Integrand size = 31, antiderivative size = 106 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+a^6 x^6\right )} \, dx=-\frac {\text {RootSum}\left [a^{18}+a^6 b^{11}-6 a^{15} \text {$\#$1}^3+15 a^{12} \text {$\#$1}^6-20 a^9 \text {$\#$1}^9+15 a^6 \text {$\#$1}^{12}-6 a^3 \text {$\#$1}^{15}+\text {$\#$1}^{18}\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b} \]
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Leaf count is larger than twice the leaf count of optimal. \(2271\) vs. \(2(106)=212\).
Time = 2.49 (sec) , antiderivative size = 2271, normalized size of antiderivative = 21.42, number of steps used = 13, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2081, 6857, 93} \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+a^6 x^6\right )} \, dx=\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {2 \sqrt [18]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-b^2} \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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{-b} a^2+b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+b^2} \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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} b^2} \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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [3]{-1} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [3]{-1} b^2} \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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} b^2} \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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [6]{-b} a^2+(-1)^{2/3} b^2} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+(-1)^{2/3} b^2} \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]{-b}\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-b^2} \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]{-b}-a x\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+b^2} \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]{-b}\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [3]{-1} b^2} \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]{-b}\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} b^2} \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]{-b}\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} b^2} \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]{-b}-(-1)^{2/3} a x\right )}{12 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+(-1)^{2/3} b^2} \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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-b^2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-b^2} \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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{-b} a^2+b^2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+b^2} \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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} b^2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-\sqrt [3]{-1} b^2} \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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [3]{-1} b^2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+\sqrt [3]{-1} b^2} \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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} b^2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{a^2 \sqrt [6]{-b}-(-1)^{2/3} b^2} \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]{-b} \sqrt [3]{x a^3+b^2}}{\sqrt [3]{a} \sqrt [3]{\sqrt [6]{-b} a^2+(-1)^{2/3} b^2}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a} (-b)^{17/18} \sqrt [3]{\sqrt [6]{-b} a^2+(-1)^{2/3} b^2} \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 (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 {\sqrt {-b}}{2 b x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\sqrt {-b}-a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b x^{2/3} \sqrt [3]{b^2+a^3 x} \left (\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 {-b}-a^3 x^3\right )} \, dx}{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 {-b}+a^3 x^3\right )} \, dx}{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]{-b} x^{2/3} \left (-\sqrt [6]{-b}-a x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 \sqrt [3]{-b} x^{2/3} \left (-\sqrt [6]{-b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 \sqrt [3]{-b} x^{2/3} \left (-\sqrt [6]{-b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{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]{-b} x^{2/3} \left (\sqrt [6]{-b}-a x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{3 \sqrt [3]{-b} x^{2/3} \left (\sqrt [6]{-b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}}+\frac {1}{3 \sqrt [3]{-b} x^{2/3} \left (\sqrt [6]{-b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{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]{-b}-a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{-b}-a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{-b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{-b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{-b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{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]{-b}-(-1)^{2/3} a x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 (-b)^{5/6} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+a^6 x^6\right )} \, dx=-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \text {RootSum}\left [a^{18}+a^6 b^{11}-6 a^{15} \text {$\#$1}^3+15 a^{12} \text {$\#$1}^6-20 a^9 \text {$\#$1}^9+15 a^6 \text {$\#$1}^{12}-6 a^3 \text {$\#$1}^{15}+\text {$\#$1}^{18}\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]
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Time = 0.80 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{18}-6 a^{3} \textit {\_Z}^{15}+15 a^{6} \textit {\_Z}^{12}-20 a^{9} \textit {\_Z}^{9}+15 a^{12} \textit {\_Z}^{6}-6 a^{15} \textit {\_Z}^{3}+a^{18}+a^{6} b^{11}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{6 b}\) | \(93\) |
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Exception generated. \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+a^6 x^6\right )} \, dx=\text {Exception raised: RuntimeError} \]
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Not integrable
Time = 2.77 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.25 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+a^6 x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a^{6} x^{6} + b\right )}\, dx \]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+a^6 x^6\right )} \, dx=\int { \frac {1}{{\left (a^{6} x^{6} + b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
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Not integrable
Time = 0.69 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+a^6 x^6\right )} \, dx=\int { \frac {1}{{\left (a^{6} x^{6} + b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
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Not integrable
Time = 5.84 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+a^6 x^6\right )} \, dx=\int \frac {1}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (a^6\,x^6+b\right )} \,d x \]
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