Integrand size = 32, antiderivative size = 300 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{-b+a x^2} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{-b^2 x^2+a^3 x^3}}\right )-\log \left (-a x+\sqrt [3]{-b^2 x^2+a^3 x^3}\right )+\frac {1}{2} \log \left (a^2 x^2+a x \sqrt [3]{-b^2 x^2+a^3 x^3}+\left (-b^2 x^2+a^3 x^3\right )^{2/3}\right )+\frac {1}{2} \text {RootSum}\left [a^6-a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-a^5 \log (x)+b^3 \log (x)+a^5 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )-b^3 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^3-a^2 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{a^3 \text {$\#$1}^2-\text {$\#$1}^5}\&\right ] \]
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Leaf count is larger than twice the leaf count of optimal. \(839\) vs. \(2(300)=600\).
Time = 0.54 (sec) , antiderivative size = 839, normalized size of antiderivative = 2.80, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2081, 920, 61, 6857, 93} \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{-b+a x^2} \, dx=-\frac {\sqrt {3} \sqrt [3]{a^3 x^3-b^2 x^2} \arctan \left (\frac {2 \sqrt [3]{x} a}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {\sqrt {3} \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \arctan \left (\frac {2 \sqrt [6]{a} \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{2 a^{5/6} x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {\sqrt {3} \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \arctan \left (\frac {2 \sqrt [6]{a} \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{2 a^{5/6} x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {\sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt {b}-\sqrt {a} x\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {\sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt {a} x+\sqrt {b}\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {\sqrt [3]{a^3 x^3-b^2 x^2} \log \left (a^3 x-b^2\right )}{2 x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {3 \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x-b^2}}-1\right )}{2 x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {3 \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt [6]{a} \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b^2}\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {3 \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt [6]{a} \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b^2}\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{a^3 x-b^2}} \]
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Rule 61
Rule 93
Rule 920
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{-b^2 x^2+a^3 x^3} \int \frac {x^{2/3} \sqrt [3]{-b^2+a^3 x}}{-b+a x^2} \, dx}{x^{2/3} \sqrt [3]{-b^2+a^3 x}} \\ & = \frac {\sqrt [3]{-b^2 x^2+a^3 x^3} \int \frac {a^3 b-a b^2 x}{\sqrt [3]{x} \left (-b^2+a^3 x\right )^{2/3} \left (-b+a x^2\right )} \, dx}{a x^{2/3} \sqrt [3]{-b^2+a^3 x}}+\frac {\left (a^2 \sqrt [3]{-b^2 x^2+a^3 x^3}\right ) \int \frac {1}{\sqrt [3]{x} \left (-b^2+a^3 x\right )^{2/3}} \, dx}{x^{2/3} \sqrt [3]{-b^2+a^3 x}} \\ & = -\frac {\sqrt {3} \sqrt [3]{-b^2 x^2+a^3 x^3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-b^2+a^3 x}}\right )}{x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (-b^2+a^3 x\right )}{2 x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {3 \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (-1+\frac {a \sqrt [3]{x}}{\sqrt [3]{-b^2+a^3 x}}\right )}{2 x^{2/3} \sqrt [3]{-b^2+a^3 x}}+\frac {\sqrt [3]{-b^2 x^2+a^3 x^3} \int \left (-\frac {a^3 b^{3/2}-\sqrt {a} b^3}{2 b \sqrt [3]{x} \left (\sqrt {b}-\sqrt {a} x\right ) \left (-b^2+a^3 x\right )^{2/3}}-\frac {a^3 b^{3/2}+\sqrt {a} b^3}{2 b \sqrt [3]{x} \left (\sqrt {b}+\sqrt {a} x\right ) \left (-b^2+a^3 x\right )^{2/3}}\right ) \, dx}{a x^{2/3} \sqrt [3]{-b^2+a^3 x}} \\ & = -\frac {\sqrt {3} \sqrt [3]{-b^2 x^2+a^3 x^3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-b^2+a^3 x}}\right )}{x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (-b^2+a^3 x\right )}{2 x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {3 \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (-1+\frac {a \sqrt [3]{x}}{\sqrt [3]{-b^2+a^3 x}}\right )}{2 x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\left (\sqrt {b} \left (a^{5/2}-b^{3/2}\right ) \sqrt [3]{-b^2 x^2+a^3 x^3}\right ) \int \frac {1}{\sqrt [3]{x} \left (\sqrt {b}-\sqrt {a} x\right ) \left (-b^2+a^3 x\right )^{2/3}} \, dx}{2 \sqrt {a} x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\left (\sqrt {b} \left (a^{5/2}+b^{3/2}\right ) \sqrt [3]{-b^2 x^2+a^3 x^3}\right ) \int \frac {1}{\sqrt [3]{x} \left (\sqrt {b}+\sqrt {a} x\right ) \left (-b^2+a^3 x\right )^{2/3}} \, dx}{2 \sqrt {a} x^{2/3} \sqrt [3]{-b^2+a^3 x}} \\ & = -\frac {\sqrt {3} \sqrt [3]{-b^2 x^2+a^3 x^3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-b^2+a^3 x}}\right )}{x^{2/3} \sqrt [3]{-b^2+a^3 x}}+\frac {\sqrt {3} \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{-b^2 x^2+a^3 x^3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{a} \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-b^2+a^3 