Integrand size = 33, antiderivative size = 71 \[ \int \frac {1}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \, dx=\frac {x \left (1-\frac {c^3 x^3}{b^3}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {c^3 x^3}{b^3}\right )}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \]
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Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {727, 252, 251} \[ \int \frac {1}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \, dx=\frac {x \left (1-\frac {c^3 x^3}{b^3}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {c^3 x^3}{b^3}\right )}{\left (b^2+b c x+c^2 x^2\right )^{2/3} (b e-c e x)^{2/3}} \]
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Rule 251
Rule 252
Rule 727
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^3 e-c^3 e x^3\right )^{2/3} \int \frac {1}{\left (b^3 e-c^3 e x^3\right )^{2/3}} \, dx}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \\ & = \frac {\left (1-\frac {c^3 x^3}{b^3}\right )^{2/3} \int \frac {1}{\left (1-\frac {c^3 x^3}{b^3}\right )^{2/3}} \, dx}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \\ & = \frac {x \left (1-\frac {c^3 x^3}{b^3}\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};\frac {c^3 x^3}{b^3}\right )}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(258\) vs. \(2(71)=142\).
Time = 10.16 (sec) , antiderivative size = 258, normalized size of antiderivative = 3.63 \[ \int \frac {1}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \, dx=-\frac {3 \sqrt [3]{e (b-c x)} \left (b-\sqrt {3} \sqrt {-b^2}+2 c x\right ) \left (\frac {3 b^2+\sqrt {3} b \sqrt {-b^2}+3 b c x-\sqrt {3} \sqrt {-b^2} c x}{3 b^2-\sqrt {3} b \sqrt {-b^2}+3 b c x+\sqrt {3} \sqrt {-b^2} c x}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},\frac {4 \sqrt {3} \sqrt {-b^2} (b-c x)}{\left (3 b+\sqrt {3} \sqrt {-b^2}\right ) \left (-b+\sqrt {3} \sqrt {-b^2}-2 c x\right )}\right )}{\left (3 b-\sqrt {3} \sqrt {-b^2}\right ) c e \left (b^2+b c x+c^2 x^2\right )^{2/3}} \]
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\[\int \frac {1}{\left (-c e x +b e \right )^{\frac {2}{3}} \left (c^{2} x^{2}+b x c +b^{2}\right )^{\frac {2}{3}}}d x\]
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\[ \int \frac {1}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \, dx=\int \frac {1}{\left (- e \left (- b + c x\right )\right )^{\frac {2}{3}} \left (b^{2} + b c x + c^{2} x^{2}\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {1}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \, dx=\int \frac {1}{{\left (b\,e-c\,e\,x\right )}^{2/3}\,{\left (b^2+b\,c\,x+c^2\,x^2\right )}^{2/3}} \,d x \]
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