Optimal. Leaf size=87 \[ \frac{6 \sqrt{3} (2-e x)^{11/2}}{11 e}-\frac{8 \sqrt{3} (2-e x)^{9/2}}{e}+\frac{288 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{384 \sqrt{3} (2-e x)^{5/2}}{5 e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0239898, antiderivative size = 87, normalized size of antiderivative = 1., 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} \[ \frac{6 \sqrt{3} (2-e x)^{11/2}}{11 e}-\frac{8 \sqrt{3} (2-e x)^{9/2}}{e}+\frac{288 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{384 \sqrt{3} (2-e x)^{5/2}}{5 e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 627
Rule 43
Rubi steps
\begin{align*} \int (2+e x)^{3/2} \left (12-3 e^2 x^2\right )^{3/2} \, dx &=\int (6-3 e x)^{3/2} (2+e x)^3 \, dx\\ &=\int \left (64 (6-3 e x)^{3/2}-16 (6-3 e x)^{5/2}+\frac{4}{3} (6-3 e x)^{7/2}-\frac{1}{27} (6-3 e x)^{9/2}\right ) \, dx\\ &=-\frac{384 \sqrt{3} (2-e x)^{5/2}}{5 e}+\frac{288 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{8 \sqrt{3} (2-e x)^{9/2}}{e}+\frac{6 \sqrt{3} (2-e x)^{11/2}}{11 e}\\ \end{align*}
Mathematica [A] time = 0.0599333, size = 59, normalized size = 0.68 \[ -\frac{2 (e x-2)^2 \sqrt{12-3 e^2 x^2} \left (105 e^3 x^3+910 e^2 x^2+3020 e x+4264\right )}{385 e \sqrt{e x+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.043, size = 52, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( 105\,{e}^{3}{x}^{3}+910\,{e}^{2}{x}^{2}+3020\,ex+4264 \right ) }{1155\,e} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{{\frac{3}{2}}} \left ( ex+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [C] time = 1.95948, size = 111, normalized size = 1.28 \begin{align*} -\frac{{\left (210 i \, \sqrt{3} e^{5} x^{5} + 980 i \, \sqrt{3} e^{4} x^{4} - 400 i \, \sqrt{3} e^{3} x^{3} - 8352 i \, \sqrt{3} e^{2} x^{2} - 9952 i \, \sqrt{3} e x + 34112 i \, \sqrt{3}\right )}{\left (e x + 2\right )} \sqrt{e x - 2}}{385 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.84532, size = 181, normalized size = 2.08 \begin{align*} -\frac{2 \,{\left (105 \, e^{5} x^{5} + 490 \, e^{4} x^{4} - 200 \, e^{3} x^{3} - 4176 \, e^{2} x^{2} - 4976 \, e x + 17056\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{385 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}}{\left (e x + 2\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]