Optimal. Leaf size=309 \[ -\frac{\sqrt [4]{3} (e x+2)^{3/4} (2-e x)^{5/4}}{2 e}+\frac{3 \sqrt [4]{3} (e x+2)^{3/4} \sqrt [4]{2-e x}}{2 e}+\frac{3 \sqrt [4]{3} \log \left (\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}-\sqrt{6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} e}-\frac{3 \sqrt [4]{3} \log \left (\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}+\sqrt{6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} e}+\frac{3 \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt{2} e}-\frac{3 \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt{2} e} \]
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Rubi [A] time = 0.335062, antiderivative size = 309, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {675, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{\sqrt [4]{3} (e x+2)^{3/4} (2-e x)^{5/4}}{2 e}+\frac{3 \sqrt [4]{3} (e x+2)^{3/4} \sqrt [4]{2-e x}}{2 e}+\frac{3 \sqrt [4]{3} \log \left (\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}-\sqrt{6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} e}-\frac{3 \sqrt [4]{3} \log \left (\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}+\sqrt{6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt{e x+2}}\right )}{2 \sqrt{2} e}+\frac{3 \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt{2} e}-\frac{3 \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt{2} e} \]
Antiderivative was successfully verified.
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Rule 675
Rule 50
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \sqrt{2+e x} \sqrt [4]{12-3 e^2 x^2} \, dx &=\int \sqrt [4]{6-3 e x} (2+e x)^{3/4} \, dx\\ &=-\frac{\sqrt [4]{3} (2-e x)^{5/4} (2+e x)^{3/4}}{2 e}+\frac{3}{2} \int \frac{\sqrt [4]{6-3 e x}}{\sqrt [4]{2+e x}} \, dx\\ &=\frac{3 \sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{2 e}-\frac{\sqrt [4]{3} (2-e x)^{5/4} (2+e x)^{3/4}}{2 e}+\frac{9}{2} \int \frac{1}{(6-3 e x)^{3/4} \sqrt [4]{2+e x}} \, dx\\ &=\frac{3 \sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{2 e}-\frac{\sqrt [4]{3} (2-e x)^{5/4} (2+e x)^{3/4}}{2 e}-\frac{6 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{4-\frac{x^4}{3}}} \, dx,x,\sqrt [4]{6-3 e x}\right )}{e}\\ &=\frac{3 \sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{2 e}-\frac{\sqrt [4]{3} (2-e x)^{5/4} (2+e x)^{3/4}}{2 e}-\frac{6 \operatorname{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{3 \sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{2 e}-\frac{\sqrt [4]{3} (2-e x)^{5/4} (2+e x)^{3/4}}{2 e}-\frac{\sqrt{3} \operatorname{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{\sqrt{3} \operatorname{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{3 \sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{2 e}-\frac{\sqrt [4]{3} (2-e x)^{5/4} (2+e x)^{3/4}}{2 e}+\frac{\left (3 \sqrt [4]{3}\right ) \operatorname{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 (3 \sqrt [4]{3}\right ) \operatorname{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 (3 \sqrt{3}\right ) \operatorname{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 (3 \sqrt{3}\right ) \operatorname{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{3 \sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{2 e}-\frac{\sqrt [4]{3} (2-e x)^{5/4} (2+e x)^{3/4}}{2 e}+\frac{3 \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 )}{2 \sqrt{2} e}-\frac{3 \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 )}{2 \sqrt{2} e}-\frac{\left (3 \sqrt [4]{3}\right ) \operatorname{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 (3 \sqrt [4]{3}\right ) \operatorname{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{3 \sqrt [4]{3} \sqrt [4]{2-e x} (2+e x)^{3/4}}{2 e}-\frac{\sqrt [4]{3} (2-e x)^{5/4} (2+e x)^{3/4}}{2 e}+\frac{3 \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt{2} e}-\frac{3 \sqrt [4]{3} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt{2} e}+\frac{3 \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 )}{2 \sqrt{2} e}-\frac{3 \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 )}{2 \sqrt{2} e}\\ \end{align*}
Mathematica [C] time = 0.0493846, size = 60, normalized size = 0.19 \[ \frac{8 \sqrt{2} (e x-2) \sqrt [4]{12-3 e^2 x^2} \, _2F_1\left (-\frac{3}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2}-\frac{e x}{4}\right )}{5 e \sqrt [4]{e x+2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.171, size = 0, normalized size = 0. \begin{align*} \int \sqrt{ex+2}\sqrt [4]{-3\,{e}^{2}{x}^{2}+12}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06396, size = 1532, normalized size = 4.96 \begin{align*} \frac{12 \cdot 3^{\frac{1}{4}} \sqrt{2} e \frac{1}{e^{4}}^{\frac{1}{4}} \arctan \left (-\frac{3^{\frac{3}{4}} \sqrt{2}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2} e^{3} \frac{1}{e^{4}}^{\frac{3}{4}} - 3^{\frac{3}{4}} \sqrt{2}{\left (e^{4} x + 2 \, e^{3}\right )} \sqrt{\frac{3^{\frac{1}{4}} \sqrt{2}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2} e \frac{1}{e^{4}}^{\frac{1}{4}} + \sqrt{3}{\left (e^{3} x + 2 \, e^{2}\right )} \sqrt{\frac{1}{e^{4}}} + \sqrt{-3 \, e^{2} x^{2} + 12}}{e x + 2}} \frac{1}{e^{4}}^{\frac{3}{4}} + 3 \, e x + 6}{3 \,{\left (e x + 2\right )}}\right ) + 12 \cdot 3^{\frac{1}{4}} \sqrt{2} e \frac{1}{e^{4}}^{\frac{1}{4}} \arctan \left (-\frac{3^{\frac{3}{4}} \sqrt{2}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2} e^{3} \frac{1}{e^{4}}^{\frac{3}{4}} - 3^{\frac{3}{4}} \sqrt{2}{\left (e^{4} x + 2 \, e^{3}\right )} \sqrt{-\frac{3^{\frac{1}{4}} \sqrt{2}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2} e \frac{1}{e^{4}}^{\frac{1}{4}} - \sqrt{3}{\left (e^{3} x + 2 \, e^{2}\right )} \sqrt{\frac{1}{e^{4}}} - \sqrt{-3 \, e^{2} x^{2} + 12}}{e x + 2}} \frac{1}{e^{4}}^{\frac{3}{4}} - 3 \, e x - 6}{3 \,{\left (e x + 2\right )}}\right ) - 3 \cdot 3^{\frac{1}{4}} \sqrt{2} e \frac{1}{e^{4}}^{\frac{1}{4}} \log \left (\frac{3^{\frac{1}{4}} \sqrt{2}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2} e \frac{1}{e^{4}}^{\frac{1}{4}} + \sqrt{3}{\left (e^{3} x + 2 \, e^{2}\right )} \sqrt{\frac{1}{e^{4}}} + \sqrt{-3 \, e^{2} x^{2} + 12}}{e x + 2}\right ) + 3 \cdot 3^{\frac{1}{4}} \sqrt{2} e \frac{1}{e^{4}}^{\frac{1}{4}} \log \left (-\frac{3^{\frac{1}{4}} \sqrt{2}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2} e \frac{1}{e^{4}}^{\frac{1}{4}} - \sqrt{3}{\left (e^{3} x + 2 \, e^{2}\right )} \sqrt{\frac{1}{e^{4}}} - \sqrt{-3 \, e^{2} x^{2} + 12}}{e x + 2}\right ) + 2 \,{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}{\left (e x + 1\right )}}{4 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \sqrt [4]{3} \int \sqrt{e x + 2} \sqrt [4]{- e^{2} x^{2} + 4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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