### 3.895 $$\int (2+e x)^{3/2} \sqrt{12-3 e^2 x^2} \, dx$$

Optimal. Leaf size=65 $-\frac{2 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{16 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}$

[Out]

(-32*(2 - e*x)^(3/2))/(Sqrt[3]*e) + (16*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) - (2*Sqrt[3]*(2 - e*x)^(7/2))/(7*e)

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Rubi [A]  time = 0.0202403, antiderivative size = 65, 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} $-\frac{2 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{16 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(2 + e*x)^(3/2)*Sqrt[12 - 3*e^2*x^2],x]

[Out]

(-32*(2 - e*x)^(3/2))/(Sqrt[3]*e) + (16*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) - (2*Sqrt[3]*(2 - e*x)^(7/2))/(7*e)

Rule 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (2+e x)^{3/2} \sqrt{12-3 e^2 x^2} \, dx &=\int \sqrt{6-3 e x} (2+e x)^2 \, dx\\ &=\int \left (16 \sqrt{6-3 e x}-\frac{8}{3} (6-3 e x)^{3/2}+\frac{1}{9} (6-3 e x)^{5/2}\right ) \, dx\\ &=-\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}+\frac{16 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{2 \sqrt{3} (2-e x)^{7/2}}{7 e}\\ \end{align*}

Mathematica [A]  time = 0.0554336, size = 50, normalized size = 0.77 $\frac{2 (e x-2) \sqrt{4-e^2 x^2} \left (15 e^2 x^2+108 e x+284\right )}{35 e \sqrt{3 e x+6}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(2 + e*x)^(3/2)*Sqrt[12 - 3*e^2*x^2],x]

[Out]

(2*(-2 + e*x)*Sqrt[4 - e^2*x^2]*(284 + 108*e*x + 15*e^2*x^2))/(35*e*Sqrt[6 + 3*e*x])

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Maple [A]  time = 0.042, size = 44, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( 15\,{e}^{2}{x}^{2}+108\,ex+284 \right ) }{105\,e}\sqrt{-3\,{e}^{2}{x}^{2}+12}{\frac{1}{\sqrt{ex+2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+2)^(3/2)*(-3*e^2*x^2+12)^(1/2),x)

[Out]

2/105*(e*x-2)*(15*e^2*x^2+108*e*x+284)*(-3*e^2*x^2+12)^(1/2)/(e*x+2)^(1/2)/e

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Maxima [C]  time = 1.71175, size = 81, normalized size = 1.25 \begin{align*} \frac{{\left (30 i \, \sqrt{3} e^{3} x^{3} + 156 i \, \sqrt{3} e^{2} x^{2} + 136 i \, \sqrt{3} e x - 1136 i \, \sqrt{3}\right )}{\left (e x + 2\right )} \sqrt{e x - 2}}{105 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(3/2)*(-3*e^2*x^2+12)^(1/2),x, algorithm="maxima")

[Out]

1/105*(30*I*sqrt(3)*e^3*x^3 + 156*I*sqrt(3)*e^2*x^2 + 136*I*sqrt(3)*e*x - 1136*I*sqrt(3))*(e*x + 2)*sqrt(e*x -
2)/(e^2*x + 2*e)

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Fricas [A]  time = 2.09546, size = 132, normalized size = 2.03 \begin{align*} \frac{2 \,{\left (15 \, e^{3} x^{3} + 78 \, e^{2} x^{2} + 68 \, e x - 568\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{105 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(3/2)*(-3*e^2*x^2+12)^(1/2),x, algorithm="fricas")

[Out]

2/105*(15*e^3*x^3 + 78*e^2*x^2 + 68*e*x - 568)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^2*x + 2*e)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \sqrt{3} \left (\int 2 \sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4}\, dx + \int e x \sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4}\, dx\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)**(3/2)*(-3*e**2*x**2+12)**(1/2),x)

[Out]

sqrt(3)*(Integral(2*sqrt(e*x + 2)*sqrt(-e**2*x**2 + 4), x) + Integral(e*x*sqrt(e*x + 2)*sqrt(-e**2*x**2 + 4),
x))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-3 \, e^{2} x^{2} + 12}{\left (e x + 2\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(3/2)*(-3*e^2*x^2+12)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(3/2), x)