x}}\right )}{2 a^{5/6} x^{2/3} \sqrt [3]{-b^2+a^3 x}}+\frac {\sqrt {3} \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{-b^2 x^2+a^3 x^3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{a} \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-b^2+a^3 x}}\right )}{2 a^{5/6} x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (\sqrt {b}-\sqrt {a} x\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (\sqrt {b}+\sqrt {a} x\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (-b^2+a^3 x\right )}{2 x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {3 \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (-1+\frac {a \sqrt [3]{x}}{\sqrt [3]{-b^2+a^3 x}}\right )}{2 x^{2/3} \sqrt [3]{-b^2+a^3 x}}+\frac {3 \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (\sqrt [6]{a} \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{x}-\sqrt [3]{-b^2+a^3 x}\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{-b^2+a^3 x}}+\frac {3 \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (\sqrt [6]{a} \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{x}-\sqrt [3]{-b^2+a^3 x}\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{-b^2+a^3 x}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{-b+a x^2} \, dx=-\frac {x^{4/3} \left (-b^2+a^3 x\right )^{2/3} \left (6 \sqrt {3} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{-b^2+a^3 x}}\right )+6 \log \left (-a \sqrt [3]{x}+\sqrt [3]{-b^2+a^3 x}\right )-3 \log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{-b^2+a^3 x}+\left (-b^2+a^3 x\right )^{2/3}\right )+\text {RootSum}\left [a^6-a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-a^5 \log (x)+b^3 \log (x)+3 a^5 \log \left (\sqrt [3]{-b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )-3 b^3 \log \left (\sqrt [3]{-b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^3-3 a^2 \log \left (\sqrt [3]{-b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^3}{-a^3 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ]\right )}{6 \left (x^2 \left (-b^2+a^3 x\right )\right )^{2/3}} \]
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Time = 0.24 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(-\ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {a^{2} x^{2}+a \left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a^{3} \textit {\_Z}^{3}+a^{6}-a \,b^{3}\right )}{\sum }\frac {\left (\textit {\_R}^{3} a^{2}-a^{5}+b^{3}\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (\textit {\_R}^{3}-a^{3}\right )}\right )}{2}\) | \(207\) |
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Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.30 (sec) , antiderivative size = 542, normalized size of antiderivative = 1.81 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{-b+a x^2} \, dx=-\frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} {\left (\sqrt {\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {-3} a x + a x\right )} {\left (\sqrt {\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} {\left (\sqrt {\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (-\frac {{\left (\sqrt {-3} a x - a x\right )} {\left (\sqrt {\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} - 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} {\left (-\sqrt {\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {-3} a x + a x\right )} {\left (-\sqrt {\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} {\left (-\sqrt {\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (-\frac {{\left (\sqrt {-3} a x - a x\right )} {\left (-\sqrt {\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} - 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \sqrt {3} \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) + \frac {1}{2} \, {\left (\sqrt {\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (-\frac {a x {\left (\sqrt {\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, {\left (-\sqrt {\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (-\frac {a x {\left (-\sqrt {\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \log \left (-\frac {a x - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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Not integrable
Time = 0.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.07 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{-b+a x^2} \, dx=\int \frac {\sqrt [3]{x^{2} \left (a^{3} x - b^{2}\right )}}{a x^{2} - b}\, dx \]
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Not integrable
Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{-b+a x^2} \, dx=\int { \frac {{\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{a x^{2} - b} \,d x } \]
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Not integrable
Time = 0.48 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{-b+a x^2} \, dx=\int { \frac {{\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{a x^{2} - b} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.11 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{-b+a x^2} \, dx=-\int \frac {{\left (a^3\,x^3-b^2\,x^2\right )}^{1/3}}{b-a\,x^2} \,d x \]
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