Optimal. Leaf size=270 \[ -\frac{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}-\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}-\sqrt{6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}+\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}+\sqrt{6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}-\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}+\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{e} \]
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Rubi [A] time = 0.262523, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {675, 47, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{e x+2}}-\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}-\sqrt{6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}+\frac{\sqrt [4]{3} \log \left (\frac{\sqrt{6-3 e x}+\sqrt{3} \sqrt{e x+2}+\sqrt{6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt{e x+2}}\right )}{\sqrt{2} e}-\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}+\frac{\sqrt{2} \sqrt [4]{3} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{e} \]
Antiderivative was successfully verified.
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Rule 675
Rule 47
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{3/2}} \, dx &=\int \frac{\sqrt [4]{6-3 e x}}{(2+e x)^{5/4}} \, dx\\ &=-\frac{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}-3 \int \frac{1}{(6-3 e x)^{3/4} \sqrt [4]{2+e x}} \, dx\\ &=-\frac{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{4-\frac{x^4}{3}}} \, dx,x,\sqrt [4]{6-3 e x}\right )}{e}\\ &=-\frac{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}+\frac{4 \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{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}+\frac{2 \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 )}{\sqrt{3} e}+\frac{2 \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 )}{\sqrt{3} e}\\ &=-\frac{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}-\frac{\sqrt [4]{3} \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 )}{\sqrt{2} e}-\frac{\sqrt [4]{3} \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 )}{\sqrt{2} e}+\frac{\sqrt{3} \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 )}{e}+\frac{\sqrt{3} \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 )}{e}\\ &=-\frac{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}-\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 ) \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 )}{e}-\frac{\left (\sqrt{2} \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 )}{e}\\ &=-\frac{4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}-\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*}
Mathematica [C] time = 0.056021, size = 60, normalized size = 0.22 \[ \frac{(e x-2) \sqrt [4]{12-3 e^2 x^2} \, _2F_1\left (\frac{5}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2}-\frac{e x}{4}\right )}{5 \sqrt{2} e \sqrt [4]{e x+2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.485, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [4]{-3\,{e}^{2}{x}^{2}+12} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}}\, 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 \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}}{{\left (e x + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.05574, size = 1593, normalized size = 5.9 \begin{align*} \text{result too large to display} \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 \frac{\sqrt [4]{- e^{2} x^{2} + 4}}{e x \sqrt{e x + 2} + 2 \sqrt{e x + 2}}\, 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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