Optimal. Leaf size=270 \[ -\frac {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}-\frac {\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e}+\frac {\sqrt {2} \sqrt [4]{3} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{2+e x}}\right )}{e}-\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}-\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e}+\frac {\sqrt [4]{3} \log \left (\frac {\sqrt {6-3 e x}+\sqrt {6} \sqrt [4]{2-e x} \sqrt [4]{2+e x}+\sqrt {3} \sqrt {2+e x}}{\sqrt {2+e x}}\right )}{\sqrt {2} e} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, 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 = {689, 49, 65,
246, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {\sqrt {2} \sqrt [4]{3} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}\right )}{e}+\frac {\sqrt {2} \sqrt [4]{3} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{2-e x}}{\sqrt [4]{e x+2}}+1\right )}{e}-\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 49
Rule 65
Rule 210
Rule 217
Rule 246
Rule 631
Rule 642
Rule 689
Rule 1176
Rule 1179
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 \text {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 \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 {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}+\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 {4 \sqrt [4]{3} \sqrt [4]{2-e x}}{e \sqrt [4]{2+e x}}-\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 {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 ) \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 {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 [A]
time = 0.37, size = 151, normalized size = 0.56 \begin {gather*} \frac {\sqrt [4]{3} \left (-4 \sqrt [4]{4-e^2 x^2}+\sqrt {4+2 e x} \tan ^{-1}\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 {4+2 e x} \tanh ^{-1}\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 \sqrt {2+e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {\left (-3 e^{2} x^{2}+12\right )^{\frac {1}{4}}}{\left (e x +2\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 492 vs.
\(2 (206) = 412\).
time = 3.33, size = 492, normalized size = 1.82 \begin {gather*} -\frac {4 \cdot 3^{\frac {1}{4}} \sqrt {2} {\left (x e^{2} + 2 \, e\right )} \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {2} {\left (x e^{4} + 2 \, e^{3}\right )} \sqrt {\frac {\sqrt {3} {\left (x e^{3} + 2 \, e^{2}\right )} e^{\left (-2\right )} + 3^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} e^{2} + 12\right )}^{\frac {1}{4}} \sqrt {x e + 2} + \sqrt {-3 \, x^{2} e^{2} + 12}}{x e + 2}} e^{\left (-3\right )} - 3^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} e^{2} + 12\right )}^{\frac {1}{4}} \sqrt {x e + 2} - 3 \, x e - 6}{3 \, {\left (x e + 2\right )}}\right ) e^{\left (-1\right )} + 4 \cdot 3^{\frac {1}{4}} \sqrt {2} {\left (x e^{2} + 2 \, e\right )} \arctan \left (\frac {3^{\frac {3}{4}} \sqrt {2} {\left (x e^{4} + 2 \, e^{3}\right )} \sqrt {\frac {\sqrt {3} {\left (x e^{3} + 2 \, e^{2}\right )} e^{\left (-2\right )} - 3^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} e^{2} + 12\right )}^{\frac {1}{4}} \sqrt {x e + 2} + \sqrt {-3 \, x^{2} e^{2} + 12}}{x e + 2}} e^{\left (-3\right )} - 3^{\frac {3}{4}} \sqrt {2} {\left (-3 \, x^{2} e^{2} + 12\right )}^{\frac {1}{4}} \sqrt {x e + 2} + 3 \, x e + 6}{3 \, {\left (x e + 2\right )}}\right ) e^{\left (-1\right )} - 3^{\frac {1}{4}} \sqrt {2} {\left (x e^{2} + 2 \, e\right )} e^{\left (-1\right )} \log \left (\frac {\sqrt {3} {\left (x e^{3} + 2 \, e^{2}\right )} e^{\left (-2\right )} + 3^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} e^{2} + 12\right )}^{\frac {1}{4}} \sqrt {x e + 2} + \sqrt {-3 \, x^{2} e^{2} + 12}}{x e + 2}\right ) + 3^{\frac {1}{4}} \sqrt {2} {\left (x e^{2} + 2 \, e\right )} e^{\left (-1\right )} \log \left (\frac {\sqrt {3} {\left (x e^{3} + 2 \, e^{2}\right )} e^{\left (-2\right )} - 3^{\frac {1}{4}} \sqrt {2} {\left (-3 \, x^{2} e^{2} + 12\right )}^{\frac {1}{4}} \sqrt {x e + 2} + \sqrt {-3 \, x^{2} e^{2} + 12}}{x e + 2}\right ) + 8 \, {\left (-3 \, x^{2} e^{2} + 12\right )}^{\frac {1}{4}} \sqrt {x e + 2}}{2 \, {\left (x e^{2} + 2 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (12-3\,e^2\,x^2\right )}^{1/4}}{{\left (e\,x+2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________