Integrand size = 28, antiderivative size = 82 \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\frac {\text {RootSum}\left [a^4-a b^3-4 a^3 \text {$\#$1}^3+6 a^2 \text {$\#$1}^6-4 a \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{4 b} \]
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Leaf count is larger than twice the leaf count of optimal. \(1208\) vs. \(2(82)=164\).
Time = 0.98 (sec) , antiderivative size = 1208, normalized size of antiderivative = 14.73, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2081, 6857, 926, 93} \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [4]{b}-\sqrt {-\sqrt {a}} x\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt {-\sqrt {a}} x+\sqrt [4]{b}\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [4]{b}-\sqrt [4]{a} x\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [4]{a} x+\sqrt [4]{b}\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}} \]
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Rule 93
Rule 926
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (-b+a x^4\right )} \, dx}{\sqrt [3]{-b x^2+a x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (-\frac {1}{2 \sqrt {b} x^{2/3} \sqrt [3]{-b+a x} \left (\sqrt {b}-\sqrt {a} x^2\right )}-\frac {1}{2 \sqrt {b} x^{2/3} \sqrt [3]{-b+a x} \left (\sqrt {b}+\sqrt {a} x^2\right )}\right ) \, dx}{\sqrt [3]{-b x^2+a x^3}} \\ & = -\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (\sqrt {b}-\sqrt {a} x^2\right )} \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (\sqrt {b}+\sqrt {a} x^2\right )} \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a x^3}} \\ & = -\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (\frac {1}{2 \sqrt [4]{b} x^{2/3} \left (\sqrt [4]{b}-\sqrt {-\sqrt {a}} x\right ) \sqrt [3]{-b+a x}}+\frac {1}{2 \sqrt [4]{b} x^{2/3} \left (\sqrt [4]{b}+\sqrt {-\sqrt {a}} x\right ) \sqrt [3]{-b+a x}}\right ) \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (\frac {1}{2 \sqrt [4]{b} x^{2/3} \left (\sqrt [4]{b}-\sqrt [4]{a} x\right ) \sqrt [3]{-b+a x}}+\frac {1}{2 \sqrt [4]{b} x^{2/3} \left (\sqrt [4]{b}+\sqrt [4]{a} x\right ) \sqrt [3]{-b+a x}}\right ) \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a x^3}} \\ & = -\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [4]{b}-\sqrt {-\sqrt {a}} x\right ) \sqrt [3]{-b+a x}} \, dx}{4 b^{3/4} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [4]{b}+\sqrt {-\sqrt {a}} x\right ) \sqrt [3]{-b+a x}} \, dx}{4 b^{3/4} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [4]{b}-\sqrt [4]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{4 b^{3/4} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [4]{b}+\sqrt [4]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{4 b^{3/4} \sqrt [3]{-b x^2+a x^3}} \\ & = \frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} \sqrt [3]{x}}\right )}{4 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} \sqrt [3]{x}}\right )}{4 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [4]{b}-\sqrt {-\sqrt {a}} x\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [4]{b}+\sqrt {-\sqrt {a}} x\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [4]{b}-\sqrt [4]{a} x\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [4]{b}+\sqrt [4]{a} x\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}}}\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}}}\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}}}\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}}}\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{-b x^2+a x^3}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\frac {x^{2/3} \sqrt [3]{-b+a x} \text {RootSum}\left [a^4-a b^3-4 a^3 \text {$\#$1}^3+6 a^2 \text {$\#$1}^6-4 a \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{4 b \sqrt [3]{x^2 (-b+a x)}} \]
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Time = 0.00 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{12}-4 a \,\textit {\_Z}^{9}+6 a^{2} \textit {\_Z}^{6}-4 a^{3} \textit {\_Z}^{3}+a^{4}-a \,b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a x -b \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{4 b}\) | \(72\) |
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Timed out. \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 1.39 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.24 \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a x - b\right )} \left (a x^{4} - b\right )}\, dx \]
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Not integrable
Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.34 \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\int { \frac {1}{{\left (a x^{4} - b\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
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Not integrable
Time = 8.48 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.34 \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\int { \frac {1}{{\left (a x^{4} - b\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.35 \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=-\int \frac {1}{\left (b-a\,x^4\right )\,{\left (a\,x^3-b\,x^2\right )}^{1/3}} \,d x \]
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