Integrand size = 32, antiderivative size = 68 \[ \int \frac {1}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\frac {\text {RootSum}\left [a^6-a^3 b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{2 b} \]
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Leaf count is larger than twice the leaf count of optimal. \(618\) vs. \(2(68)=136\).
Time = 0.24 (sec) , antiderivative size = 618, normalized size of antiderivative = 9.09, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2081, 926, 93} \[ \int \frac {1}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt {a} \sqrt [3]{x} \sqrt [3]{a^{3/2}-\sqrt {b}}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {a} b \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt {a} \sqrt [3]{x} \sqrt [3]{a^{3/2}+\sqrt {b}}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {a} b \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt {b}-a^{3/2} x\right )}{4 \sqrt {a} b \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (a^{3/2} x+\sqrt {b}\right )}{4 \sqrt {a} b \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}}}-\sqrt [3]{x}\right )}{4 \sqrt {a} b \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}}}-\sqrt [3]{x}\right )}{4 \sqrt {a} b \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}} \]
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Rule 93
Rule 926
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a^3 x} \left (-b+a^3 x^2\right )} \, dx}{\sqrt [3]{-b x^2+a^3 x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \left (-\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}-a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}}-\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}+a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}}\right ) \, dx}{\sqrt [3]{-b x^2+a^3 x^3}} \\ & = -\frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}-a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}+a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a^3 x^3}} \\ & = \frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a^3 x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a^3 x}}{\sqrt {3} \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{x}}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a^3 x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a^3 x}}{\sqrt {3} \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{x}}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt {b}-a^{3/2} x\right )}{4 \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt {b}+a^{3/2} x\right )}{4 \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}}}\right )}{4 \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}}}\right )}{4 \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.56 \[ \int \frac {1}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \text {RootSum}\left [a^6-a^3 b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{2 b \sqrt [3]{x^2 \left (-b+a^3 x\right )}} \]
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Time = 2.26 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a^{3} \textit {\_Z}^{3}+a^{6}-a^{3} b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x -b \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{2 b}\) | \(60\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 1358, normalized size of antiderivative = 19.97 \[ \int \frac {1}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\text {Too large to display} \]
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Not integrable
Time = 1.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.35 \[ \int \frac {1}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x - b\right )} \left (a^{3} x^{2} - b\right )}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{2} - b\right )}} \,d x } \]
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Not integrable
Time = 0.55 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{2} - b\right )}} \,d x } \]
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Not integrable
Time = 5.39 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=-\int \frac {1}{\left (b-a^3\,x^2\right )\,{\left (a^3\,x^3-b\,x^2\right )}^{1/3}} \,d x \]
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