Integrand size = 41, antiderivative size = 189 \[ \int \frac {-b+a^3 x^2}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{-b x^2+a^3 x^3}}\right )}{a}-\frac {\log \left (-a x+\sqrt [3]{-b x^2+a^3 x^3}\right )}{a}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{-b x^2+a^3 x^3}+\left (-b x^2+a^3 x^3\right )^{2/3}\right )}{2 a}+\text {RootSum}\left [a^6+a^3 b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(821\) vs. \(2(189)=378\).
Time = 0.53 (sec) , antiderivative size = 821, normalized size of antiderivative = 4.34, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2081, 6857, 61, 926, 93} \[ \int \frac {-b+a^3 x^2}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [3]{a^3 x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log (x)}{2 a \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt {b}-\sqrt {-a^3} x\right )}{2 \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt {-a^3} x+\sqrt {b}\right )}{2 \sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{a \sqrt [3]{x}}-1\right )}{2 a \sqrt [3]{a^3 x^3-b x^2}} \]
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Rule 61
Rule 93
Rule 926
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {-b+a^3 x^2}{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}{x^{2/3} \sqrt [3]{-b+a^3 x}}-\frac {2 b}{x^{2/3} \sqrt [3]{-b+a^3 x} \left (b+a^3 x^2\right )}\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} \sqrt [3]{-b+a^3 x}} \, dx}{\sqrt [3]{-b x^2+a^3 x^3}}-\frac {\left (2 b 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 {\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} a \sqrt [3]{x}}\right )}{a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log (x)}{2 a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-1+\frac {\sqrt [3]{-b+a^3 x}}{a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {\left (2 b x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \left (\frac {1}{2 \sqrt {b} x^{2/3} \sqrt [3]{-b+a^3 x} \left (\sqrt {b}-\sqrt {-a^3} x\right )}+\frac {1}{2 \sqrt {b} x^{2/3} \sqrt [3]{-b+a^3 x} \left (\sqrt {b}+\sqrt {-a^3} x\right )}\right ) \, dx}{\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} a \sqrt [3]{x}}\right )}{a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log (x)}{2 a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-1+\frac {\sqrt [3]{-b+a^3 x}}{a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a^3 x} \left (\sqrt {b}-\sqrt {-a^3} x\right )} \, dx}{\sqrt [3]{-b x^2+a^3 x^3}}-\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a^3 x} \left (\sqrt {b}+\sqrt {-a^3} x\right )} \, dx}{\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} a \sqrt [3]{x}}\right )}{a \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 [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{x}}\right )}{\sqrt [3]{a^3-\sqrt {-a^3} \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 [3]{a^3+\sqrt {-a^3} \sqrt {b}} \sqrt [3]{x}}\right )}{\sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}} \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log (x)}{2 a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt {b}-\sqrt {-a^3} x\right )}{2 \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt {b}+\sqrt {-a^3} x\right )}{2 \sqrt [3]{a^3+\sqrt {-a^3} \sqrt {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 [3]{a^3-\sqrt {-a^3} \sqrt {b}}}\right )}{2 \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {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 [3]{a^3+\sqrt {-a^3} \sqrt {b}}}\right )}{2 \sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}} \sqrt [3]{-b x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-1+\frac {\sqrt [3]{-b+a^3 x}}{a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{-b x^2+a^3 x^3}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.20 \[ \int \frac {-b+a^3 x^2}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{-b+a^3 x}}\right )-2 \log \left (a \left (a \sqrt [3]{x}-\sqrt [3]{-b+a^3 x}\right )\right )+\log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{-b+a^3 x}+\left (-b+a^3 x\right )^{2/3}\right )+2 a \text {RootSum}\left [a^6+a^3 b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{2 a \sqrt [3]{x^2 \left (-b+a^3 x\right )}} \]
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Time = 0.22 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x -b \right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a^{3} \textit {\_Z}^{3}+a^{6}+a^{3} b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x -b \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) a -2 \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x -b \right )\right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {a^{2} x^{2}+a \left (x^{2} \left (a^{3} x -b \right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x -b \right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 a}\) | \(174\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.28 (sec) , antiderivative size = 1185, normalized size of antiderivative = 6.27 \[ \int \frac {-b+a^3 x^2}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\text {Too large to display} \]
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Not integrable
Time = 2.81 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.16 \[ \int \frac {-b+a^3 x^2}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int \frac {a^{3} x^{2} - b}{\sqrt [3]{x^{2} \left (a^{3} x - b\right )} \left (a^{3} x^{2} + b\right )}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.22 \[ \int \frac {-b+a^3 x^2}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int { \frac {a^{3} x^{2} - b}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{2} + b\right )}} \,d x } \]
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Not integrable
Time = 0.51 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.22 \[ \int \frac {-b+a^3 x^2}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int { \frac {a^{3} x^{2} - b}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{2} + b\right )}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.22 \[ \int \frac {-b+a^3 x^2}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int -\frac {b-a^3\,x^2}{\left (a^3\,x^2+b\right )\,{\left (a^3\,x^3-b\,x^2\right )}^{1/3}} \,d x \]
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