Optimal. Leaf size=67 \[ -\frac {2 (2-e x)^{3/2}}{9 \sqrt {3} e}+\frac {16 \sqrt {2-e x}}{3 \sqrt {3} e}+\frac {32}{3 \sqrt {3} e \sqrt {2-e x}} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 67, 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 = {627, 43} \begin {gather*} -\frac {2 (2-e x)^{3/2}}{9 \sqrt {3} e}+\frac {16 \sqrt {2-e x}}{3 \sqrt {3} e}+\frac {32}{3 \sqrt {3} e \sqrt {2-e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 627
Rubi steps
\begin {align*} \int \frac {(2+e x)^{7/2}}{\left (12-3 e^2 x^2\right )^{3/2}} \, dx &=\int \frac {(2+e x)^2}{(6-3 e x)^{3/2}} \, dx\\ &=\int \left (\frac {16}{(6-3 e x)^{3/2}}-\frac {8}{3 \sqrt {6-3 e x}}+\frac {1}{9} \sqrt {6-3 e x}\right ) \, dx\\ &=\frac {32}{3 \sqrt {3} e \sqrt {2-e x}}+\frac {16 \sqrt {2-e x}}{3 \sqrt {3} e}-\frac {2 (2-e x)^{3/2}}{9 \sqrt {3} e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 43, normalized size = 0.64 \begin {gather*} -\frac {2 \sqrt {e x+2} \left (e^2 x^2+20 e x-92\right )}{9 e \sqrt {12-3 e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.40, size = 77, normalized size = 1.15 \begin {gather*} \frac {2 \sqrt {4 (e x+2)-(e x+2)^2} \left (\sqrt {3} (e x+2)^2+16 \sqrt {3} (e x+2)-128 \sqrt {3}\right )}{27 e (e x-2) \sqrt {e x+2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 47, normalized size = 0.70 \begin {gather*} \frac {2 \, {\left (e^{2} x^{2} + 20 \, e x - 92\right )} \sqrt {-3 \, e^{2} x^{2} + 12} \sqrt {e x + 2}}{27 \, {\left (e^{3} x^{2} - 4 \, e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 43, normalized size = 0.64 \begin {gather*} \frac {2 \left (e x -2\right ) \left (e^{2} x^{2}+20 e x -92\right ) \left (e x +2\right )^{\frac {3}{2}}}{3 \left (-3 e^{2} x^{2}+12\right )^{\frac {3}{2}} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 3.05, size = 36, normalized size = 0.54 \begin {gather*} \frac {2 i \, \sqrt {3} e^{2} x^{2} + 40 i \, \sqrt {3} e x - 184 i \, \sqrt {3}}{27 \, \sqrt {e x - 2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.60, size = 48, normalized size = 0.72 \begin {gather*} \frac {2\,\sqrt {12-3\,e^2\,x^2}\,\sqrt {e\,x+2}\,\left (e^2\,x^2+20\,e\,x-92\right )}{27\,e\,\left (e^2\,x^2-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________