### 3.935 $$\int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx$$

Optimal. Leaf size=106 $-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{195\ 3^{3/4} e (e x+2)^{5/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (e x+2)^{7/2}}-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (e x+2)^{9/2}}$

[Out]

-(3^(1/4)*(4 - e^2*x^2)^(5/4))/(13*e*(2 + e*x)^(9/2)) - (2*(4 - e^2*x^2)^(5/4))/(39*3^(3/4)*e*(2 + e*x)^(7/2))
- (2*(4 - e^2*x^2)^(5/4))/(195*3^(3/4)*e*(2 + e*x)^(5/2))

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Rubi [A]  time = 0.040833, antiderivative size = 106, 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 = {659, 651} $-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{195\ 3^{3/4} e (e x+2)^{5/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (e x+2)^{7/2}}-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (e x+2)^{9/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(9/2),x]

[Out]

-(3^(1/4)*(4 - e^2*x^2)^(5/4))/(13*e*(2 + e*x)^(9/2)) - (2*(4 - e^2*x^2)^(5/4))/(39*3^(3/4)*e*(2 + e*x)^(7/2))
- (2*(4 - e^2*x^2)^(5/4))/(195*3^(3/4)*e*(2 + e*x)^(5/2))

Rule 659

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1)
)/(2*c*d*(m + p + 1)), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p,
x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2
], 0]

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{9/2}} \, dx &=-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (2+e x)^{9/2}}+\frac{2}{13} \int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx\\ &=-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (2+e x)^{9/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (2+e x)^{7/2}}+\frac{2}{117} \int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx\\ &=-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{13 e (2+e x)^{9/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{39\ 3^{3/4} e (2+e x)^{7/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{195\ 3^{3/4} e (2+e x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0762167, size = 57, normalized size = 0.54 $\frac{\sqrt [4]{4-e^2 x^2} \left (2 e^3 x^3+14 e^2 x^2+37 e x-146\right )}{195\ 3^{3/4} e (e x+2)^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(9/2),x]

[Out]

((4 - e^2*x^2)^(1/4)*(-146 + 37*e*x + 14*e^2*x^2 + 2*e^3*x^3))/(195*3^(3/4)*e*(2 + e*x)^(7/2))

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Maple [A]  time = 0.042, size = 44, normalized size = 0.4 \begin{align*}{\frac{ \left ( ex-2 \right ) \left ( 2\,{e}^{2}{x}^{2}+18\,ex+73 \right ) }{585\,e}\sqrt [4]{-3\,{e}^{2}{x}^{2}+12} \left ( ex+2 \right ) ^{-{\frac{7}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(9/2),x)

[Out]

1/585*(e*x-2)*(2*e^2*x^2+18*e*x+73)*(-3*e^2*x^2+12)^(1/4)/(e*x+2)^(7/2)/e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}}}{{\left (e x + 2\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(9/2),x, algorithm="maxima")

[Out]

integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(9/2), x)

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Fricas [A]  time = 1.84171, size = 186, normalized size = 1.75 \begin{align*} \frac{{\left (2 \, e^{3} x^{3} + 14 \, e^{2} x^{2} + 37 \, e x - 146\right )}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}}{585 \,{\left (e^{5} x^{4} + 8 \, e^{4} x^{3} + 24 \, e^{3} x^{2} + 32 \, e^{2} x + 16 \, e\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(9/2),x, algorithm="fricas")

[Out]

1/585*(2*e^3*x^3 + 14*e^2*x^2 + 37*e*x - 146)*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2)/(e^5*x^4 + 8*e^4*x^3 + 24*
e^3*x^2 + 32*e^2*x + 16*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((-3*e**2*x**2+12)**(1/4)/(e*x+2)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.32091, size = 197, normalized size = 1.86 \begin{align*} -\frac{1}{9360} \cdot 3^{\frac{1}{4}}{\left (\frac{117 \,{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}{\left (\frac{4}{x e + 2} - 1\right )}}{\sqrt{x e + 2}} + \frac{130 \,{\left ({\left (x e + 2\right )}^{2} - 8 \, x e\right )}{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}}{{\left (x e + 2\right )}^{\frac{5}{2}}} - \frac{45 \,{\left ({\left (x e + 2\right )}^{3} - 12 \,{\left (x e + 2\right )}^{2} + 48 \, x e + 32\right )}{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}}{{\left (x e + 2\right )}^{\frac{7}{2}}}\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*e^2*x^2+12)^(1/4)/(e*x+2)^(9/2),x, algorithm="giac")

[Out]

-1/9360*3^(1/4)*(117*(-(x*e + 2)^2 + 4*x*e + 8)^(1/4)*(4/(x*e + 2) - 1)/sqrt(x*e + 2) + 130*((x*e + 2)^2 - 8*x
*e)*(-(x*e + 2)^2 + 4*x*e + 8)^(1/4)/(x*e + 2)^(5/2) - 45*((x*e + 2)^3 - 12*(x*e + 2)^2 + 48*x*e + 32)*(-(x*e
+ 2)^2 + 4*x*e + 8)^(1/4)/(x*e + 2)^(7/2))*e^(-1)