Integrand size = 33, antiderivative size = 196 \[ \int \frac {1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx=-\frac {\sqrt [3]{b^3 e-c^3 e x^3} \arctan \left (\frac {1-\frac {2 c \sqrt [3]{e} x}{\sqrt [3]{b^3 e-c^3 e x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c \sqrt [3]{e} \sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}}+\frac {\sqrt [3]{b^3 e-c^3 e x^3} \log \left (c \sqrt [3]{e} x+\sqrt [3]{b^3 e-c^3 e x^3}\right )}{2 c \sqrt [3]{e} \sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {727, 245} \[ \int \frac {1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx=\frac {\sqrt [3]{b^3 e-c^3 e x^3} \log \left (\sqrt [3]{b^3 e-c^3 e x^3}+c \sqrt [3]{e} x\right )}{2 c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}}-\frac {\sqrt [3]{b^3 e-c^3 e x^3} \arctan \left (\frac {1-\frac {2 c \sqrt [3]{e} x}{\sqrt [3]{b^3 e-c^3 e x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}} \]
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Rule 245
Rule 727
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{b^3 e-c^3 e x^3} \int \frac {1}{\sqrt [3]{b^3 e-c^3 e x^3}} \, dx}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \\ & = -\frac {\sqrt [3]{b^3 e-c^3 e x^3} \tan ^{-1}\left (\frac {1-\frac {2 c \sqrt [3]{e} x}{\sqrt [3]{b^3 e-c^3 e x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c \sqrt [3]{e} \sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}}+\frac {\sqrt [3]{b^3 e-c^3 e x^3} \log \left (c \sqrt [3]{e} x+\sqrt [3]{b^3 e-c^3 e x^3}\right )}{2 c \sqrt [3]{e} \sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 11.18 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx=-\frac {3 (e (b-c x))^{2/3} \sqrt [3]{\frac {b c-\sqrt {3} \sqrt {-b^2 c^2}+2 c^2 x}{3 b c-\sqrt {3} \sqrt {-b^2 c^2}}} \sqrt [3]{\frac {b c+\sqrt {3} \sqrt {-b^2 c^2}+2 c^2 x}{3 b c+\sqrt {3} \sqrt {-b^2 c^2}}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {2 c (b-c x)}{3 b c+\sqrt {3} \sqrt {-b^2 c^2}},\frac {2 c (b-c x)}{3 b c-\sqrt {3} \sqrt {-b^2 c^2}}\right )}{2 c e \sqrt [3]{b^2+b c x+c^2 x^2}} \]
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\[\int \frac {1}{\left (-c e x +b e \right )^{\frac {1}{3}} \left (c^{2} x^{2}+b x c +b^{2}\right )^{\frac {1}{3}}}d x\]
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none
Time = 37.11 (sec) , antiderivative size = 523, normalized size of antiderivative = 2.67 \[ \int \frac {1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx=\left [\frac {\sqrt {3} e \sqrt {-\frac {1}{e^{\frac {2}{3}}}} \log \left (3 \, c^{3} e x^{3} - b^{3} e - 3 \, {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}} c e^{\frac {1}{3}} x + \sqrt {3} {\left (2 \, {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}} c^{2} e x^{2} + {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}} c e^{\frac {2}{3}} x + {\left (c^{3} e x^{3} - b^{3} e\right )} e^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{e^{\frac {2}{3}}}}\right ) - e^{\frac {2}{3}} \log \left (c^{2} e x^{2} - {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}} c e^{\frac {2}{3}} x + {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}} e^{\frac {1}{3}}\right ) + 2 \, e^{\frac {2}{3}} \log \left (c e x + {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}} e^{\frac {2}{3}}\right )}{6 \, c e}, -\frac {2 \, \sqrt {3} e^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}} c e^{\frac {2}{3}} x + {\left (c^{3} e x^{3} - b^{3} e\right )} e^{\frac {1}{3}}\right )}}{3 \, {\left (c^{3} e x^{3} - b^{3} e\right )} e^{\frac {1}{3}}}\right ) + e^{\frac {2}{3}} \log \left (c^{2} e x^{2} - {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}} c e^{\frac {2}{3}} x + {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}} e^{\frac {1}{3}}\right ) - 2 \, e^{\frac {2}{3}} \log \left (c e x + {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}} e^{\frac {2}{3}}\right )}{6 \, c e}\right ] \]
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\[ \int \frac {1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx=\int \frac {1}{\sqrt [3]{- e \left (- b + c x\right )} \sqrt [3]{b^{2} + b c x + c^{2} x^{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx=\int \frac {1}{{\left (b\,e-c\,e\,x\right )}^{1/3}\,{\left (b^2+b\,c\,x+c^2\,x^2\right )}^{1/3}} \,d x \]
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