3.903 $$\int \sqrt{2+e x} (12-3 e^2 x^2)^{3/2} \, dx$$

Optimal. Leaf size=65 $-\frac{2 (2-e x)^{9/2}}{\sqrt{3} e}+\frac{48 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e}$

[Out]

(-96*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (48*Sqrt[3]*(2 - e*x)^(7/2))/(7*e) - (2*(2 - e*x)^(9/2))/(Sqrt[3]*e)

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Rubi [A]  time = 0.0196335, 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} $-\frac{2 (2-e x)^{9/2}}{\sqrt{3} e}+\frac{48 \sqrt{3} (2-e x)^{7/2}}{7 e}-\frac{96 \sqrt{3} (2-e x)^{5/2}}{5 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-96*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (48*Sqrt[3]*(2 - e*x)^(7/2))/(7*e) - (2*(2 - e*x)^(9/2))/(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 \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 = 0.0447866, size = 52, normalized size = 0.8 $-\frac{2 (e x-2)^2 \sqrt{4-e^2 x^2} \left (35 e^2 x^2+220 e x+428\right )}{35 e \sqrt{3 e x+6}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*(-2 + e*x)^2*Sqrt[4 - e^2*x^2]*(428 + 220*e*x + 35*e^2*x^2))/(35*e*Sqrt[6 + 3*e*x])

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Maple [A]  time = 0.043, size = 44, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( 35\,{e}^{2}{x}^{2}+220\,ex+428 \right ) }{315\,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]

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

[Out]

2/315*(e*x-2)*(35*e^2*x^2+220*e*x+428)*(-3*e^2*x^2+12)^(3/2)/e/(e*x+2)^(3/2)

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Maxima [C]  time = 1.99547, size = 96, normalized size = 1.48 \begin{align*} -\frac{{\left (70 i \, \sqrt{3} e^{4} x^{4} + 160 i \, \sqrt{3} e^{3} x^{3} - 624 i \, \sqrt{3} e^{2} x^{2} - 1664 i \, \sqrt{3} e x + 3424 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)^(1/2)*(-3*e^2*x^2+12)^(3/2),x, algorithm="maxima")

[Out]

-1/105*(70*I*sqrt(3)*e^4*x^4 + 160*I*sqrt(3)*e^3*x^3 - 624*I*sqrt(3)*e^2*x^2 - 1664*I*sqrt(3)*e*x + 3424*I*sqr
t(3))*(e*x + 2)*sqrt(e*x - 2)/(e^2*x + 2*e)

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Fricas [A]  time = 1.86763, size = 155, normalized size = 2.38 \begin{align*} -\frac{2 \,{\left (35 \, e^{4} x^{4} + 80 \, e^{3} x^{3} - 312 \, e^{2} x^{2} - 832 \, e x + 1712\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)^(1/2)*(-3*e^2*x^2+12)^(3/2),x, algorithm="fricas")

[Out]

-2/105*(35*e^4*x^4 + 80*e^3*x^3 - 312*e^2*x^2 - 832*e*x + 1712)*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*} 3 \sqrt{3} \left (\int 4 \sqrt{e x + 2} \sqrt{- e^{2} x^{2} + 4}\, dx + \int - e^{2} x^{2} \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)**(1/2)*(-3*e**2*x**2+12)**(3/2),x)

[Out]

3*sqrt(3)*(Integral(4*sqrt(e*x + 2)*sqrt(-e**2*x**2 + 4), x) + Integral(-e**2*x**2*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{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{3}{2}} \sqrt{e x + 2}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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