### 3.894 $$\int (2+e x)^{5/2} \sqrt{12-3 e^2 x^2} \, dx$$

Optimal. Leaf size=87 $\frac{2 (2-e x)^{9/2}}{3 \sqrt{3} e}-\frac{24 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{128 (2-e x)^{3/2}}{\sqrt{3} e}$

[Out]

(-128*(2 - e*x)^(3/2))/(Sqrt[3]*e) + (96*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) - (24*Sqrt[3]*(2 - e*x)^(7/2))/(7*e) +
(2*(2 - e*x)^(9/2))/(3*Sqrt[3]*e)

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Rubi [A]  time = 0.0267335, 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} $\frac{2 (2-e x)^{9/2}}{3 \sqrt{3} e}-\frac{24 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{128 (2-e x)^{3/2}}{\sqrt{3} e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(2 + e*x)^(5/2)*Sqrt[12 - 3*e^2*x^2],x]

[Out]

(-128*(2 - e*x)^(3/2))/(Sqrt[3]*e) + (96*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) - (24*Sqrt[3]*(2 - e*x)^(7/2))/(7*e) +
(2*(2 - e*x)^(9/2))/(3*Sqrt[3]*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)^{5/2} \sqrt{12-3 e^2 x^2} \, dx &=\int \sqrt{6-3 e x} (2+e x)^3 \, dx\\ &=\int \left (64 \sqrt{6-3 e x}-16 (6-3 e x)^{3/2}+\frac{4}{3} (6-3 e x)^{5/2}-\frac{1}{27} (6-3 e x)^{7/2}\right ) \, dx\\ &=-\frac{128 (2-e x)^{3/2}}{\sqrt{3} e}+\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e}-\frac{24 \sqrt{3} (2-e x)^{7/2}}{7 e}+\frac{2 (2-e x)^{9/2}}{3 \sqrt{3} e}\\ \end{align*}

Mathematica [A]  time = 0.0752209, size = 58, normalized size = 0.67 $\frac{2 (e x-2) \sqrt{4-e^2 x^2} \left (35 e^3 x^3+330 e^2 x^2+1284 e x+2552\right )}{105 e \sqrt{3 e x+6}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(2 + e*x)^(5/2)*Sqrt[12 - 3*e^2*x^2],x]

[Out]

(2*(-2 + e*x)*Sqrt[4 - e^2*x^2]*(2552 + 1284*e*x + 330*e^2*x^2 + 35*e^3*x^3))/(105*e*Sqrt[6 + 3*e*x])

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Maple [A]  time = 0.043, size = 52, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( 35\,{e}^{3}{x}^{3}+330\,{e}^{2}{x}^{2}+1284\,ex+2552 \right ) }{315\,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)^(5/2)*(-3*e^2*x^2+12)^(1/2),x)

[Out]

2/315*(e*x-2)*(35*e^3*x^3+330*e^2*x^2+1284*e*x+2552)*(-3*e^2*x^2+12)^(1/2)/e/(e*x+2)^(1/2)

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Maxima [C]  time = 1.71542, size = 96, normalized size = 1.1 \begin{align*} \frac{{\left (70 i \, \sqrt{3} e^{4} x^{4} + 520 i \, \sqrt{3} e^{3} x^{3} + 1248 i \, \sqrt{3} e^{2} x^{2} - 32 i \, \sqrt{3} e x - 10208 i \, \sqrt{3}\right )}{\left (e x + 2\right )} \sqrt{e x - 2}}{315 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(5/2)*(-3*e^2*x^2+12)^(1/2),x, algorithm="maxima")

[Out]

1/315*(70*I*sqrt(3)*e^4*x^4 + 520*I*sqrt(3)*e^3*x^3 + 1248*I*sqrt(3)*e^2*x^2 - 32*I*sqrt(3)*e*x - 10208*I*sqrt
(3))*(e*x + 2)*sqrt(e*x - 2)/(e^2*x + 2*e)

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Fricas [A]  time = 2.11747, size = 154, normalized size = 1.77 \begin{align*} \frac{2 \,{\left (35 \, e^{4} x^{4} + 260 \, e^{3} x^{3} + 624 \, e^{2} x^{2} - 16 \, e x - 5104\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{315 \,{\left (e^{2} x + 2 \, e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(5/2)*(-3*e^2*x^2+12)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*e^4*x^4 + 260*e^3*x^3 + 624*e^2*x^2 - 16*e*x - 5104)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^2*x + 2*
e)

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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]

integrate((e*x+2)**(5/2)*(-3*e**2*x**2+12)**(1/2),x)

[Out]

Timed out

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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{5}{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(5/2)*(-3*e^2*x^2+12)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-3*e^2*x^2 + 12)*(e*x + 2)^(5/2), x)