Optimal. Leaf size=270 \[ -\frac {(2-e x)^{3/4} \sqrt [4]{e x+2}}{\sqrt [4]{3} e}-\frac {\log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} \sqrt [4]{3} e}+\frac {\log \left (\frac {\sqrt {6-3 e x}+\sqrt {3} \sqrt {e x+2}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{e x+2}}{\sqrt {e x+2}}\right )}{\sqrt {2} \sqrt [4]{3} e}+\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt [4]{3} e}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt [4]{3} e} \]
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Rubi [A] time = 0.24, antiderivative size = 270, normalized size of antiderivative = 1.00, 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, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {(2-e x)^{3/4} \sqrt [4]{e x+2}}{\sqrt [4]{3} e}-\frac {\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} \sqrt [4]{3} e}+\frac {\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} \sqrt [4]{3} e}+\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{\sqrt [4]{3} e}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{\sqrt [4]{3} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 204
Rule 297
Rule 331
Rule 617
Rule 628
Rule 675
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {\sqrt {2+e x}}{\sqrt [4]{12-3 e^2 x^2}} \, dx &=\int \frac {\sqrt [4]{2+e x}}{\sqrt [4]{6-3 e x}} \, dx\\ &=-\frac {(2-e x)^{3/4} \sqrt [4]{2+e x}}{\sqrt [4]{3} e}+\int \frac {1}{\sqrt [4]{6-3 e x} (2+e x)^{3/4}} \, dx\\ &=-\frac {(2-e x)^{3/4} \sqrt [4]{2+e x}}{\sqrt [4]{3} e}-\frac {4 \operatorname {Subst}\left (\int \frac {x^2}{\left (4-\frac {x^4}{3}\right )^{3/4}} \, dx,x,\sqrt [4]{6-3 e x}\right )}{3 e}\\ &=-\frac {(2-e x)^{3/4} \sqrt [4]{2+e x}}{\sqrt [4]{3} e}-\frac {4 \operatorname {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 )}{3 e}\\ &=-\frac {(2-e x)^{3/4} \sqrt [4]{2+e x}}{\sqrt [4]{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 )}{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 )}{3 e}\\ &=-\frac {(2-e x)^{3/4} \sqrt [4]{2+e x}}{\sqrt [4]{3} e}-\frac {\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 {\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 {\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} \sqrt [4]{3} e}-\frac {\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} \sqrt [4]{3} e}\\ &=-\frac {(2-e x)^{3/4} \sqrt [4]{2+e x}}{\sqrt [4]{3} e}-\frac {\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} \sqrt [4]{3} e}+\frac {\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} \sqrt [4]{3} e}-\frac {\sqrt {2} \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 [4]{3} e}+\frac {\sqrt {2} \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 [4]{3} e}\\ &=-\frac {(2-e x)^{3/4} \sqrt [4]{2+e x}}{\sqrt [4]{3} e}+\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt [4]{3} e}-\frac {\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{\sqrt [4]{3} e}-\frac {\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} \sqrt [4]{3} e}+\frac {\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} \sqrt [4]{3} e}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 60, normalized size = 0.22 \begin {gather*} \frac {4 \sqrt {2} (e x-2) \sqrt [4]{e x+2} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2}-\frac {e x}{4}\right )}{3 e \sqrt [4]{12-3 e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 206, normalized size = 0.76 \begin {gather*} -\frac {\left (4 (e x+2)-(e x+2)^2\right )^{3/4}}{\sqrt [4]{3} e \sqrt {e x+2}}+\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e x+2} \sqrt [4]{4 (e x+2)-(e x+2)^2}}{-e x+\sqrt {4 (e x+2)-(e x+2)^2}-2}\right )}{\sqrt [4]{3} e}+\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\frac {e x+2}{\sqrt {2}}+\frac {\sqrt {4 (e x+2)-(e x+2)^2}}{\sqrt {2}}}{\sqrt {e x+2} \sqrt [4]{4 (e x+2)-(e x+2)^2}}\right )}{\sqrt [4]{3} e} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 651, normalized size = 2.41 \begin {gather*} \frac {12 \, \sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \frac {1}{e^{4}}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} e^{3} \frac {1}{e^{4}}^{\frac {3}{4}} + e^{2} x^{2} - \sqrt {3} \sqrt {2} \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (e^{5} x^{2} - 4 \, e^{3}\right )} \sqrt {\frac {\sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} e \frac {1}{e^{4}}^{\frac {1}{4}} + 3 \, \sqrt {\frac {1}{3}} {\left (e^{4} x^{2} - 4 \, e^{2}\right )} \sqrt {\frac {1}{e^{4}}} - \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 2\right )}}{e^{2} x^{2} - 4}} \frac {1}{e^{4}}^{\frac {3}{4}} - 4}{e^{2} x^{2} - 4}\right ) + 12 \, \sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \frac {1}{e^{4}}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} e^{3} \frac {1}{e^{4}}^{\frac {3}{4}} - e^{2} x^{2} - \sqrt {3} \sqrt {2} \left (\frac {1}{3}\right )^{\frac {3}{4}} {\left (e^{5} x^{2} - 4 \, e^{3}\right )} \sqrt {-\frac {\sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} e \frac {1}{e^{4}}^{\frac {1}{4}} - 3 \, \sqrt {\frac {1}{3}} {\left (e^{4} x^{2} - 4 \, e^{2}\right )} \sqrt {\frac {1}{e^{4}}} + \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 2\right )}}{e^{2} x^{2} - 4}} \frac {1}{e^{4}}^{\frac {3}{4}} + 4}{e^{2} x^{2} - 4}\right ) - 3 \, \sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \frac {1}{e^{4}}^{\frac {1}{4}} \log \left (\frac {3 \, {\left (\sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} e \frac {1}{e^{4}}^{\frac {1}{4}} + 3 \, \sqrt {\frac {1}{3}} {\left (e^{4} x^{2} - 4 \, e^{2}\right )} \sqrt {\frac {1}{e^{4}}} - \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 2\right )}\right )}}{e^{2} x^{2} - 4}\right ) + 3 \, \sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (e^{2} x + 2 \, e\right )} \frac {1}{e^{4}}^{\frac {1}{4}} \log \left (-\frac {3 \, {\left (\sqrt {2} \left (\frac {1}{3}\right )^{\frac {1}{4}} {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2} e \frac {1}{e^{4}}^{\frac {1}{4}} - 3 \, \sqrt {\frac {1}{3}} {\left (e^{4} x^{2} - 4 \, e^{2}\right )} \sqrt {\frac {1}{e^{4}}} + \sqrt {-3 \, e^{2} x^{2} + 12} {\left (e x + 2\right )}\right )}}{e^{2} x^{2} - 4}\right ) - 2 \, {\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {3}{4}} \sqrt {e x + 2}}{6 \, {\left (e^{2} x + 2 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {e x + 2}}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {e x +2}}{\left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {e x + 2}}{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {e\,x+2}}{{\left (12-3\,e^2\,x^2\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {3^{\frac {3}{4}} \int \frac {\sqrt {e x + 2}}{\sqrt [4]{- e^{2} x^{2} + 4}}\, dx}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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