### 3.920 $$\int \frac{(2+e x)^{7/2}}{(12-3 e^2 x^2)^{3/2}} \, dx$$

Optimal. Leaf size=67 $-\frac{2 (2-e x)^{3/2}}{9 \sqrt{3} e}+\frac{16 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{32}{3 \sqrt{3} e \sqrt{2-e x}}$

[Out]

32/(3*Sqrt[3]*e*Sqrt[2 - e*x]) + (16*Sqrt[2 - e*x])/(3*Sqrt[3]*e) - (2*(2 - e*x)^(3/2))/(9*Sqrt[3]*e)

________________________________________________________________________________________

Rubi [A]  time = 0.021156, antiderivative size = 67, 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} $-\frac{2 (2-e x)^{3/2}}{9 \sqrt{3} e}+\frac{16 \sqrt{2-e x}}{3 \sqrt{3} e}+\frac{32}{3 \sqrt{3} e \sqrt{2-e x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(2 + e*x)^(7/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0643718, size = 43, normalized size = 0.64 $-\frac{2 \sqrt{e x+2} \left (e^2 x^2+20 e x-92\right )}{9 e \sqrt{12-3 e^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(2 + e*x)^(7/2)/(12 - 3*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[2 + e*x]*(-92 + 20*e*x + e^2*x^2))/(9*e*Sqrt[12 - 3*e^2*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.042, size = 43, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ({e}^{2}{x}^{2}+20\,ex-92 \right ) }{3\,e} \left ( ex+2 \right ) ^{{\frac{3}{2}}} \left ( -3\,{e}^{2}{x}^{2}+12 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+2)^(7/2)/(-3*e^2*x^2+12)^(3/2),x)

[Out]

2/3*(e*x-2)*(e^2*x^2+20*e*x-92)*(e*x+2)^(3/2)/e/(-3*e^2*x^2+12)^(3/2)

________________________________________________________________________________________

Maxima [C]  time = 1.68838, size = 49, normalized size = 0.73 \begin{align*} \frac{2 i \, \sqrt{3} e^{2} x^{2} + 40 i \, \sqrt{3} e x - 184 i \, \sqrt{3}}{27 \, \sqrt{e x - 2} e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(7/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="maxima")

[Out]

1/27*(2*I*sqrt(3)*e^2*x^2 + 40*I*sqrt(3)*e*x - 184*I*sqrt(3))/(sqrt(e*x - 2)*e)

________________________________________________________________________________________

Fricas [A]  time = 1.85742, size = 111, normalized size = 1.66 \begin{align*} \frac{2 \,{\left (e^{2} x^{2} + 20 \, e x - 92\right )} \sqrt{-3 \, e^{2} x^{2} + 12} \sqrt{e x + 2}}{27 \,{\left (e^{3} x^{2} - 4 \, e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(7/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="fricas")

[Out]

2/27*(e^2*x^2 + 20*e*x - 92)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^3*x^2 - 4*e)

________________________________________________________________________________________

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)**(7/2)/(-3*e**2*x**2+12)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+2)^(7/2)/(-3*e^2*x^2+12)^(3/2),x, algorithm="giac")

[Out]

sage0*x