-a^6\,x^6\right )\,{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}} \,d x \]
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