### 3.911 $$\int \frac{(2+e x)^{7/2}}{\sqrt{12-3 e^2 x^2}} \, dx$$

Optimal. Leaf size=85 $\frac{2 (2-e x)^{7/2}}{7 \sqrt{3} e}-\frac{8 \sqrt{3} (2-e x)^{5/2}}{5 e}+\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}-\frac{128 \sqrt{2-e x}}{\sqrt{3} e}$

[Out]

(-128*Sqrt[2 - e*x])/(Sqrt[3]*e) + (32*(2 - e*x)^(3/2))/(Sqrt[3]*e) - (8*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (2*(
2 - e*x)^(7/2))/(7*Sqrt[3]*e)

________________________________________________________________________________________

Rubi [A]  time = 0.0233863, antiderivative size = 85, 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} $\frac{2 (2-e x)^{7/2}}{7 \sqrt{3} e}-\frac{8 \sqrt{3} (2-e x)^{5/2}}{5 e}+\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}-\frac{128 \sqrt{2-e x}}{\sqrt{3} e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-128*Sqrt[2 - e*x])/(Sqrt[3]*e) + (32*(2 - e*x)^(3/2))/(Sqrt[3]*e) - (8*Sqrt[3]*(2 - e*x)^(5/2))/(5*e) + (2*(
2 - e*x)^(7/2))/(7*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}}{\sqrt{12-3 e^2 x^2}} \, dx &=\int \frac{(2+e x)^3}{\sqrt{6-3 e x}} \, dx\\ &=\int \left (\frac{64}{\sqrt{6-3 e x}}-16 \sqrt{6-3 e x}+\frac{4}{3} (6-3 e x)^{3/2}-\frac{1}{27} (6-3 e x)^{5/2}\right ) \, dx\\ &=-\frac{128 \sqrt{2-e x}}{\sqrt{3} e}+\frac{32 (2-e x)^{3/2}}{\sqrt{3} e}-\frac{8 \sqrt{3} (2-e x)^{5/2}}{5 e}+\frac{2 (2-e x)^{7/2}}{7 \sqrt{3} e}\\ \end{align*}

Mathematica [A]  time = 0.0827437, size = 57, normalized size = 0.67 $\frac{2 (e x-2) \sqrt{e x+2} \left (5 e^3 x^3+54 e^2 x^2+284 e x+1416\right )}{35 e \sqrt{12-3 e^2 x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(-2 + e*x)*Sqrt[2 + e*x]*(1416 + 284*e*x + 54*e^2*x^2 + 5*e^3*x^3))/(35*e*Sqrt[12 - 3*e^2*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 52, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ex-4 \right ) \left ( 5\,{e}^{3}{x}^{3}+54\,{e}^{2}{x}^{2}+284\,ex+1416 \right ) }{35\,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)^(7/2)/(-3*e^2*x^2+12)^(1/2),x)

[Out]

2/35*(e*x-2)*(5*e^3*x^3+54*e^2*x^2+284*e*x+1416)*(e*x+2)^(1/2)/e/(-3*e^2*x^2+12)^(1/2)

________________________________________________________________________________________

Maxima [C]  time = 1.90065, size = 61, normalized size = 0.72 \begin{align*} -\frac{2 i \, \sqrt{3}{\left (5 \, e^{4} x^{4} + 44 \, e^{3} x^{3} + 176 \, e^{2} x^{2} + 848 \, e x - 2832\right )}}{105 \, \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)^(1/2),x, algorithm="maxima")

[Out]

-2/105*I*sqrt(3)*(5*e^4*x^4 + 44*e^3*x^3 + 176*e^2*x^2 + 848*e*x - 2832)/(sqrt(e*x - 2)*e)

________________________________________________________________________________________

Fricas [A]  time = 1.81925, size = 135, normalized size = 1.59 \begin{align*} -\frac{2 \,{\left (5 \, e^{3} x^{3} + 54 \, e^{2} x^{2} + 284 \, e x + 1416\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)^(7/2)/(-3*e^2*x^2+12)^(1/2),x, algorithm="fricas")

[Out]

-2/105*(5*e^3*x^3 + 54*e^2*x^2 + 284*e*x + 1416)*sqrt(-3*e^2*x^2 + 12)*sqrt(e*x + 2)/(e^2*x + 2*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)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + 2\right )}^{\frac{7}{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)^(7/2)/(-3*e^2*x^2+12)^(1/2),x, algorithm="giac")

[Out]

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