Integrand size = 24, antiderivative size = 269 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=\frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}+\frac {\sqrt {2} \sqrt [4]{3} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e}-\frac {\sqrt {2} \sqrt [4]{3} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e}+\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e}-\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e} \]
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Time = 0.18 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {689, 52, 65, 246, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=\frac {\sqrt {2} \sqrt [4]{3} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}-\frac {\sqrt {2} \sqrt [4]{3} \arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{e}+\frac {\sqrt [4]{3} \sqrt [4]{2-e x} (e x+2)^{3/4}}{e}+\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} e}-\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} e} \]
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Rule 52
Rule 65
Rule 210
Rule 217
Rule 246
Rule 631
Rule 642
Rule 689
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}} \, dx \\ & = \frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}+3 \int \frac {1}{(6-3 e x)^{3/4} \sqrt [4]{2+e x}} \, dx \\ & = \frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}-\frac {4 \text {Subst}\left (\int \frac {1}{\sqrt [4]{4-\frac {x^4}{3}}} \, dx,x,\sqrt [4]{6-3 e x}\right )}{e} \\ & = \frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}-\frac {4 \text {Subst}\left (\int \frac {1}{1+\frac {x^4}{3}} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{e} \\ & = \frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}-\frac {2 \text {Subst}\left (\int \frac {\sqrt {3}-x^2}{1+\frac {x^4}{3}} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {3} e}-\frac {2 \text {Subst}\left (\int \frac {\sqrt {3}+x^2}{1+\frac {x^4}{3}} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {3} e} \\ & = \frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}+\frac {\sqrt [4]{3} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{3}+2 x}{-\sqrt {3}-\sqrt {2} \sqrt [4]{3} x-x^2} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} e}+\frac {\sqrt [4]{3} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{3}-2 x}{-\sqrt {3}+\sqrt {2} \sqrt [4]{3} x-x^2} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} e}-\frac {\sqrt {3} \text {Subst}\left (\int \frac {1}{\sqrt {3}-\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{e}-\frac {\sqrt {3} \text {Subst}\left (\int \frac {1}{\sqrt {3}+\sqrt {2} \sqrt [4]{3} x+x^2} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{e} \\ & = \frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}+\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {2-e x}-\sqrt {2} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e}-\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {2-e x}+\sqrt {2} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e}-\frac {\left (\sqrt {2} \sqrt [4]{3}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e}+\frac {\left (\sqrt {2} \sqrt [4]{3}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e} \\ & = \frac {\sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{e}+\frac {\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e}-\frac {\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e}+\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {2-e x}-\sqrt {2} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e}-\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {2-e x}+\sqrt {2} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=\frac {\sqrt [4]{3} \left (\sqrt {2+e x} \sqrt [4]{4-e^2 x^2}-\sqrt {2} \arctan \left (\frac {\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}{2+e x-\sqrt {4-e^2 x^2}}\right )-\sqrt {2} \text {arctanh}\left (\frac {2+e x+\sqrt {4-e^2 x^2}}{\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}\right )\right )}{e} \]
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\[\int \frac {\left (-3 x^{2} e^{2}+12\right )^{\frac {1}{4}}}{\sqrt {e x +2}}d x\]
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Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=-\frac {3^{\frac {1}{4}} e \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (\frac {3^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} + {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e x + 2}\right ) - 3^{\frac {1}{4}} e \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {3^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e x + 2}\right ) + i \cdot 3^{\frac {1}{4}} e \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (\frac {3^{\frac {1}{4}} {\left (i \, e^{2} x + 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} + {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e x + 2}\right ) - i \cdot 3^{\frac {1}{4}} e \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (\frac {3^{\frac {1}{4}} {\left (-i \, e^{2} x - 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} + {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e x + 2}\right ) - {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}} \sqrt {e x + 2}}{e} \]
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\[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=\sqrt [4]{3} \int \frac {\sqrt [4]{- e^{2} x^{2} + 4}}{\sqrt {e x + 2}}\, dx \]
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\[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=\int { \frac {{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}}}{\sqrt {e x + 2}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=-\frac {3^{\frac {1}{4}} {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \sqrt {2} \log \left (\sqrt {2} {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {1}{4}} + \sqrt {\frac {4}{e x + 2} - 1} + 1\right ) - \sqrt {2} \log \left (-\sqrt {2} {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {1}{4}} + \sqrt {\frac {4}{e x + 2} - 1} + 1\right ) - 2 \, {\left (e x + 2\right )} {\left (\frac {4}{e x + 2} - 1\right )}^{\frac {1}{4}}\right )}}{2 \, e} \]
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Timed out. \[ \int \frac {\sqrt [4]{12-3 e^2 x^2}}{\sqrt {2+e x}} \, dx=\int \frac {{\left (12-3\,e^2\,x^2\right )}^{1/4}}{\sqrt {e\,x+2}} \,d x \]
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