3.934 \(\int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{7/2}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{\left (4-e^2 x^2\right )^{5/4}}{15\ 3^{3/4} e (e x+2)^{5/2}}-\frac{\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} e (e x+2)^{7/2}} \]

[Out]

-(4 - e^2*x^2)^(5/4)/(3*3^(3/4)*e*(2 + e*x)^(7/2)) - (4 - e^2*x^2)^(5/4)/(15*3^(3/4)*e*(2 + e*x)^(5/2))

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Rubi [A]  time = 0.0260016, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac{\left (4-e^2 x^2\right )^{5/4}}{15\ 3^{3/4} e (e x+2)^{5/2}}-\frac{\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} e (e x+2)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(7/2),x]

[Out]

-(4 - e^2*x^2)^(5/4)/(3*3^(3/4)*e*(2 + e*x)^(7/2)) - (4 - e^2*x^2)^(5/4)/(15*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)^{7/2}} \, dx &=-\frac{\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} e (2+e x)^{7/2}}+\frac{1}{9} \int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{5/2}} \, dx\\ &=-\frac{\left (4-e^2 x^2\right )^{5/4}}{3\ 3^{3/4} e (2+e x)^{7/2}}-\frac{\left (4-e^2 x^2\right )^{5/4}}{15\ 3^{3/4} e (2+e x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0637389, size = 45, normalized size = 0.63 \[ \frac{(e x-2) (e x+7) \sqrt [4]{4-e^2 x^2}}{15\ 3^{3/4} e (e x+2)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(7/2),x]

[Out]

((-2 + e*x)*(7 + e*x)*(4 - e^2*x^2)^(1/4))/(15*3^(3/4)*e*(2 + e*x)^(5/2))

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Maple [A]  time = 0.042, size = 35, normalized size = 0.5 \begin{align*}{\frac{ \left ( ex-2 \right ) \left ( ex+7 \right ) }{45\,e}\sqrt [4]{-3\,{e}^{2}{x}^{2}+12} \left ( ex+2 \right ) ^{-{\frac{5}{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)^(7/2),x)

[Out]

1/45*(e*x-2)*(e*x+7)*(-3*e^2*x^2+12)^(1/4)/(e*x+2)^(5/2)/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{7}{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)^(7/2),x, algorithm="maxima")

[Out]

integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(7/2), x)

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Fricas [A]  time = 1.75313, size = 143, normalized size = 2.01 \begin{align*} \frac{{\left (e^{2} x^{2} + 5 \, e x - 14\right )}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}}{45 \,{\left (e^{4} x^{3} + 6 \, e^{3} x^{2} + 12 \, e^{2} x + 8 \, 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)^(7/2),x, algorithm="fricas")

[Out]

1/45*(e^2*x^2 + 5*e*x - 14)*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2)/(e^4*x^3 + 6*e^3*x^2 + 12*e^2*x + 8*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)**(7/2),x)

[Out]

Timed out

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Giac [A]  time = 1.26122, size = 124, normalized size = 1.75 \begin{align*} -\frac{1}{180} \cdot 3^{\frac{1}{4}}{\left (\frac{9 \,{\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{5 \,{\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}}}\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)^(7/2),x, algorithm="giac")

[Out]

-1/180*3^(1/4)*(9*(-(x*e + 2)^2 + 4*x*e + 8)^(1/4)*(4/(x*e + 2) - 1)/sqrt(x*e + 2) + 5*((x*e + 2)^2 - 8*x*e)*(
-(x*e + 2)^2 + 4*x*e + 8)^(1/4)/(x*e + 2)^(5/2))*e^(-1)