Integrand size = 24, antiderivative size = 340 \[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {5\ 3^{3/4} (2-e x)^{3/4} \sqrt [4]{2+e x}}{2 e}-\frac {3^{3/4} (2-e x)^{3/4} (2+e x)^{5/4}}{2 e}-\frac {(2-e x)^{3/4} (2+e x)^{9/4}}{3 \sqrt [4]{3} e}+\frac {5\ 3^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} e}-\frac {5\ 3^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} e}-\frac {5\ 3^{3/4} \log \left (\frac {\sqrt {6-3 e x}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{2 \sqrt {2} e}+\frac {5\ 3^{3/4} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{2 \sqrt {2} e} \]
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Time = 0.21 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {689, 52, 65, 338, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=\frac {5\ 3^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt {2} e}-\frac {5\ 3^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt {2} e}-\frac {(2-e x)^{3/4} (e x+2)^{9/4}}{3 \sqrt [4]{3} e}-\frac {3^{3/4} (2-e x)^{3/4} (e x+2)^{5/4}}{2 e}-\frac {5\ 3^{3/4} (2-e x)^{3/4} \sqrt [4]{e x+2}}{2 e}-\frac {5\ 3^{3/4} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{2 \sqrt {2} e}+\frac {5\ 3^{3/4} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{2 \sqrt {2} e} \]
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Rule 52
Rule 65
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 689
Rule 1176
Rule 1179
Rubi steps \begin{align*} \text {integral}& = \int \frac {(2+e x)^{9/4}}{\sqrt [4]{6-3 e x}} \, dx \\ & = -\frac {(2-e x)^{3/4} (2+e x)^{9/4}}{3 \sqrt [4]{3} e}+3 \int \frac {(2+e x)^{5/4}}{\sqrt [4]{6-3 e x}} \, dx \\ & = -\frac {3^{3/4} (2-e x)^{3/4} (2+e x)^{5/4}}{2 e}-\frac {(2-e x)^{3/4} (2+e x)^{9/4}}{3 \sqrt [4]{3} e}+\frac {15}{2} \int \frac {\sqrt [4]{2+e x}}{\sqrt [4]{6-3 e x}} \, dx \\ & = -\frac {5\ 3^{3/4} (2-e x)^{3/4} \sqrt [4]{2+e x}}{2 e}-\frac {3^{3/4} (2-e x)^{3/4} (2+e x)^{5/4}}{2 e}-\frac {(2-e x)^{3/4} (2+e x)^{9/4}}{3 \sqrt [4]{3} e}+\frac {15}{2} \int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx \\ & = -\frac {5\ 3^{3/4} (2-e x)^{3/4} \sqrt [4]{2+e x}}{2 e}-\frac {3^{3/4} (2-e x)^{3/4} (2+e x)^{5/4}}{2 e}-\frac {(2-e x)^{3/4} (2+e x)^{9/4}}{3 \sqrt [4]{3} e}-\frac {10 \text {Subst}\left (\int \frac {x^2}{\left (4-\frac {x^4}{3}\right )^{3/4}} \, dx,x,\sqrt [4]{6-3 e x}\right )}{e} \\ & = -\frac {5\ 3^{3/4} (2-e x)^{3/4} \sqrt [4]{2+e x}}{2 e}-\frac {3^{3/4} (2-e x)^{3/4} (2+e x)^{5/4}}{2 e}-\frac {(2-e x)^{3/4} (2+e x)^{9/4}}{3 \sqrt [4]{3} e}-\frac {10 \text {Subst}\left (\int \frac {x^2}{1+\frac {x^4}{3}} \, dx,x,\frac {\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}}\right )}{e} \\ & = -\frac {5\ 3^{3/4} (2-e x)^{3/4} \sqrt [4]{2+e x}}{2 e}-\frac {3^{3/4} (2-e x)^{3/4} (2+e x)^{5/4}}{2 e}-\frac {(2-e x)^{3/4} (2+e x)^{9/4}}{3 \sqrt [4]{3} e}+\frac {5 \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 )}{e}-\frac {5 \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 )}{e} \\ & = -\frac {5\ 3^{3/4} (2-e x)^{3/4} \sqrt [4]{2+e x}}{2 e}-\frac {3^{3/4} (2-e x)^{3/4} (2+e x)^{5/4}}{2 e}-\frac {(2-e x)^{3/4} (2+e x)^{9/4}}{3 \sqrt [4]{3} e}-\frac {15 \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 )}{2 e}-\frac {15 \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 )}{2 e}-\frac {\left (5\ 3^{3/4}\right ) \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 )}{2 \sqrt {2} e}-\frac {\left (5\ 3^{3/4}\right ) \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 )}{2 \sqrt {2} e} \\ & = -\frac {5\ 3^{3/4} (2-e x)^{3/4} \sqrt [4]{2+e x}}{2 e}-\frac {3^{3/4} (2-e x)^{3/4} (2+e x)^{5/4}}{2 e}-\frac {(2-e x)^{3/4} (2+e x)^{9/4}}{3 \sqrt [4]{3} e}-\frac {5\ 3^{3/4} \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 )}{2 \sqrt {2} e}+\frac {5\ 3^{3/4} \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 )}{2 \sqrt {2} e}-\frac {\left (5\ 3^{3/4}\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 )}{\sqrt {2} e}+\frac {\left (5\ 3^{3/4}\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 )}{\sqrt {2} e} \\ & = -\frac {5\ 3^{3/4} (2-e x)^{3/4} \sqrt [4]{2+e x}}{2 e}-\frac {3^{3/4} (2-e x)^{3/4} (2+e x)^{5/4}}{2 e}-\frac {(2-e x)^{3/4} (2+e x)^{9/4}}{3 \sqrt [4]{3} e}+\frac {5\ 3^{3/4} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} e}-\frac {5\ 3^{3/4} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt {2} e}-\frac {5\ 3^{3/4} \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 )}{2 \sqrt {2} e}+\frac {5\ 3^{3/4} \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 )}{2 \sqrt {2} e} \\ \end{align*}
Time = 1.02 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.50 \[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=\frac {-\left (4-e^2 x^2\right )^{3/4} \left (71+17 e x+2 e^2 x^2\right )-45 \sqrt {4+2 e x} \arctan \left (\frac {\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}{2+e x-\sqrt {4-e^2 x^2}}\right )+45 \sqrt {4+2 e x} \text {arctanh}\left (\frac {2+e x+\sqrt {4-e^2 x^2}}{\sqrt {4+2 e x} \sqrt [4]{4-e^2 x^2}}\right )}{6 \sqrt [4]{3} e \sqrt {2+e x}} \]
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\[\int \frac {\left (e x +2\right )^{\frac {5}{2}}}{\left (-3 x^{2} e^{2}+12\right )^{\frac {1}{4}}}d x\]
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Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.08 \[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=-\frac {45 \cdot 27^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (\frac {27^{\frac {1}{4}} {\left (e^{3} x^{2} - 4 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} + {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{e^{2} x^{2} - 4}\right ) - 45 \cdot 27^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {27^{\frac {1}{4}} {\left (e^{3} x^{2} - 4 \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{e^{2} x^{2} - 4}\right ) + 45 \cdot 27^{\frac {1}{4}} {\left (-i \, e^{2} x - 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {27^{\frac {1}{4}} {\left (i \, e^{3} x^{2} - 4 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{e^{2} x^{2} - 4}\right ) + 45 \cdot 27^{\frac {1}{4}} {\left (i \, e^{2} x + 2 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {27^{\frac {1}{4}} {\left (-i \, e^{3} x^{2} + 4 i \, e\right )} \left (-\frac {1}{e^{4}}\right )^{\frac {1}{4}} - {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{e^{2} x^{2} - 4}\right ) + {\left (2 \, e^{2} x^{2} + 17 \, e x + 71\right )} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{18 \, {\left (e^{2} x + 2 \, e\right )}} \]
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\[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=\frac {3^{\frac {3}{4}} \left (\int \frac {4 \sqrt {e x + 2}}{\sqrt [4]{- e^{2} x^{2} + 4}}\, dx + \int \frac {4 e x \sqrt {e x + 2}}{\sqrt [4]{- e^{2} x^{2} + 4}}\, dx + \int \frac {e^{2} x^{2} \sqrt {e x + 2}}{\sqrt [4]{- e^{2} x^{2} + 4}}\, dx\right )}{3} \]
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\[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=\int { \frac {{\left (e x + 2\right )}^{\frac {5}{2}}}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}}} \,d x } \]
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Exception generated. \[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(2+e x)^{5/2}}{\sqrt [4]{12-3 e^2 x^2}} \, dx=\int \frac {{\left (e\,x+2\right )}^{5/2}}{{\left (12-3\,e^2\,x^2\right )}^{1/4}} \,d x \]
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