Integrand size = 28, antiderivative size = 120 \[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {(d-3 e x)^{2/3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e}+\frac {\log (d+e x)}{4 d^{2/3} e}-\frac {3 \log \left (-\frac {(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e} \]
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Time = 0.02 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {124} \[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {(d-3 e x)^{2/3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e}+\frac {\log (d+e x)}{4 d^{2/3} e}-\frac {3 \log \left (-\frac {(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e} \]
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Rule 124
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {(d-3 e x)^{2/3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e}+\frac {\log (d+e x)}{4 d^{2/3} e}-\frac {3 \log \left (-\frac {(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(439\) vs. \(2(120)=240\).
Time = 1.25 (sec) , antiderivative size = 439, normalized size of antiderivative = 3.66 \[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d-3 e x}}{2^{2/3} \sqrt [3]{d}+\sqrt [3]{d-3 e x}-\sqrt [3]{2} \sqrt [3]{d+3 e x}}\right )-4 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d-3 e x}}{-2 2^{2/3} \sqrt [3]{d}+\sqrt [3]{d-3 e x}+2 \sqrt [3]{2} \sqrt [3]{d+3 e x}}\right )-4 \log \left (2^{2/3} \sqrt [3]{d}+\sqrt [3]{d-3 e x}-\sqrt [3]{2} \sqrt [3]{d+3 e x}\right )-2 \log \left (-2^{2/3} \sqrt [3]{d}+2 \sqrt [3]{d-3 e x}+\sqrt [3]{2} \sqrt [3]{d+3 e x}\right )+\log \left (2 \sqrt [3]{2} d^{2/3}+4 (d-3 e x)^{2/3}-2 \sqrt [3]{2} \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x}+2^{2/3} (d+3 e x)^{2/3}+2 \sqrt [3]{d} \left (2^{2/3} \sqrt [3]{d-3 e x}-2 \sqrt [3]{d+3 e x}\right )\right )+2 \log \left (2 \sqrt [3]{2} d^{2/3}+(d-3 e x)^{2/3}+\sqrt [3]{2} \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x}+2^{2/3} (d+3 e x)^{2/3}-\sqrt [3]{d} \left (2^{2/3} \sqrt [3]{d-3 e x}+4 \sqrt [3]{d+3 e x}\right )\right )}{8 d^{2/3} e} \]
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\[\int \frac {1}{\left (-3 e x +d \right )^{\frac {1}{3}} \left (e x +d \right ) \left (3 e x +d \right )^{\frac {1}{3}}}d x\]
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Timed out. \[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\int \frac {1}{\sqrt [3]{d - 3 e x} \left (d + e x\right ) \sqrt [3]{d + 3 e x}}\, dx \]
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\[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\int { \frac {1}{{\left (3 \, e x + d\right )}^{\frac {1}{3}} {\left (e x + d\right )} {\left (-3 \, e x + d\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\int { \frac {1}{{\left (3 \, e x + d\right )}^{\frac {1}{3}} {\left (e x + d\right )} {\left (-3 \, e x + d\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\int \frac {1}{\left (d+e\,x\right )\,{\left (d-3\,e\,x\right )}^{1/3}\,{\left (d+3\,e\,x\right )}^{1/3}} \,d x \]
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