Optimal. Leaf size=65 \[ -\frac {96 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {48 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {2 (2-e x)^{9/2}}{\sqrt {3} e} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {641, 45}
\begin {gather*} -\frac {2 (2-e x)^{9/2}}{\sqrt {3} e}+\frac {48 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {96 \sqrt {3} (2-e x)^{5/2}}{5 e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 641
Rubi steps
\begin {align*} \int \sqrt {2+e x} \left (12-3 e^2 x^2\right )^{3/2} \, dx &=\int (6-3 e x)^{3/2} (2+e x)^2 \, dx\\ &=\int \left (16 (6-3 e x)^{3/2}-\frac {8}{3} (6-3 e x)^{5/2}+\frac {1}{9} (6-3 e x)^{7/2}\right ) \, dx\\ &=-\frac {96 \sqrt {3} (2-e x)^{5/2}}{5 e}+\frac {48 \sqrt {3} (2-e x)^{7/2}}{7 e}-\frac {2 (2-e x)^{9/2}}{\sqrt {3} e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.17, size = 52, normalized size = 0.80 \begin {gather*} -\frac {2 (-2+e x)^2 \sqrt {4-e^2 x^2} \left (428+220 e x+35 e^2 x^2\right )}{35 e \sqrt {6+3 e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.48, size = 46, normalized size = 0.71
method | result | size |
gosper | \(\frac {2 \left (e x -2\right ) \left (35 e^{2} x^{2}+220 e x +428\right ) \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}}}{315 e \left (e x +2\right )^{\frac {3}{2}}}\) | \(44\) |
default | \(-\frac {2 \sqrt {-3 e^{2} x^{2}+12}\, \left (e x -2\right )^{2} \left (35 e^{2} x^{2}+220 e x +428\right )}{105 \sqrt {e x +2}\, e}\) | \(46\) |
risch | \(\frac {2 \sqrt {\frac {-3 e^{2} x^{2}+12}{e x +2}}\, \sqrt {e x +2}\, \left (35 e^{4} x^{4}+80 e^{3} x^{3}-312 e^{2} x^{2}-832 e x +1712\right ) \left (e x -2\right )}{35 \sqrt {-3 e^{2} x^{2}+12}\, e \sqrt {-3 e x +6}}\) | \(88\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] Result contains complex when optimal does not.
time = 0.51, size = 71, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (-35 i \, \sqrt {3} x^{4} e^{4} - 80 i \, \sqrt {3} x^{3} e^{3} + 312 i \, \sqrt {3} x^{2} e^{2} + 832 i \, \sqrt {3} x e - 1712 i \, \sqrt {3}\right )} {\left (x e + 2\right )} \sqrt {x e - 2}}{105 \, {\left (x e^{2} + 2 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.84, size = 60, normalized size = 0.92 \begin {gather*} -\frac {2 \, {\left (35 \, x^{4} e^{4} + 80 \, x^{3} e^{3} - 312 \, x^{2} e^{2} - 832 \, x e + 1712\right )} \sqrt {-3 \, x^{2} e^{2} + 12} \sqrt {x e + 2}}{105 \, {\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*} 3 \sqrt {3} \left (\int 4 \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\, dx + \int \left (- e^{2} x^{2} \sqrt {e x + 2} \sqrt {- e^{2} x^{2} + 4}\right )\, dx\right ) \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 [B]
time = 0.19, size = 71, normalized size = 1.09 \begin {gather*} \frac {2\,\sqrt {12-3\,e^2\,x^2}\,\sqrt {e\,x+2}\,\left (-35\,e^3\,x^3-10\,e^2\,x^2+332\,e\,x+168\right )}{105\,e}-\frac {4096\,\sqrt {12-3\,e^2\,x^2}}{105\,e\,\sqrt {e\,x+2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________