3.936 \(\int \frac{\sqrt [4]{12-3 e^2 x^2}}{(2+e x)^{11/2}} \, dx\)

Optimal. Leaf size=141 \[ -\frac{2 \left (4-e^2 x^2\right )^{5/4}}{1105\ 3^{3/4} e (e x+2)^{5/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{221\ 3^{3/4} e (e x+2)^{7/2}}-\frac{3 \sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{221 e (e x+2)^{9/2}}-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{17 e (e x+2)^{11/2}} \]

[Out]

-(3^(1/4)*(4 - e^2*x^2)^(5/4))/(17*e*(2 + e*x)^(11/2)) - (3*3^(1/4)*(4 - e^2*x^2)^(5/4))/(221*e*(2 + e*x)^(9/2
)) - (2*(4 - e^2*x^2)^(5/4))/(221*3^(3/4)*e*(2 + e*x)^(7/2)) - (2*(4 - e^2*x^2)^(5/4))/(1105*3^(3/4)*e*(2 + e*
x)^(5/2))

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Rubi [A]  time = 0.0577388, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {659, 651} \[ -\frac{2 \left (4-e^2 x^2\right )^{5/4}}{1105\ 3^{3/4} e (e x+2)^{5/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{221\ 3^{3/4} e (e x+2)^{7/2}}-\frac{3 \sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{221 e (e x+2)^{9/2}}-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{17 e (e x+2)^{11/2}} \]

Antiderivative was successfully verified.

[In]

Int[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(11/2),x]

[Out]

-(3^(1/4)*(4 - e^2*x^2)^(5/4))/(17*e*(2 + e*x)^(11/2)) - (3*3^(1/4)*(4 - e^2*x^2)^(5/4))/(221*e*(2 + e*x)^(9/2
)) - (2*(4 - e^2*x^2)^(5/4))/(221*3^(3/4)*e*(2 + e*x)^(7/2)) - (2*(4 - e^2*x^2)^(5/4))/(1105*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)^{11/2}} \, dx &=-\frac{\sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{17 e (2+e x)^{11/2}}+\frac{3}{17} \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}}{17 e (2+e x)^{11/2}}-\frac{3 \sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{221 e (2+e x)^{9/2}}+\frac{6}{221} \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}}{17 e (2+e x)^{11/2}}-\frac{3 \sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{221 e (2+e x)^{9/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{221\ 3^{3/4} e (2+e x)^{7/2}}+\frac{2}{663} \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}}{17 e (2+e x)^{11/2}}-\frac{3 \sqrt [4]{3} \left (4-e^2 x^2\right )^{5/4}}{221 e (2+e x)^{9/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{221\ 3^{3/4} e (2+e x)^{7/2}}-\frac{2 \left (4-e^2 x^2\right )^{5/4}}{1105\ 3^{3/4} e (2+e x)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0773567, size = 65, normalized size = 0.46 \[ \frac{\sqrt [4]{4-e^2 x^2} \left (2 e^4 x^4+18 e^3 x^3+65 e^2 x^2+123 e x-682\right )}{1105\ 3^{3/4} e (e x+2)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(12 - 3*e^2*x^2)^(1/4)/(2 + e*x)^(11/2),x]

[Out]

((4 - e^2*x^2)^(1/4)*(-682 + 123*e*x + 65*e^2*x^2 + 18*e^3*x^3 + 2*e^4*x^4))/(1105*3^(3/4)*e*(2 + e*x)^(9/2))

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Maple [A]  time = 0.041, size = 52, normalized size = 0.4 \begin{align*}{\frac{ \left ( ex-2 \right ) \left ( 2\,{e}^{3}{x}^{3}+22\,{e}^{2}{x}^{2}+109\,ex+341 \right ) }{3315\,e}\sqrt [4]{-3\,{e}^{2}{x}^{2}+12} \left ( ex+2 \right ) ^{-{\frac{9}{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)^(11/2),x)

[Out]

1/3315*(e*x-2)*(2*e^3*x^3+22*e^2*x^2+109*e*x+341)*(-3*e^2*x^2+12)^(1/4)/(e*x+2)^(9/2)/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{11}{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)^(11/2),x, algorithm="maxima")

[Out]

integrate((-3*e^2*x^2 + 12)^(1/4)/(e*x + 2)^(11/2), x)

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Fricas [A]  time = 1.82339, size = 225, normalized size = 1.6 \begin{align*} \frac{{\left (2 \, e^{4} x^{4} + 18 \, e^{3} x^{3} + 65 \, e^{2} x^{2} + 123 \, e x - 682\right )}{\left (-3 \, e^{2} x^{2} + 12\right )}^{\frac{1}{4}} \sqrt{e x + 2}}{3315 \,{\left (e^{6} x^{5} + 10 \, e^{5} x^{4} + 40 \, e^{4} x^{3} + 80 \, e^{3} x^{2} + 80 \, e^{2} x + 32 \, 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)^(11/2),x, algorithm="fricas")

[Out]

1/3315*(2*e^4*x^4 + 18*e^3*x^3 + 65*e^2*x^2 + 123*e*x - 682)*(-3*e^2*x^2 + 12)^(1/4)*sqrt(e*x + 2)/(e^6*x^5 +
10*e^5*x^4 + 40*e^4*x^3 + 80*e^3*x^2 + 80*e^2*x + 32*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)**(11/2),x)

[Out]

Timed out

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Giac [A]  time = 1.3661, size = 284, normalized size = 2.01 \begin{align*} -\frac{1}{212160} \cdot 3^{\frac{1}{4}}{\left (\frac{663 \,{\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{1105 \,{\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{765 \,{\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}}} + \frac{195 \,{\left ({\left (x e + 2\right )}^{4} - 16 \,{\left (x e + 2\right )}^{3} + 96 \,{\left (x e + 2\right )}^{2} - 256 \, x e - 256\right )}{\left (-{\left (x e + 2\right )}^{2} + 4 \, x e + 8\right )}^{\frac{1}{4}}}{{\left (x e + 2\right )}^{\frac{9}{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)^(11/2),x, algorithm="giac")

[Out]

-1/212160*3^(1/4)*(663*(-(x*e + 2)^2 + 4*x*e + 8)^(1/4)*(4/(x*e + 2) - 1)/sqrt(x*e + 2) + 1105*((x*e + 2)^2 -
8*x*e)*(-(x*e + 2)^2 + 4*x*e + 8)^(1/4)/(x*e + 2)^(5/2) - 765*((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) + 195*((x*e + 2)^4 - 16*(x*e + 2)^3 + 96*(x*e + 2)^2 - 256*x*e -
256)*(-(x*e + 2)^2 + 4*x*e + 8)^(1/4)/(x*e + 2)^(9/2))*e^(-1)