### 3.912 $$\int \frac{(2+e x)^{5/2}}{\sqrt{12-3 e^2 x^2}} \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0199924, 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} $-\frac{2 (2-e x)^{5/2}}{5 \sqrt{3} e}+\frac{16 (2-e x)^{3/2}}{3 \sqrt{3} e}-\frac{32 \sqrt{2-e x}}{\sqrt{3} e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0591318, size = 49, normalized size = 0.75 $\frac{2 (e x-2) \sqrt{e x+2} \left (3 e^2 x^2+28 e x+172\right )}{15 e \sqrt{12-3 e^2 x^2}}$

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[2 + e*x]*(172 + 28*e*x + 3*e^2*x^2))/(15*e*Sqrt[12 - 3*e^2*x^2])

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Maple [A]  time = 0.043, size = 44, normalized size = 0.7 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( 3\,{e}^{2}{x}^{2}+28\,ex+172 \right ) }{15\,e}\sqrt{ex+2}{\frac{1}{\sqrt{-3\,{e}^{2}{x}^{2}+12}}}} \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]

2/15*(e*x-2)*(3*e^2*x^2+28*e*x+172)*(e*x+2)^(1/2)/e/(-3*e^2*x^2+12)^(1/2)

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Maxima [C]  time = 1.88108, size = 63, normalized size = 0.97 \begin{align*} -\frac{6 i \, \sqrt{3} e^{3} x^{3} + 44 i \, \sqrt{3} e^{2} x^{2} + 232 i \, \sqrt{3} e x - 688 i \, \sqrt{3}}{45 \, \sqrt{e x - 2} e} \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]

-1/45*(6*I*sqrt(3)*e^3*x^3 + 44*I*sqrt(3)*e^2*x^2 + 232*I*sqrt(3)*e*x - 688*I*sqrt(3))/(sqrt(e*x - 2)*e)

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Fricas [A]  time = 1.82014, size = 113, normalized size = 1.74 \begin{align*} -\frac{2 \,{\left (3 \, e^{2} x^{2} + 28 \, e x + 172\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{45 \,{\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]

-2/45*(3*e^2*x^2 + 28*e*x + 172)*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 \frac{{\left (e x + 2\right )}^{\frac{5}{2}}}{\sqrt{-3 \, e^{2} x^{2} + 12}}\,{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((e*x + 2)^(5/2)/sqrt(-3*e^2*x^2 + 12), x)