Integrand size = 53, antiderivative size = 564 \[ \int \frac {1}{(d+e x) \sqrt [3]{-c^2 d^2+b c d e+2 b^2 e^2+9 b c e^2 x+9 c^2 e^2 x^2}} \, dx=-\frac {\sqrt {3} \sqrt [3]{-c e (c d-2 b e)+3 c^2 e^2 x} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x} \arctan \left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{2} \left (-c e (c d-2 b e)+3 c^2 e^2 x\right )^{2/3}}{\sqrt {3} \sqrt [3]{c} \sqrt [3]{e} \sqrt [3]{2 c d-b e} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x}}\right )}{2 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-((c d-2 b e) (c d+b e))+9 b c e^2 x+9 c^2 e^2 x^2}}-\frac {\sqrt [3]{-c e (c d-2 b e)+3 c^2 e^2 x} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x} \log (d+e x)}{2 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-((c d-2 b e) (c d+b e))+9 b c e^2 x+9 c^2 e^2 x^2}}+\frac {3 \sqrt [3]{-c e (c d-2 b e)+3 c^2 e^2 x} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x} \log \left (-\frac {\sqrt [3]{\frac {3}{2}} \left (-c e (c d-2 b e)+3 c^2 e^2 x\right )^{2/3}}{\sqrt [3]{c} \sqrt [3]{e} \sqrt [3]{2 c d-b e}}-\sqrt [3]{6} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x}\right )}{4 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-((c d-2 b e) (c d+b e))+9 b c e^2 x+9 c^2 e^2 x^2}} \]
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Time = 0.28 (sec) , antiderivative size = 564, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {769, 124} \[ \int \frac {1}{(d+e x) \sqrt [3]{-c^2 d^2+b c d e+2 b^2 e^2+9 b c e^2 x+9 c^2 e^2 x^2}} \, dx=-\frac {\sqrt {3} \sqrt [3]{3 c^2 e^2 x-c e (c d-2 b e)} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x} \arctan \left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{2} \left (3 c^2 e^2 x-c e (c d-2 b e)\right )^{2/3}}{\sqrt {3} \sqrt [3]{c} \sqrt [3]{e} \sqrt [3]{2 c d-b e} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x}}\right )}{2 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-(c d-2 b e) (b e+c d)+9 b c e^2 x+9 c^2 e^2 x^2}}-\frac {\log (d+e x) \sqrt [3]{3 c^2 e^2 x-c e (c d-2 b e)} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x}}{2 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-(c d-2 b e) (b e+c d)+9 b c e^2 x+9 c^2 e^2 x^2}}+\frac {3 \sqrt [3]{3 c^2 e^2 x-c e (c d-2 b e)} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x} \log \left (-\frac {\sqrt [3]{\frac {3}{2}} \left (3 c^2 e^2 x-c e (c d-2 b e)\right )^{2/3}}{\sqrt [3]{c} \sqrt [3]{e} \sqrt [3]{2 c d-b e}}-\sqrt [3]{6} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x}\right )}{4 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-(c d-2 b e) (b e+c d)+9 b c e^2 x+9 c^2 e^2 x^2}} \]
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Rule 124
Rule 769
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [3]{9 b c e^2-3 c e (2 c d-b e)+18 c^2 e^2 x} \sqrt [3]{9 b c e^2+3 c e (2 c d-b e)+18 c^2 e^2 x}\right ) \int \frac {1}{(d+e x) \sqrt [3]{9 b c e^2-3 c e (2 c d-b e)+18 c^2 e^2 x} \sqrt [3]{9 b c e^2+3 c e (2 c d-b e)+18 c^2 e^2 x}} \, dx}{\sqrt [3]{-c^2 d^2+b c d e+2 b^2 e^2+9 b c e^2 x+9 c^2 e^2 x^2}} \\ & = -\frac {\sqrt {3} \sqrt [3]{-c e (c d-2 b e)+3 c^2 e^2 x} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{2} \left (-c e (c d-2 b e)+3 c^2 e^2 x\right )^{2/3}}{\sqrt {3} \sqrt [3]{c} \sqrt [3]{e} \sqrt [3]{2 c d-b e} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x}}\right )}{2 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-((c d-2 b e) (c d+b e))+9 b c e^2 x+9 c^2 e^2 x^2}}-\frac {\sqrt [3]{-c e (c d-2 b e)+3 c^2 e^2 x} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x} \log (d+e x)}{2 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-((c d-2 b e) (c d+b e))+9 b c e^2 x+9 c^2 e^2 x^2}}+\frac {3 \sqrt [3]{-c e (c d-2 b e)+3 c^2 e^2 x} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x} \log \left (-\frac {\sqrt [3]{\frac {3}{2}} \left (-c e (c d-2 b e)+3 c^2 e^2 x\right )^{2/3}}{\sqrt [3]{c} \sqrt [3]{e} \sqrt [3]{2 c d-b e}}-\sqrt [3]{6} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x}\right )}{4 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-((c d-2 b e) (c d+b e))+9 b c e^2 x+9 c^2 e^2 x^2}} \\ \end{align*}
Time = 1.55 (sec) , antiderivative size = 473, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(d+e x) \sqrt [3]{-c^2 d^2+b c d e+2 b^2 e^2+9 b c e^2 x+9 c^2 e^2 x^2}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{2 c d-b e} \sqrt [3]{2 b^2 e^2+b c e (d+9 e x)-c^2 \left (d^2-9 e^2 x^2\right )}}{-2 \sqrt [3]{2} b e+\sqrt [3]{2} c (d-3 e x)+\sqrt [3]{2 c d-b e} \sqrt [3]{2 b^2 e^2+b c e (d+9 e x)-c^2 \left (d^2-9 e^2 x^2\right )}}\right )+2 \log \left (\sqrt {e} \left (-\sqrt [3]{2} c d+\sqrt [3]{2} e (2 b+3 c x)+2 \sqrt [3]{2 c d-b e} \sqrt [3]{2 b^2 e^2+b c e (d+9 e x)-c^2 \left (d^2-9 e^2 x^2\right )}\right )\right )-\log \left (2^{2/3} c^2 d^2 e-4\ 2^{2/3} b c d e^2+4\ 2^{2/3} b^2 e^3-6\ 2^{2/3} c^2 d e^2 x+12\ 2^{2/3} b c e^3 x+9\ 2^{2/3} c^2 e^3 x^2+2 e \sqrt [3]{4 c d-2 b e} (c d-2 b e-3 c e x) \sqrt [3]{2 b^2 e^2+b c e (d+9 e x)-c^2 \left (d^2-9 e^2 x^2\right )}+4 e (2 c d-b e)^{2/3} \left (2 b^2 e^2+b c e (d+9 e x)-c^2 \left (d^2-9 e^2 x^2\right )\right )^{2/3}\right )}{4 \sqrt [3]{2} e (2 c d-b e)^{2/3}} \]
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\[\int \frac {1}{\left (e x +d \right ) \left (9 c^{2} e^{2} x^{2}+9 b c \,e^{2} x +2 b^{2} e^{2}+b c d e -c^{2} d^{2}\right )^{\frac {1}{3}}}d x\]
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Timed out. \[ \int \frac {1}{(d+e x) \sqrt [3]{-c^2 d^2+b c d e+2 b^2 e^2+9 b c e^2 x+9 c^2 e^2 x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(d+e x) \sqrt [3]{-c^2 d^2+b c d e+2 b^2 e^2+9 b c e^2 x+9 c^2 e^2 x^2}} \, dx=\int \frac {1}{\sqrt [3]{\left (b e + c d + 3 c e x\right ) \left (2 b e - c d + 3 c e x\right )} \left (d + e x\right )}\, dx \]
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\[ \int \frac {1}{(d+e x) \sqrt [3]{-c^2 d^2+b c d e+2 b^2 e^2+9 b c e^2 x+9 c^2 e^2 x^2}} \, dx=\int { \frac {1}{{\left (9 \, c^{2} e^{2} x^{2} + 9 \, b c e^{2} x - c^{2} d^{2} + b c d e + 2 \, b^{2} e^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}} \,d x } \]
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\[ \int \frac {1}{(d+e x) \sqrt [3]{-c^2 d^2+b c d e+2 b^2 e^2+9 b c e^2 x+9 c^2 e^2 x^2}} \, dx=\int { \frac {1}{{\left (9 \, c^{2} e^{2} x^{2} + 9 \, b c e^{2} x - c^{2} d^{2} + b c d e + 2 \, b^{2} e^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(d+e x) \sqrt [3]{-c^2 d^2+b c d e+2 b^2 e^2+9 b c e^2 x+9 c^2 e^2 x^2}} \, dx=\int \frac {1}{\left (d+e\,x\right )\,{\left (2\,b^2\,e^2+b\,c\,d\,e+9\,b\,c\,e^2\,x-c^2\,d^2+9\,c^2\,e^2\,x^2\right )}^{1/3}} \,d x \]
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